Continuous Variable Quantum Advantages and Applications in Quantum Optics
CContinuous Variable Quantum Advantages and Applications in Quantum Optics
ByU
LYSSE C HABAUD
Laboratoire d’Informatique de Paris 6S
ORBONNE U NIVERSITÉ
A dissertation submitted to Sorbonne Université in accor-dance with the requirements of the degree of D
OCTOR OF P HILOSOPHY , under the supervision of Damian Markhamand Elham Kashefi.Members of the jury: Anthony Leverrier, Andreas Winter,Sébastien Tanzili, Perola Milman and Gerardo Adesso. J ULY a r X i v : . [ qu a n t - ph ] F e b OREWORD T he three years leading up to the writing of this dissertation have been incredibly rich.Intellectually, I found a stimulus that I had been missing for years. I also met and interactedwith so many outstanding people! It has been truly an incredible time for which I feellucky and grateful. My heartfelt thanks to my doctoral advisors Damian and Elham, who wereincredibly supportive and distilled the perfect blend of guidance and freedom that allowed me toflourish as a researcher. I am also thankful to the members of my jury: Andreas Winter, PerolaMilman, Sébastien Tanzilli, Gerardo Adesso and especially Anthony Leverrier for his continuedsupport.I share the love of continuous variable quantum information with Frédéric and we have hadmany exciting and inspiring discussions from his first “coffee-break attack”, which have alwaysbeen a pleasure. The friends and colleagues of the QI team have provided the best environmentI could think of and I have to thank them all for their openness and passion, as I have beenable to interact with them with pleasure, both in research and outside the lab: Eleni, Tomand Shane, Pierre-Emmanuel, Raja, Luka, Robert and Alisa—the members of the “cool guysoffice”—Clément, Francesco, Nathan, Léo, Luis, Matthieu, Anu, Niraj, Andrea, Shraddha, Rhea,Victor, Federico, Adrien, Simon, Verena, Yao, Shouvik, Dominik, Gözde, Damien, Harold, Matteoand Cyril. My discussions with the LKB experimental team: Ganaël, Mattia, Nicolas, Valentina,who agreed to listen to a mad theorist, were also great times and it was always a blast to jointhe Edinburgh team during our wild retreats : Alex, Atul, Ellen, Daniel, Brian, Rawad, Mina,Mahshid, Theodoros and Petros. I would like to thank especially Andru for his kindness andthoughtfulness.These years have been marked by exciting trips around the world and I am deeply grateful toThomas Vidick, Scott Aaronson, Andrew Childs, Aram Harrow and Kae Nemoto for hosting myvisits. I also had the opportunity and pleasure of interacting and working with Giulia Ferrini,Raul García-Patrón, Peter van Loock, Antoine Joux, Iordanis Kerenidis, Jens Eisert, DominikHangleiter, Nathan Walk, Ingo Roth and Adel Sohbi. I am also thankful to all the researchers Ihad the opportunity to meet at the lab, during workshops or at bigger conferences, for sharingtheir time and knowledge with me.Thanks to my long-time friends Pierre, Safia, Oscar, Jim, Thomas, Aurélien, Alice, Vincent,Ziyad, Gaël, Balthazar, Maxence, Alexia, François, Nicolas, Alexandre, Arthur and Baptiste formaking the moments outside of the quantum information universe extremely enjoyable, togetherwith my brother, my parents and my grandfathers for their curiosity and interest, despite mysometimes foggy explanations.My unlimited thanks to my wife Léonie for her love and support through all these years.Thank you for being in my life and helping me to become a better person. I can’t wait to seewhat’s next together with you! i BSTRACT Q uantum physics has led to a revolution in our conception of the nature of our world and isnow bringing about a technological revolution. The use of quantum information promisesindeed applications that outperform those of today’s so-called classical devices. Continuousvariable quantum information theory refers to the study of quantum information encoded incontinuous degrees of freedom of quantum systems. This theory extends the mathematicalstudy of quantum information to quantum states in Hilbert spaces of infinite dimension. Itoffers different perspectives compared to discrete variable quantum information theory and isparticularly suitable for the description of quantum states of light. Quantum optics is thus anatural experimental platform for developing quantum applications in continuous variable.This thesis focuses on three main questions: where does a quantum advantage , that is, theability of quantum machines to outperform classical machines, come from? How to ensure theproper functioning of a quantum machine? What advantages can be gained in practice from theuse of quantum information? These three questions are at the heart of the development of futurequantum technologies and we provide several answers within the frameworks of continuousvariable quantum information and linear quantum optics.Quantum advantage in continuous variable comes in particular from the use of so-called non-Gaussian quantum states. We introduce the stellar formalism to characterize these states.We then study the transition from classically simulable models to models universal for quantumcomputing. We show that quantum computational supremacy , the dramatic speedup of quantumcomputers over their classical counterparts, may be realised with non-Gaussian states andGaussian measurements.Quantum certification denotes the methods seeking to verify the correct functioning of aquantum machine. We consider certification of quantum states in continuous variable, intro-ducing several protocols according to the assumptions made on the tested state. We developefficient methods for the verification of a large class of multimode quantum states, including theoutput states of the Boson Sampling model, enabling the experimental verification of quantumsupremacy with photonic quantum computing.We give several new examples of practical applications of quantum information in linearquantum optics. Generalising the swap test , we highlight a connection between the ability todistinguish two quantum states and the ability to perform universal programmable quantummeasurements, for which we give various implementations in linear optics, based on the useof single photons or coherent states. Finally, we obtain, thanks to linear optics, the first imple-mentation of a quantum protocol for weak coin flipping , a building block for many cryptographicapplications. iii
ABLE OF C ONTENTS
PageIntroduction 1
Motivation and context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Additional remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 W function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.2 Glauber–Sudarshan P function . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.3 Husimi Q function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3 Gaussian states and processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3.1 Gaussian unitary operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3.2 Single-mode Gaussian pure states . . . . . . . . . . . . . . . . . . . . . . . . 221.3.3 Multimode case: the symplectic formalism . . . . . . . . . . . . . . . . . . . . 231.4 Linear optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.4.1 Quantum states of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.4.2 Quantum optical measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 261.4.3 Linear interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.4.4 Hong–Ou–Mandel effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.4.5 Boson Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.5 Segal–Bargmann formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.5.2 Properties of holomorphic functions . . . . . . . . . . . . . . . . . . . . . . . . 35v ABLE OF CONTENTS
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Conclusion and outlook 241Bibliography 243 vii
NTRODUCTION Q uantum mechanics has deepened our understanding of the world. It has led us to rethinkthe very notion of reality—how can a cat be neither dead nor alive?—by putting forthintriguing properties such as entanglement and superposition . Nowadays, new informationprocessing devices using quantum properties are being developed, such as quantum computers,and it is fascinating and maybe incumbent to see whether and to what extent these quantumtechnologies may outperform conventional technologies. Motivation and context
While classical mechanics, as opposed to quantum, has been quite successful in describing theworld at our scale, quantum mechanics has proven to be a very powerful tool for understandingthe world at the particle scale. Interesting effects appear at this scale, and the challenge posedby the development of quantum technologies is not only to understand these effects but alsoto harness them. Quantum information—that is, information encoded in quantum degreesof freedom of physical systems—promises advantages over classical information notably forcomputing, communication, cryptography and sensing. That the use of quantum mechanicsmay provide an advantage over classical mechanics for information processing is an excitingperspective, which raises the following question:
What leads to a quantum advantage?
This profound question has attracted enormous attention and so far has only partial answers.From a foundational point of view, this question asks what differentiates the quantum from theclassical and what makes nature fundamentally nonclassical. While shedding light on the verynature of our world, answering this question also enables the development of new technologiesexploiting quantum properties to gain an advantage over classical machines.In order to understand the possible origins of a quantum advantage it is worthwhile to highlightsome of the differences between quantum and classical information and in particular quantumfeatures that are inherently nonclassical.Properties of quantum systems are intrinsically random prior to being measured and thisrandomness is lost whenever the quantum system is measured—hence the infamous Schrödinger’s1
NTRODUCTION cat thought experiment, in which a cat is locked in a box with a device that kills the animalwith some probability: before opening the box, the cat is neither dead nor alive, but rather in asuperposition of these two states, and opening the box collapses the state of the cat to either deador alive. In a more general fashion, the state of a quantum system can be mathematically describedby a wave function consisting of complex-valued probability amplitudes. The probabilities for thepossible results of measurements made on the system can be derived from these amplitudes. Astheir name indicates, wave functions behave qualitatively like mechanical waves: they satisfy alinear wave equation and may interfere. This interference of probability amplitudes is a strikingexample of nonclassical phenomena. A quantum computer outperforming its classical counterpartwould crucially interfere various branches of a computation.The linear evolution of probability amplitudes also has striking consequences: it impliesthat an arbitrary quantum state cannot be perfectly cloned [WZ82]. This contrasts with thefact that classical information is trivial to copy. This quantum no-cloning property can also bederived from the uncertainty principle, which asserts that complementary quantum observables—such as position and momentum—cannot be simultaneously measured with arbitrary precision:measuring one of the two collapses the state of the measured quantum system such that thevalue of the other becomes uniformly random. If one was able to perfectly clone a quantum state,one could measure the position of the first copy and the momentum of the second and infer bothquantities for the original state, thus contradicting the uncertainty principle. While uncertaintyand no-cloning may be seen as limitations of quantum information, quantum advantage incryptography notably comes from exploiting these properties to hide information from a possibleeavesdropper [BB84b].These quantum properties may be witnessed already for a single system. On the otherhand, multiple systems may display correlations and it turns out that quantum systems may becorrelated in a way classical systems cannot, as a consequence of entanglement. A quantum stateover multiple subsystems is said to be entangled if it cannot be separated into the individualstates of its subsystems. An important consequence of entanglement is the nonlocality of quantumtheory, i.e., the fact that correlations displayed by spatially separated quantum systems cannotbe reproduced locally by classical means [Bel64]—what Einstein famously described as “spookyaction at a distance”. While these nonclassical correlations may be exploited for the so-calledquantum teleportation [BBC + + NTRODUCTION technological applications is decoherence, i.e., the loss of coherence of the information encoded ina physical system, due to the interaction of that system with its environment. Quantum deviceswill inevitably interact with their environment and suffer the effect of noise. How to mitigatethe consequences of decoherence is an active domain of research [Pre98a]. In theory, quantumcomputations may be performed fault-tolerantly, even though this results in a huge overhead interms of physical systems needed for the computation. It is also not obvious how one can mitigatenoise in other quantum information processing tasks, such as sensing or simulations, where thefault-tolerant quantum computing approach is not natural. Hence, another question that ariseswhen looking for an advantage using a quantum device is the following:
How do we check the correct functioning of a quantum device?
Answering this second question is a timely problem in the absence of fault-tolerant mechanisms,for benchmarking existing and upcoming quantum devices. It has also attracted a lot of attention[EHW + + + + NTRODUCTION
The demonstration of quantum supremacy, that is the convincing demonstration of a quantumcomputation beating what is possible classically, is however only a milestone, and what is at stakein the development of quantum technologies is to obtain advantages for real-world applications.It is thus natural to ask the following question:
What useful advantages can we obtain from the use of quantum information?
Depending on the application considered, a quantum advantage may take different forms: toobtain the result of a computation faster [Sho94, Gro98], to communicate more messages withinthe same physical system [BW92] or in a more secured fashion [BB84a], or to perform a measure-ment with a better precision [GLM11], for example. Answering this third question amounts todevelopping new theoretical quantum algorithms as well as deriving realistic implementationsfor existing ones, for example with linear quantum optics and quantum states of light.This section has provided an overview of the different contexts on which the work of this thesisis based. Motivated by the three very general questions above—origin of quantum advantage,certification of quantum devices and useful quantum advantages—this dissertation exploresvarious directions, with particular emphasis on continuous variable quantum information theoryand optical quantum information processing. The next section presents a technical summary ofthe content of the thesis.
Summary of results
After a preliminary chapter 1, chapters 2 and 3 deal with continuous variable quantum informa-tion theory and computing. Chapters 4 and 5 consider the probems of quantum state certificationand testing, in the continuous variable regime and using quantum optics. Chapter 6 discusses theimplementation of a quantum cryptography protocol with quantum optics. We detail the contentof each chapter in what follows. The dependencies between the chapters are indicated in Fig. 0.1.
Chapter 1.
After briefly introducing preliminary material on quantum information theory, thischapter presents the formalisms of continuous variable quantum information theory used inthis thesis. Phase-space formalism is discussed. A description of Gaussian states and processesfollows, together with the symplectic formalism. Then, quantum linear optics is presented withan exposition of Boson Sampling [AA13]. Finally, the Segal–Bargmann formalism is introduced.
Chapter 2.
This chapter investigates the origin of quantum advantage for continuous variablequantum computing. Continuous variable quantum states are separated into two broad families:Gaussian and non-Gaussian. While Gaussian states feature interesting properties such asentanglement, non-Gaussian states are crucial for a variety of quantum information tasks[ESP02, Fiu02, GC02, WHG +
03, GPFC +
04, GS07, NFC09, ADDS +
09, BDE + NTRODUCTION
Chapter 4
Certification of continuous variable quantum states
Chapter 3
Beyond-classicalquantum continuousvariable models
Chapter 2
Stellar representation of non-Gaussian quantum states
Chapter 5
Programmable measurements with linear optics
Chapter 6
Quantum weak coin flipping with linear optics
Chapter 1
Continuous variable quantum informationformalisms F IGURE + Chapter 3.
In this chapter, we explore the quantum advantage transition for continuous vari-ables, i.e., the boundary between classically simulable quantum computational models andmodels capable of outperforming their classical counterparts, in terms of non-Gaussian resources.We give classical simulation algorithms for several quantum models and computational tasks,including linear optics with adaptive measurements and Gaussian circuits with non-Gaussianinput states. Then, we introduce a subuniversal family of continuous variable circuits related toBoson Sampling called Continuous Variable Sampling from photon-added or photon-subtractedsqueezed states (CVS) circuits. We show that the continuous output probability densities of thesecircuits are on average hard to sample exactly classically, by relating their output probabilities topermanents of (0, 1)-matrices. The main results of this chapter are classical simulation algorithmsfor Gaussian circuits with weakly non-Gaussian input states, as well as showing how quantumsupremacy may be achieved with non-Gaussian states, together with Gaussian operations and5
NTRODUCTION measurements.
Chapter 4.
This chapter considers the certification of continuous variable quantum states.Determining an unknown quantum state is difficult especially in continuous variables, where itis described by possibly infinitely many complex parameters. Existing methods like homodynequantum state tomography require many different measurement settings and heavy classicalpost-processing [LR09]. This chapter shows how continuous variable quantum states can beefficiently verified: we introduce a reliable method for performing continuous variable quantumstate state tomography using a single Gaussian measurement, namely heterodyne detection,which can be implemented with quantum optics ; then, we show how this tomography methodmay be promoted to a state certification protocol under the i.i.d. assumption, by adding anenergy test. We also derive a similar protocol for continuous variable quantum state verification,making no assumption whatsoever on the state preparation method, using a de Finetti reductionfor infinite-dimensional systems [RC09]. We further show that this protocol extends to themultimode case and allows us to efficiently verify output states of Boson Sampling and CVSinterferometers. The main result of this chapter is a flexible protocol for building trust for a largeclass of multimode mode continuous variable quantum states with Gaussian measurements,which provides analytical confidence intervals and allow for a reliable verification of quantumcomputational supremacy with photonic quantum computing.
Chapter 5.
On top of being a promising candidate for the demonstration of quantum supremacywith Boson Sampling, quantum optics provides an exciting experimental platform for near-termquantum applications, as well as for probing quantum behaviours. This chapter discusses therelations between quantum state discrimination, quantum state identity testing and universalprogrammable projective measurements and proposes implementations in linear optics. A gener-alisation of the swap test [BCWDW01] is introduced, together with its implementation in linearoptics using single-photon encoding. We show how this allows us to construct universal quantum-programmable projective measurements, based on a simple classical post-processing of samplesfrom number-resolving or parity detectors. In order to simplify the experimental requirements, analternative scheme is derived which uses a coherent state encoding, a simpler interferometer andsingle-photon threshold detectors, with applications to optical quantum communication protocols.
Chapter 6.
Cryptographic protocols are built from a selection of simpler functionalities, calledprimitives. Remarkably, quantum mechanics allows for the implementation of some primi-tives with information-theoretic security which can only be achieved with conditional securityclassically, i.e., by relying on computational assumptions. The so-called coin flipping by tele-phone [Blu83], or weak coin flipping, is one of such cryptographic primitives. It refers to thecryptographic scenario in which two mistrustful and distant parties want to agree on a randombit, while they favor opposite outcomes. The use of quantum mechanics allows for achieving bettersecurity than classical mechanics. However, even though various quantum weak coin flippingprotocols have been theorised [SR02, KN04, Moc05, Moc07, ARV19], no practical implementation6
NTRODUCTION has been proposed so far. This chapter introduces an implementation in linear optics of quantumweak coin flipping. The proposed implementation relies on adapting a theoretical protocol forquantum weak coin flipping [SR02] to linear optics, using the so-called dual-rail encoding, i.e.,encoding a qubit with a photon in two spatial modes. The protocol can be implemented withcurrent technology and may display quantum advantage over any classical protocol for the sametask.
Additional remarks
This thesis is intended to be accessible to a reader familiar with the basics of quantum informationand computing with discrete and continuous variables. Good introductions to the field of quantuminformation theory include [Pre98b] and [NC02], while [BvL05] provides a comprehensive reviewof quantum information with continuous variables. Pointers to the relevant literature are alsodisplayed throughout the thesis.This thesis is based on several previous works.•
Chapter 2.
This chapter is mainly based on a joint work with D. Markham and F.Grosshans [CMG20], and section 2.3.3 is based on a joint work with with G. Roland,M. Walschaers, F. Grosshans, V. Parigi, D. Markham and N. Treps [CRW + Chapter 3.
Section 3.2 is based on a joint work with A. Sohbi and D. Markham [CMS20],sections 3.3.1 and 3.3.2 on a joint work with G. Ferrini, F. Grosshans and D. Markham [CFGM20],and section 3.3.3 on a joint work with T. Douce, D. Markham, P. van Loock, E. Kashefi andG. Ferrini [CDM + Chapter 4.
Section 4.1 is based on a joint work with J. Eisert, D. Hangleiter, N. Walk, I.Roth, D. Markham, R. Parekh, and E. Kashefi [EHW + + + Chapter 5.
Sections 5.1 and 5.2 are based on a joint work with E. Diamanti, D. Markham,E. Kashefi and A. Joux [CDM + + Chapter 6.
This chapter is based on a joint work with M. Bozzio, E. Diamanti and I.Kerenidis [BCKD20]. 7 H A PT ER C ONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS C ontinuous variable quantum information theory refers to the study of information encodedin quantum physical systems with continuous degrees of freedom. The approach of thework presented in this dissertation for studying continuous variable quantum informationis to use different mathematical formalisms as different ways of gaining intuition. Jugglingseveral representations of the same mathematical object is indeed an excellent way to getinsights about this object. In this chapter, we briefly review the formalisms for continuousvariable quantum information theory used throughout the rest of the thesis. These include phasespace formalism for continuous variable quantum states and operators, symplectic formalismfor Gaussian states, quantum optics and Boson Sampling, and Segal–Bargmann formalism forcontinuous variable quantum states. The sets (cid:78) , (cid:82) and (cid:67) are the usual sets of natural, real and complex numbers, with a ∗ exponentwhen 0 is removed from the set. The size of a set X is denoted by | X | . The natural logarithm isdenoted log.We write complexity classes with sans serif font: P , NP ...The number of subsystems or modes will generally be denoted by m ∈ (cid:78) ∗ . Hilbert spaces aredenoted by H or K . The expressions | φ 〉 , | ψ 〉 denote pure states, and ρ and σ denote densityoperators of possibly mixed quantum states.For vectors and operators, we denote by a ∗ exponent the complex conjugate, by a T exponentthe transpose and by a † exponent the transpose complex conjugate (adjoint). Matrices are9 HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS denoted by capital letters and covariance matrices will be denoted by V . Operators are indicatedby a hat, with the exception of density operators, positive-operator valued measure elements andidentity operator (cid:49) . In particular, ˆ a and ˆ a † denote the annihilation and creation operators and ˆ q and ˆ p denote the position-like and momentum-like quadrature operators. The identity matrix isalso denoted (cid:49) , sometimes with an index indicating its size. The zero matrix is similarly denoted (cid:48) . The trace is denoted by Tr and the determinant by Det.Pr denotes a probability, while (cid:69) denotes an expected value. A function δ may stand for theKronecker symbol or a Dirac delta, depending on the context. The letters α , β and γ are used forcoherent state amplitudes or complex amplitudes, while the letters ξ and ζ are used for squeezingparameters. The letter z denotes a complex variable.We write ⊗ and ⊕ for the tensor product and the direct sum, respectively. We use bold mathfor multimode states, vectors and multi-index notations. Let m , n ∈ (cid:78) ∗ . We define = (0, . . . , 0)and = (1, . . . , 1), and we write n = (0, . . . , 0) ∈ (cid:78) n or n = (1, . . . , 1) ∈ (cid:78) n to avoid ambiguity. For all k ∈ {
1, . . . , m } , we also define k = (0, . . . , 0, 1, 0, . . . , 0), where the k th entry is 1 and all the other m − z = ( z , . . . , z m ) ∈ (cid:67) m , all z (cid:48) = ( z (cid:48) , . . . , z (cid:48) m ) ∈ (cid:67) m and all p = ( p , . . . , p m ) ∈ (cid:78) m we write z ∗ = ( z ∗ , . . . , z ∗ m ) − z = ( − z , . . . , − z m )˜ z = z ⊕ z ∗ = ( z , . . . , z m , z ∗ , . . . , z ∗ m ) | z 〉 = | z . . . z m 〉(cid:107) z (cid:107) = | z | + · · · + | z m | z p = z p . . . z p m m z + z (cid:48) = ( z + z (cid:48) , . . . , z m + z (cid:48) m ) z ≤ z (cid:48) ⇔ z k ≤ z (cid:48) k ∀ k ∈ {
1, . . . , m } p ! = p ! . . . p m ! | p | = p + · · · + p m ∂ p = ∂ p . . . ∂ p m m (cid:181) ∂∂ z (cid:182) p = ∂ | p | ∂ z p · · · ∂ z p m m . (1.1)We will use for brevity the notations c χ = cosh χ , s χ = sinh χ and t χ = tanh χ , for all χ ∈ (cid:67) . Thecommutator is denoted by [ , ] and the anticommutator by { , } . Finally we adopt the convention (cid:126) = .1. PRELIMINARY MATERIAL The presentation given here is very succinct and good introductions to the field of quantuminformation theory include [Pre98b] and [NC02].In quantum information theory, we identify two notions of randomness. On the one hand, thereis an inherent randomness in the formalism of quantum measurements, which we call quan-tum randomness. On the other hand, classical randomness corresponds to the usual notion ofrandomness to which we refer, for example, when we draw a card from a shuffled deck of cardsor when we roll a die. In practice, a quantum system can manifest both classical and quantumrandomness.The properties of a quantum system are described by its quantum state. Quantum states withno classical randomness are called pure states. These pure quantum states are representedmathematically as normalised vectors in a separable Hilbert space H . We adopt Dirac bra-ketnotation [Dir81] in what follows: a column vector ψ is represented as the ket | ψ 〉 and its adjoint(transpose complex conjugate) line vector is represented as the bra 〈 ψ | . In particular, the projectoronto | ψ 〉 is expressed as | ψ 〉〈 ψ | and the inner product of two states | φ 〉 and | ψ 〉 is denoted by 〈 φ | ψ 〉 ,with 〈 φ | ψ 〉 = 〈 ψ | φ 〉 ∗ . The quantity | 〈 φ | ψ 〉 | is referred to as the overlap of the states | φ 〉 and | ψ 〉 .The simplest nontrivial example is a Hilbert space of dimension 2. In that case, quantumstates are referred to as qubit states, states in a Hilbert space of finite dimension d > | 〉 , | 〉 ) of a Hilbert space of dimension 2,a qubit state | ψ 〉 is expressed as | ψ 〉 = a | 〉 + b | 〉 , | a | + | b | =
1, (1.2)for a , b ∈ (cid:67) , with 〈 ψ | = a ∗ 〈 | + b ∗ 〈 | . The coefficients a and b are the complex amplitudes of thequbit state | ψ 〉 . If a (cid:54)= b (cid:54)=
0, the state | ψ 〉 is said to be in a superposition of the states | 〉 and | 〉 .The basis ( | 〉 , | 〉 ) is referred to as the computational basis. On the other hand, setting |±〉 = (cid:112) ( | 〉 ± | 〉 ), the states ( |+〉 , |−〉 ) also form an orthonormal basis, referred to as the diagonalbasis.Observable physical quantities, or simply observables, are represented mathematically by self-adjoint (hermitian) operators ˆ O = ˆ O † . Such operators have an orthonormal basis of eigenvectorsand measuring the observable ˆ O gives an outcome sampled from the list of its eigenvalues. Theprobability of each outcome is determined by the Born rule:Pr[ λ ] = 〈 ψ | Π λ | ψ 〉 , (1.3)where λ is the eigenvalue, | ψ 〉 is the state of the measured quantum system and Π λ is a projectoronto the eigenvector corresponding to the eigenvalue λ . Equivalently, we say that we measurein a specific orthonormal basis to say that we measure an observable which has this basis as11 HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS an eigenbasis. In particular, Eq. (1.2) may be interpreted as follows: 〈 ψ | Π | ψ 〉 = 〈 ψ | 〉〈 | ψ 〉 =| 〈 | ψ 〉 | = | a | (resp. | b | ) is the probability that we obtain the outcome 0 (resp. 1) when measuringthe state | ψ 〉 in the ( | 〉 , | 〉 ) basis. The two probabilities sum to 1, corresponding to the fact thatthe measurement will yield an outcome, either 0 or 1. The measurement outcome is randomwhen a (cid:54)= b (cid:54)=
0, i.e., quantum randomness manifests when the measured state is ina superposition of eigenvectors of the observable. Measuring a quantum state collapses thestate onto the eigenvector corresponding to the outcome obtained. In particular, any subsequentmeasurement of the same observable will yield the same result with probability 1.The most general notion of quantum measurement is captured by positive-operator valuedmeasures (POVM). A POVM is a set of semidefinite operators { Π i } i ∈ I whose elements sumto the identity operator, indexed by a set of outcomes I . The operator Π i is associated to themeasurement outcome i ∈ I and the probability for this outcome is given by Eq. (1.3), replacing λ by i . The case where the operators Π i are projectors, as in Eq. (1.3), corresponds to projection-valued measures (PVM).Quantum systems can also exhibit classical randomness. When that is the case, we refer to thequantum state as mixed. A mixed quantum state is represented mathematically by a so-calleddensity operator, i.e., a hermitian operator with trace 1 acting on a Hilbert space. The densityoperator for a pure state | ψ 〉 is simply a projector | ψ 〉〈 ψ | . A mixed quantum state can be writtenas a convex combination, or mixture, of pure states. For example, the state obtained by flippingan unbiaised coin and choosing the state | 〉 for tails and | 〉 for heads is a mixed state expressedas | 〉〈 | + | 〉〈 | , which is different from the pure superposition (cid:112) ( | 〉 + | 〉 ), whose densityoperator is given by | 〉〈 | + | 〉〈 | + | 〉〈 | + | 〉〈 | . The Born rule for a mixed state ρ readsPr[ i ] = Tr ( Π i ρ ), (1.4)where { Π i } i ∈ I is a POVM over a set of outcomes I . Setting ρ = | ψ 〉〈 ψ | , we retrieve the Bornrule for pure states in Eq. (1.3). Writing the semidefinite operator Π i = M † i M i , the state after ameasurement with outcome i ∈ I is given by ρ ( i ) = M i ρ M † i Tr ( Π i ρ ) . (1.5)Note that the choice of M i is not unique and this choice reflects different possible ways ofphysically implementing the same POVM. Given an observable ˆ O , the quantity Tr ( ˆ O ρ ) is theexpectation of the operator ˆ O for the quantum state ρ and is alternatively denoted 〈 ˆ O 〉 ρ .The global state of two independent quantum systems with states | φ 〉 and | ψ 〉 in two Hilbertspaces H and H (cid:48) , respectively, lies in the tensor product H ⊗ H (cid:48) and is obtained by taking thetensor product | φ 〉 ⊗ | ψ 〉 of both states. We will usually write | φ 〉 ⊗ | ψ 〉 = | φψ 〉 when there is noambiguity. The dimension of the Hilbert space H ⊗ H (cid:48) is the product of the dimensions of theHilbert spaces H and H (cid:48) , implying in particular that the computational basis of n -qubit stateshas size 2 n . 12 .1. PRELIMINARY MATERIAL Two quantum systems may not be independent and a pure quantum state which cannotbe written as a tensor product of quantum states is called entangled. For example, the state (cid:112) ( | 〉+| 〉 ) is entangled while the state ( | 〉+| 〉 ) ⊗ ( | 〉+| 〉 ) is separable. A (mixed) quantumstate is called separable if it can be written as a mixture of separable pure states, and entangledotherwise.Entanglement may be conceived as the quantum version of classical correlation [Wer89]: themixed quantum state | 〉〈 | + | 〉〈 | is classically correlated—the measurements of eachsubsystem in the ( | 〉 , | 〉 ) basis will always yield the same outcomes—but not entangled, since itis a mixture of product states. On the other hand, the pure state (cid:112) ( | 〉 + | 〉 ) = (cid:112) ( |++〉 + |−−〉 )is entangled. This state is ‘more’ correlated than the previous one in the following sense: not onlythe measurements of each subsystem in the ( | 〉 , | 〉 ) basis will always yield the same outcomesbut measuring each subsystem in the ( |+〉 , |−〉 ) basis will also always yield the same outcomes.Given a state ρ over two subsystems in H and H (cid:48) , the reduced state of the first subsystemis obtained by tracing out, or taking the partial trace over, the second subsystem Tr H (cid:48) ( ρ ). Aseparable state is fully described by the reduced states of its individual subsystems, while this isno longer the case for an entangled state.The simplest example of evolution of a quantum system is a unitary evolution over a time t ,described by a unitary operator ˆ U with ˆ U † ˆ U = (cid:49) , generated by a Hamiltonian H with H † = H ,such that ˆ U = e − iHt . If the system is in a pure state | ψ 〉 , then the state after the evolution is anormalised pure state ˆ U | ψ 〉 . If the system is in a mixed state ρ , then the state after the evolutionis a mixed state with density operator ˆ U ρ ˆ U † .More general quantum evolutions are described by quantum channels, i.e., completely positivetrace-preserving maps (CPTP). By Stinespring dilation theorem, CPTP maps can be expressed asunitaries acting on a larger space. Formally, if E is a CPTP map acting on a Hilbert space H ,then there exist a Hilbert space H (cid:48) and a unitary operator ˆ U such that for all density operators ρ , E ( ρ ) = Tr H (cid:48) [ U ( ρ ⊗ | 〉〈 | ) U † ]. (1.6)In other words, any quantum channel can be obtained by tensoring with a second system in afixed state, a unitary evolution and a reduction to a subsystem. Naimark’s theorem provides asimilar result for decomposing a POVM as a unitary followed by a PVM on a larger space.The most general physical evolutions are described by quantum operations, i.e., completelypositive trace-decreasing maps (CPTD). These operations can be obtained as obtained by tensoringwith a second system in a fixed state, a unitary evolution, a PVM and a reduction to a subsystem.Non-CPTD maps are referred to as unphysical operations. Such operations can be approximatedby quantum operations, for example when they act as CPTD maps on a subset of the Hilbertspace.A quantum computation is composed of the three following steps: input, evolution and mea-surement. With the above, one may conceive elaborate quantum computations as building a13 HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS highly entangled state from a simple input product state via a unitary evolution and samplingfrom a probability distribution given by the Born rule and the choice of measurement. Quantumcomputations can be looked at in the circuit picture, in which the unitary evolution is decomposedas a product of gates acting on at most two subsystems at a time.Discrimination of quantum states is a central element in many quantum information processingtasks [NC02] and various measures are available [FVDG99]. We review two measures used exten-sively in the thesis: the fidelity and the trace distance. The properties outlined are independentof the dimension of the Hilbert space.The fidelity between two states ρ and σ is defined as F ( ρ , σ ) = Tr (cid:181)(cid:113) (cid:112) σ ρ (cid:112) σ (cid:182) . (1.7)Note that the definition used here is the square of the definition in [FVDG99, NC02]. Eventhough it is not apparent with the above equation, the fidelity is symmetric in its arguments ρ and σ . When at least one of the two states is a pure state, this expression reduces to F (cid:161) ψ , ρ (cid:162) = Tr( | ψ 〉〈 ψ | ρ ) = 〈 ψ | ρ | ψ 〉 . (1.8)In particular when both states are pure F (cid:161) φ , ψ (cid:162) = | 〈 φ | ψ 〉 | .We write the Schatten 1-norm of a bounded operator T as (cid:107) T (cid:107) = Tr (cid:179)(cid:112) T † T (cid:180) = Tr( | T | ). (1.9)The trace distance between two states ρ , σ is defined as D ( ρ , σ ) = (cid:107) ρ − σ (cid:107) =
12 Tr( | ρ − σ | ). (1.10)It is jointly convex in its two arguments. The fidelity is related to the trace distance by theFuchs-van de Graaf inequalities [FVDG99]1 − (cid:112) F ( ρ , σ ) ≤ D ( ρ , σ ) ≤ (cid:112) − F ( ρ , σ ) . (1.11)When one of the states is pure, the lower bound may be refined as1 − F ( ψ , ρ ) ≤ D ( ψ , ρ ). (1.12)When both states are pure, the upper bound in Eq. (1.11) becomes an equality: D ( φ , ψ ) = (cid:112) − F ( φ , ψ ) = (cid:113) − | 〈 φ | ψ 〉 | . (1.13)14 .1. PRELIMINARY MATERIAL The fidelity is nondecreasing under quantum operations and the trace distance is nonincreasingunder quantum operations. The total variation distance of two probability distributions P and Q over a sample space S is defined as (cid:107) P − Q (cid:107) tvd = (cid:88) s ∈ S | P ( s ) − Q ( s ) | . (1.14)A similar definition holds for probability densities over a continuous sample space, by replacingthe discrete sum by a continuous sum. The trace distance verifies D ( ρ , σ ) = max ˆ O (cid:107) P ˆ O ρ − P ˆ O σ (cid:107) tvd , (1.15)where P ˆ O ρ (resp. P ˆ O σ ) is the probability distribution associated to measuring the observable ˆ O forthe state ρ (resp. σ ) and where the maximum of the total variation distance is taken over allobservables. The trace distance thus has an operational significance: if two states are close intrace distance, then any computation taking as input one of the two states is indistinguishablefrom the same computation taking as input the other state. Moreover, with Eq. (1.11), lowerbounds on the fidelity also give upper bounds on the total variation distance, which are tightwhen the states are pure, by Eq. (1.13).In what follows, we consider the case of infinite-dimensional Hilbert spaces, allowing for thedescription of quantum systems with continuous degrees of freedom. Discrete variables can beencoded in continuous degrees of freedom and finite-dimensional Hilbert spaces may be embeddedin infinite-dimensional ones. Despite its discrete character, we will also refer to the study of suchembedded discrete variable quantum information in an infinite-dimensional Hilbert space ascontinuous variable quantum information theory, since the same mathematical formalisms areemployed in both case. We refer the reader to the first chapters of [BvL05, FOP05, ARL14] for a further introductionon the material presented in this section. While the presentation that follows is quite technical,it avoids many of the subtleties which appear when dealing with infinite-dimensional Hilbertspaces. The interested reader will find an example of a formal treatment in [DlM05].The continuous variable equivalent of a qubit or qudit is the qumode, or simply mode. Single-mode continuous variable quantum states are mathematically described as normalised complexvectors in an infinite-dimensional separable Hilbert space, with an infinite countable orthonormalbasis { | n 〉 } n ∈ (cid:78) referred to as the Fock basis, or photon-number basis in the context of opticalquantum information processing. In particular, | 〉 is referred to as the vacuum state and | 〉 asthe single-photon state. A single-mode pure state | ψ 〉 can be written in Fock basis as | ψ 〉 = (cid:88) n ≥ ψ n | n 〉 , (1.16)15 HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS where ψ n ∈ (cid:67) for all n ∈ (cid:78) , with the normalisation condition (cid:80) +∞ n = | ψ n | =
1. The Fock basis comeswith canonical adjoint operators ˆ a and ˆ a † referred to as annihilation and creation operators,respectively, or photon subtraction and photon addition operators in the context of opticalquantum information processing. These operators are defined by their action on the Fock basis asˆ a | n 〉 = (cid:112) n | n − 〉 , for n ∈ (cid:78) ∗ ,ˆ a | 〉 = a † | n 〉 = (cid:112) n + | n + 〉 , for n ∈ (cid:78) , (1.17)and follow the canonical commutation relation[ ˆ a , ˆ a † ] = (cid:49) , (1.18)where (cid:49) is the identity operator. The eigenstates of the annihilation operator are the coherentstates { | α 〉 } α ∈ (cid:67) , defined as | α 〉 = e − | α | (cid:88) n ≥ α n (cid:112) n ! | n 〉 , (1.19)for all α ∈ (cid:67) . Alternatively, defining the displacement operator asˆ D ( α ) = e α ˆ a † − α ∗ ˆ a , (1.20)for all α ∈ (cid:67) , the coherent state of amplitude α ∈ (cid:67) is obtained from the vacuum state as | α 〉 = ˆ D ( α ) | 〉 . (1.21)The inner product of two coherent states | α 〉 and | β 〉 is given by 〈 α | β 〉 = e α ∗ β − ( | α | +| β | ) , (1.22)for all α , β ∈ (cid:67) . In particular, two coherent states have nonzero overlap. These states form anovercomplete family: (cid:90) α ∈ (cid:67) | α 〉〈 α | d απ = (cid:49) , (1.23)where d α = d ℜ ( α ) d ℑ ( α ). The canonical position-like and momentum-like operators ˆ q and ˆ p aredefined as ˆ q = (cid:112) a + ˆ a † ),ˆ p = i (cid:112) a − ˆ a † ). (1.24)These hermitian operators, also referred to as quadrature operators in the context of opticalquantum information processing, follow the canonical commutation relation[ ˆ q , ˆ p ] = i (cid:49) . (1.25)16 .2. PHASE SPACE FORMALISM They satisfy Heisenberg uncertainty principle [Hei85] σ ˆ q σ ˆ p ≥
12 , (1.26)where σ ˆ q and σ ˆ p denote the standard deviation of position and momentum, respectively, i.e., theycannot be measured both with arbitrary precision for the same quantum state: measuring onerandomises the other.The eigenstates of ˆ q (resp. ˆ p ) form a continuous family of unnormalisable states { | q 〉 } q ∈ (cid:82) (resp. { | p 〉 } p ∈ (cid:82) ), thus technically lying outside of the Hilbert space. These states may be treatedformally as an infinite uncountable basis of the Hilbert space, the so-called position basis (resp.momentum basis). Expanding a single-mode pure state | ψ 〉 in the position basis gives | ψ 〉 = (cid:90) q ∈ (cid:82) ψ ( q ) | q 〉 dq , (1.27)where ψ ( q ) = 〈 q | ψ 〉 is the position wave function of the state | ψ 〉 , with the normalisation conditionfor the position probability distribution (cid:82) q ∈ (cid:82) | ψ ( q ) | dq =
1. A similar expansion holds in themomentum basis with the momentum wave function. The position and momentum bases arerelated by a Fourier transform: | q 〉 = (cid:112) π (cid:90) p ∈ (cid:82) e − iqp | p 〉 d p , (1.28)and | p 〉 = (cid:112) π (cid:90) q ∈ (cid:82) e iqp | q 〉 dq . (1.29)Note that the Fock state | n = 〉 and the coherent state | α = 〉 are equal, but different from theposition state | q = 〉 and the momentum state | p = 〉 , themselves distinct. We refer the reader to [CG69b, CG69a] for an introduction to the material presented in thissection. In particular, we restrict to single-mode states and operators.The expectation values of the position and momentum operators lie in the so-called phase space,which is the quantum analogue of classical phase space. Continuous variable quantum statesand operators can be alternatively described by a phase space representation. This formulationidentifies a quantum state with a normalised distribution over phase space.This allows for a simple and experimentally relevant classification of quantum states: thosewith a Gaussian phase space distribution are called Gaussian states and the others non-Gaussianstates. By extension, operations mapping Gaussian states to Gaussian states are also calledGaussian. These Gaussian operations and states are the ones implementable with linear opticsand quadratic non-linearities [BvL05], and are hence relatively easy to construct experimentally.17
HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS
Hereafter, we identify the single-mode phase space with (cid:67) , where the real part correspondsto expectation values of the position operator and the imaginary part to expectation valuesmomentum operator. We adopt the convention α = (cid:112) ( q + i p ) ∈ (cid:67) , with d απ = d ℜ ( α ) d ℑ ( α ) π = dqdp π .There exists a continuum of equivalent phase space distributions representing the sameoperator in phase space. This continuum of representations is parametrized by a real parameter s ≤
1. For all s ≤
1, let us define the operatorˆ T ( α , s ) : = (cid:90) β ∈ (cid:67) ˆ D ( β ) exp (cid:179) αβ ∗ − α ∗ β + s | β | (cid:180) d βπ , (1.30)for all α ∈ (cid:67) . The phase space representation with parameter s of an operator ˆ O is defined as W ˆ O ( α , s ) = Tr (cid:163) ˆ T ( α , s ) ˆ O (cid:164) . (1.31)This expression should be treated formally for unbounded operators and the case s = s → − . The same definition holds for density operators, in which case therepresentation is real-valued and corresponds to the expectation value of the operator ˆ T . Thephase space representations are normalised as (cid:90) α ∈ (cid:67) W ρ ( α , s ) d απ = Tr ( ρ ), (1.32)for any density operator ρ and any s ≤
1. As the parameter s decreases, the phase space represen-tation smoothens. This is captured by the following relation: W ( α , s ) = t − s (cid:90) β ∈ (cid:67) W ( β , t ) exp (cid:181) − | α − β | t − s (cid:182) d βπ , (1.33)for all s < t ≤
1, i.e., the representation with lower parameter is obtained from the representationwith higher parameter by a Gaussian convolution. In particular, if one representation is aGaussian function, then all representations are Gaussian. Moreover, for all operators ˆ O and ˆ O ,Tr (cid:161) ˆ O ˆ O (cid:162) = (cid:90) α ∈ (cid:67) W ˆ O ( α , − s ) W ˆ O ( α , s ) d απ , (1.34)for all s ∈ [ −
1, 1]. This important property allows one to retrieve information about quantumsystems by probing their phase space representation: if one of the two operators in the aboveequation is a density operator, the expectation value is obtained asTr (cid:161) ˆ O ρ (cid:162) = (cid:90) α ∈ (cid:67) W ˆ O ( α , − s ) W ρ ( α , s ) d απ , (1.35)for all s ∈ [ −
1, 1].In what follows, we detail some properties of the three most prominent representations inthe literature: the Wigner W function [Wig97], the Glauber–Sudarshan P function [Sud63,Gla63] and the Husimi Q function [Hus40], corresponding to the values s = s = s = − .2. PHASE SPACE FORMALISM respectively (Fig. 1.1). In particular, we will make extensive use of the Husimi representationthroughout the first chapters of the thesis. We adopt the normalising conventions W ( α ) = π W ( α , 0), P ( α ) = π W ( α , 1), Q ( α ) = π W ( α , − α ∈ (cid:67) , so that the W , P and Q functions are normalised to 1 for normalised states (note thedifference of normalisation with [CG69a] for the Wigner and Husimi functions). Smoother More singularGaussianconvolution Gaussianconvolution
Figure 1.1: A pictorial representation of the continuum of phase space representations. W function The Wigner function is a nonsingular distribution for all states and is referred to as a quasiprob-ability distribution, as it is a normalised distribution which can take negative values. Thiscontrasts with classical phase space probability distributions.By virtue of Hudson’s theorem [Hud74, SC83], a pure quantum state is non-Gaussian if andonly if its Wigner function has negative values. In other words, if a pure quantum state has apositive Wigner function, then it is a Gaussian state. Various notions relating to negativity of theWigner function have been introduced for measuring how much non-Gaussian a quantum stateis [K ˙Z04, AGPF18].The Wigner function can be expressed as [Roy77] W ˆ O ( α ) = π Tr (cid:104) ˆ D ( α ) ˆ Π ˆ D † ( α ) ˆ O (cid:105) , (1.37)for all α ∈ (cid:67) and for any operator ˆ O , where ˆ Π = ( − ˆ a † ˆ a = (cid:80) n ≥ ( − n | n 〉〈 n | is the parity operatorand ˆ D ( α ) = e α ˆ a † − α ∗ ˆ a is a displacement operator of amplitude α ∈ (cid:67) . In particular, the Wignerfunction of a quantum state is related to the expectation value of displaced parity operators. P function The Glauber–Sudarshan P function is the most singular phase space representation. For quantumstates, it is actually always a singular distribution.19 HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS
The P function gives a convenient diagonal representation of a state in coherent state basisas ρ = (cid:90) α ∈ (cid:67) P ρ ( α ) | α 〉〈 α | d α , (1.38)and this representation is unique. The P function can be expressed formally as [Meh67] P ˆ O ( α ) = e | α | π (cid:90) β ∈ (cid:67) 〈− β | ˆ O | β 〉 exp (cid:161) αβ ∗ − α ∗ β + | β | (cid:162) d βπ , (1.39)for all α ∈ (cid:67) and for any operator ˆ O . Q function The Husimi Q function is a smoother version of the Wigner function and the Glauber–Sudarshan P function. It is given by Q ˆ O ( α ) = π 〈 α | ˆ O | α 〉 , (1.40)for all α ∈ (cid:67) and for any operator ˆ O , where | α 〉 is the coherent state of amplitude α ∈ (cid:67) . TheHusimi Q function of a state thus is always nonnegative and normalised. However, it does notrepresent probabilities of mutually exclusive states since the overlap between two coherent statesis always nonzero.For any state ρ and any operator ˆ O we have, with Eq. (1.41), the so-called optical equivalencetheorem for antinormal ordering:Tr (cid:161) ˆ O ρ (cid:162) = π (cid:90) α ∈ (cid:67) Q ρ ( α ) P ˆ O ( α ) d α . (1.41)Hudson’s theorem may be formulated as follows for the Husimi function [LB95]: a pure quantumstate is non-Gaussian if and only if its Husimi function has zeros. In other words, a pure quantumstate is non-Gaussian if and only if it is orthogonal to at least one coherent state. Gaussian states and processes have been defined in the previous section, the former as thestates having a Gaussian phase space representation and the latter as the processes mappingGaussian states to Gaussian states. Ubiquitous in quantum physics, they are well understoodtheoretically [FOP05, WPGP +
12, ARL14] and routinely implemented experimentally [GCP07].We review Gaussian processes and states in the following sections, restricting to pure states,unitary operations and projectors.
The displacement operator of amplitude α ∈ (cid:67) has been introduced in the previous section andreads ˆ D ( α ) = e α ˆ a † − α ∗ ˆ a . (1.42)20 .3. GAUSSIAN STATES AND PROCESSES It satisfies the relations ˆ D † ( α ) = ˆ D ( − α ),ˆ D ( α ) ˆ a ˆ D † ( α ) = ˆ a − α (cid:49) ,ˆ D ( α ) ˆ a † ˆ D † ( α ) = ˆ a † − α ∗ (cid:49) ,ˆ D ( α ) ˆ D ( β ) = e ( αβ ∗ − α ∗ β ) ˆ D ( α + β ), (1.43)for all α , β ∈ (cid:67) . We denote a tensor product of m single-mode displacements by ˆ D ( α ) = (cid:78) mi = ˆ D ( α i )for all α = ( α , . . . , α m ) ∈ (cid:67) .The squeezing operator is defined asˆ S ( ξ ) = e ( ξ ˆ a − ξ ∗ ˆ a † ) , (1.44)for all ξ ∈ (cid:67) . The parameter ξ is called squeezing parameter. The squeezing operator satisfies therelations ˆ S † ( ξ ) = ˆ S ( − ξ ),ˆ S ( ξ ) ˆ a ˆ S † ( ξ ) = cosh r ˆ a + e − i θ sinh r ˆ a † ,ˆ S ( ξ ) ˆ a † ˆ S † ( ξ ) = cosh r ˆ a † + e i θ sinh r ˆ a , (1.45)for all ξ = re i θ ∈ (cid:67) . We denote a tensor product of m single-mode squeezings by ˆ S ( ξ ) = (cid:78) mi = ˆ S ( ξ i )for all ξ = ( ξ , . . . , ξ m ) ∈ (cid:67) .The displacement and squeezing operators may be conceived as acting on a state by displac-ing and squeezing its phase space representation, respectively, as their name indicates. Thisgeometrical intuition holds in particular for the Wigner quasiprobability distribution.Any single-mode Gaussian unitary operation may be written as a squeezing and a displace-ment operator. The ordering is only a convention, since the displacement and squeezing operatorssatisfy the braiding relation [NT97]ˆ D ( α ) ˆ S ( ξ ) = ˆ S ( ξ ) ˆ D ( γ ), γ = α cosh r + α ∗ e − i θ sinh r , (1.46)for all α ∈ (cid:67) and all ξ = re i θ ∈ (cid:67) .Passive linear transformation over m modes are defined as the unitary transformations ˆ U which act unitarily on the creation operators of the modes ˆ a † , . . . , ˆ a † m as well as on the annihilationoperators ˆ a , . . . , ˆ a m . Any such transformation ˆ U is associated to an m × m unitary matrix U whichtransforms the creation operators of the modes as ˆ a † ...ˆ a † m → U ˆ a † ...ˆ a † m , (1.47)and the annihilation operators of the modes as ˆ a ...ˆ a m → U ∗ ˆ a ...ˆ a m . (1.48)21 HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS
These transformations map the multimode vacuum state onto itself.Finally, Gaussian projectors are identified with projections onto Gaussian pure states, whichwe review in what follows.
General single-mode Gaussian pure states are obtained from the vacuum with a Gaussian unitaryoperation. They are the squeezed coherent states (or alternatively the displaced squeezed vacuumstates): ˆ S ( ξ ) ˆ D ( α ) | 〉 , (1.49)for α , ξ ∈ (cid:67) . Setting ξ = α ∈ (cid:67) , while setting α = ξ ∈ (cid:67) .The phase space representation of a coherent state is a Gaussian displaced in phase space,while the phase space representation of a squeezed vacuum state is a Gaussian centered at 0,squeezed in a direction depending on the phase of the squeezing parameter. The strength of thesqueezing depends on the modulus of the squeezing parameter (Fig. 1.2). In particular, positionand momentum eigenstates can be conceived formally as infinitely squeezed vacuum states,displaced by a finite amplitude [SEMC13]. q
We present a short introduction to the symplectic formalism and refer to [ARL14] for a detailedexposition.Any m -mode Gaussian state ρ can be described by a 2 m × m covariance matrix V (cid:82) contain-ing its second canonical moments and a displacement vector d (cid:82) of size m containing its firstcanonical moments. The coefficients of the covariance matrix are defined, for k , l ∈ {
1, . . . , 2 m } ,by V (cid:82) kl = 〈 R k R l + R l R k 〉 ρ − 〈 R k 〉 ρ 〈 R l 〉 ρ where R = ( ˆ q , . . . , ˆ q m , ˆ p , . . . , ˆ p m ). The coefficients of thedisplacement vector are given by d (cid:82) j = 〈 R j 〉 ρ for all j ∈ {
1, . . . , 2 m } . Alternatively and more con-veniently, one can describe covariance matrices and displacement vectors in the complex basis λ = ( ˆ a , . . . , ˆ a m , ˆ a † , . . . , ˆ a † m ). We write V and ˜ d the covariance matrix and displacement vector inthat basis, with V = Ω V (cid:82) Ω † , ˜ d = Ω d (cid:82) , (1.50)where Ω = (cid:112) (cid:195) (cid:49) m i (cid:49) m (cid:49) m − i (cid:49) m (cid:33) . (1.51)The complex covariance matrix has the structure V = (cid:195) A BB ∗ A ∗ (cid:33) , (1.52)with A = A † and B = B T , so that V † = V . The displacement vector has the structure˜ d = (cid:195) dd ∗ (cid:33) . (1.53)We will also refer to the above vector d as the displacement vector.Gaussian multimode unitary operations are generated by Hamiltonians that are at mostquadratic in the annihilation and creation operators of the modes. As a consequence, they induceaffine transformations of the annihilation and creation operators which preserve their canoni-cal commutation relations, i.e., symplectic linear transfomations, together with displacements.The evolution of a Gaussian state during a Gaussian evolution (excluding displacements) is de-scribed by a complex symplectic transformation of its complex covariance matrix and its complexdisplacement vector: ( V , ˜ d ) → ( S V S † , S ˜ d ), (1.54)where a complex symplectic matrix S satisfies S Ω J Ω † S † = Ω J Ω † , (1.55)where J = (cid:195) (cid:48) m (cid:49) m − (cid:49) m (cid:48) m (cid:33) and where the matrix Ω is defined in Eq. (1.51). We will use the notations S ξ ≡ (cid:195) D c ( ξ ) D s ( ξ ) D s ( ξ ) D c ( ξ ) (cid:33) (1.56)23 HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS for all ξ = ( ξ , . . . , ξ m ) ∈ (cid:67) m , with D c ( ξ ) = Diag( c ξ , . . . , c ξ m ) and D s ( ξ ) = Diag( s ξ , . . . , s ξ m ), where c χ = cosh χ and s χ = sinh χ for the symplectic matrices that implement squeezing and S U ≡ (cid:195) U ∗ (cid:48) m (cid:48) m U (cid:33) (1.57)for the symplectic matrix associated with a passive linear transformations with m × m unitary ma-trix U . A displacement does not affect the covariance matrix and only translates the displacementvector.The so-called Bloch-Messiah or Euler decomposition implies that any 2 m × m complexsymplectic matrix can be written as S U S ξ S V for some m × m unitary matrices U and V andsome squeezing parameters ξ = ( ξ , . . . , ξ m ) ∈ (cid:67) m . In particular, any multimode Gaussian unitaryoperation can be decomposed as a passive linear transformation followed by a product of single-mode squeezings, followed by another passive linear transformation, together with single-modedisplacements.Since any multimode Gaussian pure quantum state may be engineered from the vacuum witha Gaussian unitary operation, by virtue of Williamson decomposition, and since the vacuum ismapped onto itself by passive linear transformations, this means that any multimode Gaussianpure quantum state can be written as a tensor product of single-mode Gaussian states (displacedsqueezed vacuum states) followed by a single passive linear transformation. Linear optics covers the manipulation of light by unitary transformations whose exponent is atmost quadratic in the field operator [WM07], i.e., Gaussian unitaries. It induces transformationsof quantum states of light which are divided in two categories, passive and active transformations,depending on whether these transformations change the total number of photons of the inputstate. In what follows, we review a few examples of quantum states of light and quantum opticalmeasurements, and we detail passive linear optical transformations, implemented by unitaryinterferometers, with the examples of the Hong-Ou-Mandel effect [HOM87] and its generalisationBoson Sampling [AA13].
We briefly list single-mode quantum states that are common in the literature, some of whichwere already introduced in the previous sections, and which we will encounter in the followingchapters.• Photon-number states: these states form the orthonormal Fock basis and are obtained fromthe vacuum as | n 〉 = ( ˆ a † ) n (cid:112) n ! | 〉 , (1.58)24 .4. LINEAR OPTICS for all n ∈ (cid:78) . Taking n = n >
0. They are the eigenstates of the photon-number operator ˆ n = ˆ a † ˆ a : for all n ∈ (cid:78) , ˆ n | n 〉 = n | n 〉 .• Coherent states: these Gaussian states are expressed as | α 〉 = e − | α | (cid:88) n ≥ α n (cid:112) n ! | n 〉 , (1.59)for all α ∈ (cid:67) . These states are a good approximation of the quantum state of a laser andare sometimes referred to as classical states, because their behaviour resembles that of aclassical harmonic oscillator. They are the eigenstates of the annihilation operator ˆ a : for all α ∈ (cid:67) , ˆ a | α 〉 = α | α 〉 .• Squeezed vacuum states: these Gaussian states are expressed as | ξ 〉 = (cid:112) cosh r (cid:88) n ≥ ( − e − i θ tanh r ) n (cid:112) (2 n )!2 n n ! | n 〉 , (1.60)for all ξ = re i θ ∈ (cid:67) . They display reduced variance for one quadrature, but increased variancefor the conjugate quadrature, in accordance with the uncertainty principle.• Photon-subtracted/added states: these states are obtained by applying the annihilation/creationoperator to a state (and renormalising). These unphysical operations cannot be implementeddeterminisically and are implemented probabilistically in practice. For example, a pho-ton subtraction may be implemented by mixing the input state with the vacuum on abeam splitter with near unity reflectance. Then, conditioned on a successful single-photonheralding of the transmitted light, the reflected state has been photon-subtracted.• Cat states: named after Schrödinger’s cat, these states are superpositions of two coherentstates of equal amplitudes, | α 〉 and |− α 〉 , like the cat in Schrödinger’s thought experimentis in a superposition of two classical states, dead and alive. Varying the relative phasebetween the coherent states in the superposition gives different cat states. In particular,we introduce the cat + and cat − states: | cat ± α 〉 = (cid:113) N ± α ( | α 〉 ± |− α 〉 ), (1.61)for all α ∈ (cid:67) , where N ± α = ± e − | α | ) is a normalisation factor.• GKP states: finally, let us mention the Gottesman-Kitaev-Preskill (GKP) states which forma family of unphysical states with periodic wave functions [GKP01]. These states are formalperiodic superpositions of infinitely squeezed states and their physical approximations haveapplications for continuous variable quantum error correction [TBMS20].25 HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS
We list various (idealised) single-mode measurements in what follows: homodyne detection, bal-anced heterodyne detection, unbalanced heterodyne detection, single-photon threshold detection,photon number parity detection and photon-number resolving detection. Detailed informationon these detection methods can be found, e.g., in [FOP05]. We will only consider multimodedetections that are tensor products of such single-mode detections. -- ⇢
1. The POVM elements for unbalanced heterodyne detection withunbalancing parameter ξ ∈ (cid:67) are given by Π ξα = π | α , ξ 〉〈 α , ξ | , (1.64)for all α ∈ (cid:67) , where | α , ξ 〉 = ˆ S ( ξ ) ˆ D ( α ) | 〉 is a squeezed coherent state. Writing ξ = re i θ , theunbalalancing parameter is related to the optical setup by r = (cid:175)(cid:175) log (cid:161) TR (cid:162)(cid:175)(cid:175) , with θ being the phase ofthe local oscillator [CDM + Q function. Setting ξ = | ξ | = r to infinity gives homodyne detection. Any Gaussian measurementcan thus be implemented by Gaussian unitary operations and heterodyne detection only, since itcan be implemented by Gaussian unitary operations and homodyne detection only [GC02, EP03].27 HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS
Additionnally, we introduce three non-Gaussian measurements, each giving more informationabout the photon number of the measured state. The first is single-photon threshold detec-tion [Had09], or simply threshold detection, whose POVM elements are given by Π = | 〉〈 | , Π = (cid:49) − | 〉〈 | . (1.65)This binary measurement only distinguishes the vacuum state from other states. The second isphoton number parity detection [HBR07], or simply parity detection, whose POVM elements aregiven by Π + = (cid:88) n ≥ | n 〉〈 n | , Π − = (cid:88) n ≥ | n + 〉〈 n + | . (1.66)This is a binary measurement of the parity operator ˆ Π = ( − ˆ a † ˆ a yielding, as its name indicates,the parity of the number of photons of the measured state. The third is photon number-resolvingdetection [DMB + Π n = | n 〉〈 n | , (1.67)for all n ∈ (cid:78) , i.e., projections onto Fock states. Linear optical unitary interferometers are composed of beam splitters and phase shifters andimplement passive linear transformations of the modes. In particular, any passive linear trans-formation ˆ U over m modes with m × m unitary matrix U can be implemented by a linearinterferometer with at most m ( m − balanced beam splitters and m phase shifters [RZBB94]. Thecorresponding unitary interferometer is described by the same unitary matrix U = ( u i j ) ≤ i , j ≤ m .Unlike in the circuit picture, the matrix U does not act on the computational basis, which in thiscase is the infinite multimode Fock basis, but rather describes the linear evolution of the creationoperator of each mode. More precisely, ˆ a † ...ˆ a † m → U ˆ a † ...ˆ a † m = (cid:80) mk = u k ˆ a † k ... (cid:80) mk = u mk ˆ a † m . (1.68)In that picture, the direct sum plays the role of the tensor product in the computational basis:taking the direct sum of two unitaries corresponds to putting linear optical elements in parallel,while multiplying unitaries corresponds to putting linear optical elements in sequence.Multimode coherent states have a specific evolution through linear interferometers: they aremapped onto coherent states and do not become entangled, unlike other states. If U is the unitarymatrix describing an interferometer which implements a passive linear transformation ˆ U , aninput coherent state | α 〉 is mapped to an output coherent state ˆ U | α 〉 = | U α 〉 , where the vector of28 .4. LINEAR OPTICS output amplitudes U α is obtained by multiplying the vector of input amplitudes α by the unitarymatrix U .Remarkable quantum effects may be witnessed when the input to linear optical unitaryinterferometers are single-photon Fock states instead of coherent states. The celebrated Knill–Laflamme–Milburn scheme [KLM01] shows that single photons and linear optics are enough toachieve universal quantum computing together with adaptive measurements (making the rest ofthe computation depend on the result of intermediate measurements). Already without adaptivemeasurements, interesting effects can be observed. We give two notable examples in the followingsections: the Hong–Ou–Mandel effect and Boson Sampling. The Hong–Ou–Mandel effect, or photon bunching, refers to the bosonic behaviour of indistin-guishable photons which bunch together when mixed on a balanced beamsplitter (Fig. 1.5). Abalanced beam splitter is a unitary interferometer over two modes, with unitary matrix H = (cid:112) (cid:195) − (cid:33) . (1.69)Figure 1.5: Hong-Ou-Mandel effect. The dashed red line represents a balanced beam splitter. Thenumber of photons is detected for both output arms. If the input single photons are indistinguish-able, the outcomes (20) and (02) occur with the same probability and the outcome (11) neveroccurs.The input state is composed of two single photons, one in each mode. Labelling the modes u and d , for ‘up’ and ‘down’, let ˆ a † u , ˆ a † d and ˆ b † u , ˆ b † d be the creation operators of the input and outputmodes, respectively. The balanced beam splitter acts on the input creation operators as (cid:195) ˆ b † u ˆ b † d (cid:33) = H (cid:195) ˆ a † u ˆ a † d (cid:33) . (1.70)29 HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS
The input state thus evolves as | 〉 = ˆ a † u ˆ a † d | 〉 H →
12 ( ˆ b † u + ˆ b † d )( ˆ b † u − ˆ b † d ) | 〉=
12 ( ˆ b † u − ˆ b † d ) | 〉= (cid:112) | 〉 − | 〉 ), (1.71)where we used ˆ b † u ˆ b † d = ˆ b † d ˆ b † u . In particular, measuring the photon number in both output modeswill always yield 0 for one of the modes: the outcome (11) is never witnessed if the photons areindistinguishable, i.e., the photons have bunched together. Let m ∈ (cid:78) ∗ and n ∈ (cid:78) , with m ≥ n . Boson Sampling, introduced in [AA13], is a generalisation ofthe Hong–Ou–Mandel setup, where the balanced beam splitter is replaced by a general unitaryinterferomer over m modes with m × m unitary matrix U and the input is composed of n singlephotons in the first n modes and vacuum in the remaining m − n modes, the photon number of alloutput modes being measured (Fig. 1.6).Even though Boson Sampling has been formulated for general bosonic particles, linear opticsprovides a convenient way of looking at it. Boson Sampling is a subuniversal model of quantumcomputation, believed to be hard to simulate by classical computers while not possessing thecomputational power of a universal quantum computer. We review this model in what follows andwe refer to [AA13] for a detailed version of the material presented in this section. In particular,we do not discuss the theoretical use of postselection.We denote photon number states over m modes by | s 〉 = | s . . . s m 〉 = ( ˆ a † ) s (cid:112) s ! · · · ( ˆ a † m ) s m (cid:112) s m ! | 〉 ⊗ m , (1.72)where s k and ˆ a † k are respectively the number of photons and the creation operator of the k th mode. We identify these states with m -tuples of integers s = ( s , . . . , s m ) ∈ (cid:78) m (see section 1.1.1 formulti-index notations). The input state with n single photons in the first n modes and vacuum inthe other modes is denoted | t 〉 , with t = ( n , m − n ). We introduce, Φ m , n : = { s ∈ (cid:78) m , | s | = n } . (1.73)This set corresponds to the m -mode Fock states with total number of photons equal to n . We have | Φ m , n | = (cid:161) m + n − n (cid:162) and t ∈ Φ m , n . 30 .4. LINEAR OPTICS . . .. . . . . .. . . | i
1, . . . , m , and where the permanent of an n × n matrix A = ( a i j ) ≤ i , j ≤ n isdefined as Per A = (cid:88) σ ∈ S n n (cid:89) i = a i σ ( i ) , (1.77)where the sum is over the permutations of the set {
1, . . . , n } .31 HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS
We write Pr m , n [. | t ] the probability distribution of the outputs over Φ m , n of the unitaryinterferometer U acting on the input | t 〉 . With the previous notations we obtain, for all s , t ∈ Φ m , n ,Pr m , n [ s | t ] = (cid:175)(cid:175) Per (cid:161) U s , t (cid:162)(cid:175)(cid:175) s ! t ! . (1.78)With t = ( n , m − n ) we have t ! = m , n [ s | t ] = s ! (cid:175)(cid:175) Per (cid:161) U s , t (cid:162)(cid:175)(cid:175) . (1.79)The output photon-number distribution of a Boson Sampling interferometer with n photonsover m modes thus is related to the modulus squared of the permanent of an n × n matrix withcomplex entries. This matrix is obtained from the unitary matrix U describing the interferometerby discarding its last m − n columns and repeating its lines according to the detection pattern s .The permanent defined in Eq. (1.77) is a ‘hard’ quantity to compute. In order to appreciate thishardness, let us take a brief and informal detour through the realm of complexity theory [Man01].A formal introduction to the complexity classes presented here is given in [AA13].A complexity class is a set of computational problems. These problems may be of differenttypes: in particular, a decision problem is a problem with yes or no answers, a function problemis a problem with more general answers (e.g., natural, real or complex numbers), and a samplingproblem consists in outputting samples from a target probability distribution, either exactly orapproximately.In the language of complexity theory, an efficient algorithm is an algorithm which takes anumber of steps which is polynomial in the size of its input (its number of bits), and the genericmodel for a classical computer is a deterministic Turing machine.Given a complexity class C , a problem p is said to be C -hard if any problem in C can berephrased efficiently as an instance of the problem p . Roughly speaking, this means that theproblem p is harder than any of the problems in C . If the problem p is also in C , it is referred toas C -complete.The class of decision problems that can be solved efficiently by a classical computer is denoted P . The class of decision problems whose solution can be verified efficiently by a classical computeris denoted NP . A great open problem in complexity theory is whether these two complexity classesare equal or if P (cid:54)= NP , the latter being widely believed.An oracle for a given computational problem is a black box which is able to produce a solutionfor any instance of this problem. An oracle for a complexity class is a black box which, given anyproblem in the complexity class, is able to produce a solution for any instance of this problem.The access to an oracle is denoted with an exponent. For example, a problem which can be solvedefficiently when given access to an oracle for an NP -complete problem is in the class P NP .The polynomial hierarchy PH is a tower of complexity classes generalising P and NP . It canbe defined inductively based on an oracle construction, where the level 0 is P , the level 1 contains NP , the level 2 contains NP NP , and so on. Each level is contained in the next one and if two32 .4. LINEAR OPTICS consecutive levels are equal, then they are also equal to all of the above levels—we talk abouta collapse of the polynomial hierarchy. The conjecture that the polynomial hierarchy does notcollapse, i.e., that all levels within the hierarchy are distinct, is a stronger version of the P (cid:54)= NP conjecture.The class of decision problems that can be solved efficiently by a classical computer withaccess to a genuine random number source is denoted BPP . It lies at the second level of thepolynomial hierarchy PH [Lau83].The class of function problems which consist in counting the number of solutions of an NP problem is denoted P . Its equivalent complexity class of decision problems is denoted P P and byToda’s theorem [Tod91] we have PH ⊂ P P , i.e., counting the solutions of NP problems is harderthan any problem in the whole polynomial hierarchy of complexity classes.With these elements introduced, we are now in position to discuss the hardness of the permanent:computing exactly the permanent of matrices with (0, 1) entries is a P -complete problem [Val79]and hence PH -hard. Moreover, approximating the permanent of real matrices up to multiplicativeerror, i.e., outputting an estimate ˜ P such that (1 −
1/ poly m ) P ≤ ˜ P ≤ (1 + / poly m ) P where P is thepermanent of a square matrix of size m with real entries, is also P -hard [AA13].The computational problem ‘Boson Sampling’ corresponds to the task of sampling from theoutput probability distribution in Eq. (1.79), given the description U of the Boson Samplinginterferometer.Making use of the hardness of the permanent and the connection between the output proba-bilities of a Boson Sampling interferometer and the permanent, two main results are derivedin [AA13] about the hardness of classically solving two versions of the Boson Sampling problem,which we refer to as exact hardness and approximate hardness.Exact hardness corresponds to the following result: let O be an oracle which, given a unitarymatrix U and a random string as its unique source of randomness, samples exactly from theoutput probability distribution of the Boson Sampling interferometer U . Then PH ⊂ BPP NP O . Inparticular, an efficient classical simulation of exact Boson Sampling collapses the polynomialhierarchy to its third level.This result uses the fact that a single output probability of a Boson Sampling interferometeris hard to approximate up to multiplicative error and that being able to sample efficiently from aprobability distribution allows one to obtain a multiplicative approximation of the probabilityof any outcome in FBPP NP (where FBPP is the class of function problems that can be solvedefficiently using a
BPP machine) thanks to Stockmeyer’s approximate counting algorithm [Sto85].In that case, an oracle which samples from an exact Boson Sampling probability distribution isrequired.On the other hand, approximate sampling refers to the task of sampling from a probabilitydistribution which has a given constant total variation distance with a target distribution (seeEq. (1.14)). Approximate hardness of Boson Sampling is more elaborate than exact hardness33
HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS and relies on two plausible but unproven conjectures, even though the statement of the result isnearly identical: let O be an oracle which, given a unitary matrix U and a random string as itsunique source of randomness, samples approximately from the output probability distribution ofthe Boson Sampling interferometer U . Then PH ⊂ BPP NP O . In particular, an efficient classicalsimulation of approximate Boson Sampling collapses the polynomial hierarchy to its third level.Unlike for exact sampling, one cannot apply directly Stockmeyer’s approximate countingalgorithm in order to obtain multiplicative estimates of the probabilities of the target distribution.This is because the oracle now only outputs samples from an approximate probability distribution,i.e., a probability distribution which is very close to the correct one for most of the samples butnot all samples. In the worst case, the probability that we are trying to estimate could be theprobability of one of these ‘bad samples’, and estimating this probability would merely give us avery bad estimate of the permanent, which is not hard to achieve. The trick to get around thatproblem is to hide the instance of the permanent that we wish to estimate into the probability ofa random output of a Boson Sampling interferometer: given a classical machine which correctlyperforms the sampling for most of the samples, it would then correctly sample our instance withhigh probability. In the worst case, this effectively averages the constant total variation errorover the sample space, allowing for a much more precise approximation of the permanent usingStockmeyer’s algorithm.This hiding procedure is based on the fact that small enough submatrices of random unitarymatrices are very close to random complex Gaussian matrices. In order to restrict to matricesthat do not have repeated lines, the so-called antibunching regime n = O ( (cid:112) m ) is chosen, whichensures a negligible probability of detecting more than one photon in the same output mode. Theprocedure outlined above then allows one to prove that the problem | GPE | ± which consists inapproximating up to additive error the square modulus of the permanent of random complexGaussian matrices is in FBPP NP O , where O is an oracle for approximate Boson Sampling.The proof of approximate hardness then relies on two conjectures about the permanent ofrandom complex Gaussian matrices in order to bridge the gap between additive approximationsof the square modulus of the permanent of random complex Gaussian matrices and collapseof the polynomial hierarchy: the permanent of Gaussians conjecture and the permanent anti-concentration conjecture. The former conjecture states that the problem GPE × which consists inapproximating the permanent of random complex Gaussian matrices up to multiplicative error is P -hard. The latter conjecture states that with high probability the permanent of a randomlychosen complex Gaussian matrix is not too small. This implies in turn that the problem | GPE | ± of additive approximation of the square modulus of the permanent of random complex Gaussianmatrices is as hard as the problem GPE × of multiplicative approximation of the permanent ofrandom complex Gaussian matrices. With these two conjectures and the above argument, weobtain PH ⊂ P P ⊂ GPE × = | GPE | ± ⊂ FBPP NP O , which concludes the proof.Assuming that the polynomial hierarchy does not collapse, Boson Sampling is hard to simu-34 .5. SEGAL–BARGMANN FORMALISM late exactly classically, and even approximately with additional mathematical conjectures. Theapproximate hardness of Boson Sampling is important since it opens the way for an experimentaldemonstration of quantum supremacy. Indeed, it is unrealistic to expect that an experimentalBoson Sampling device would sample exactly from the ideal Boson Sampling distribution. More-over, given the nature of the computational task at hand, i.e., outputting samples from a givenprobability distribution, there is no hope of being able to verify that an exact sampling has beenperformed. On the other hand, verifying that approximate Boson Sampling has been performedcould be possible and indeed we derive such a verification protocol in chapter 4. The Segal–Bargmann formalism [Bar61, SM63] associates to every quantum state an analyticalfunction over the complex plane. It has been used to study quantum chaos [LV90, ABB96,KMW97, BS99], and the completeness of sequences of coherent states [Per71, BGZ75, BZ78]. Wegive hereafter a quick introduction to this formalism. Further details may be found in chapter 2and in [Vou06].
We introduce below the analytical function, which we refer to as the stellar function. This functionhas been recently studied, in the context of non-Gaussian quantum state engineering [GG19], inorder to simplify calculations related to photon-subtracted Gaussian states.
Definition 1.1 (Stellar function) . Let | ψ 〉 = (cid:80) n ≥ ψ n | n 〉 ∈ H be a normalised state. The stellarfunction of the state | ψ 〉 is defined as F (cid:63) ψ ( z ) = e | z | 〈 z ∗ | ψ 〉 = (cid:88) n ≥ ψ n z n (cid:112) n ! , (1.80)for all z ∈ (cid:67) , where | z 〉 = e − | z | (cid:80) n ≥ z n (cid:112) n ! | n 〉 is the coherent state of amplitude z .The stellar function is a holomorphic function over the complex plane, which provides an analyticrepresentation of a quantum state. A holomorphic function is a complex-valued function of one or more complex variables that is, atevery point of its domain, complex differentiable in a neighbourhood of the point. As it turns out,the set of holomorphic functions is equal to the set of analytic functions, i.e., the functions thatcan be written as a convergent power series in a neighbourhood of each point of their domain.When their domain is the whole complex plane, they are called entire functions. In what followswe consider univariate entire functions. 35
HAPTER 1. CONTINUOUS VARIABLE QUANTUM INFORMATION FORMALISMS
These functions provide a natural extension of univariate complex polynomials and variousproperties of polynomials extend to entire functions. In particular, Liouville’s theorem states thatany bounded entire function is constant. The principle of permanence asserts that the zeros of ananalytic function are isolated or this function is identically 0. Furthermore, the number of zerosof an analytic function f inside some contour is given by Cauchy’s argument principle. Theorem 1.1 (Cauchy’s argument principle) . Let f be an analytic function and let C be a contourin the domain of f . Then, Z C ( f ) = i π (cid:73) C f (cid:48) ( z ) f ( z ) dz , (1.81) where Z C ( f ) is the number of zeros of f inside the contour C, counted with multiplicity. The growth of an analytic function is described by a pair of non-negative numbers ρ , σ called theorder and the type. They are defined as [Boa54] ρ = lim r →+∞ sup ln ln M ( r )ln r , σ = lim r →+∞ sup ln M ( r ) r ρ , (1.82)where M ( r ) is the maximum value of the modulus of the funtion on the circle | z | = r . Forpolynomials, the growth is deeply related to the number of zeros—the degree. For entire functions,the growth is related to the density of zeros (see, e.g., [SS10] for more details). An entire functioncan also be factorized into a possibly infinite product involving its zeros, thanks to Weierstrassfactorization theorem. For entire functions of finite order, this result is refined by Hadamard-Weierstrass factorization theorem. Theorem 1.2 (Hadamard-Weierstrass factorization theorem) . Let f be an entire function of finiteorder ρ . Let m ∈ (cid:78) be the multiplicity of as a root of f . Let { z n } n ∈ (cid:78) be the non-zero roots of f ,counted with multiplicity. Then, there exist p , q ∈ (cid:78) , with p , q ≤ ρ , and a polynomial P of degree qsuch that, for all z ∈ (cid:67) , f ( z ) = z m e P ( z ) +∞ (cid:89) n = E p (cid:181) zz n (cid:182) , (1.83) where E p ( z ) = (1 − z ) e z + z /2 +···+ z p / p . (1.84)36 H A PT ER S TELLAR REPRESENTATION OF NON -G AUSSIAN QUANTUM STATES N on-Gaussian states are crucial for a variety of quantum information tasks [ESP02, Fiu02,GC02, WHG +
03, GPFC +
04, GS07, NFC09, ADDS +
09, BDE + + Q function inphase space. We use of this representation in order to derive an infinite hierarchy of single-modestates based on the number of zeros of the Husimi Q function, the stellar hierarchy. We givean operational characterisation of the states in this hierarchy with the minimal number ofsingle-photon additions needed to engineer them and derive equivalence classes under Gaussianunitary operations. We study in detail the topological properties of this hierarchy with respect tothe trace norm, and discuss implications for the robustness of the states in the stellar hierarchyand for non-Gaussian state engineering.This chapter is based on [CMG20, CRW + In continuous variable quantum information, quantum states are mathematically describedby vectors in a separable Hilbert space of infinite dimension (see section 1.1.3). Alternatively,phase space formalism allows us to describe quantum states conveniently using generalisedquasi-probability distributions [CG69a], among which are the Husimi Q function, the Wigner37 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES W function, and the Glauber–Sudarshan P function (see section 1.2). The states that have aGaussian Wigner or Husimi function are called Gaussian states, while all the other states arecalled non-Gaussian. By extension, the operations mapping Gaussian states to Gaussian statesare called Gaussian operations, and measurements projecting onto Gaussian states are calledGaussian measurements (see section 1.3).Hudson [Hud74] has notably shown that a single-mode pure quantum state is non-Gaussianif and only if its Wigner function has negative values and this result has been generalised tomultimode states by Soto and Claverie [SC83]. This characterization is an interesting startingpoint for studying non-Gaussian states. From this result, one can introduce measures of astate being non-Gaussian using Wigner negativity, e.g., the negative volume [K ˙Z04], that areinvariant under Gaussian operations. However, computing these quantities from experimentaldata is complicated in practice. Other measures and witnesses for non-Gaussian states have beenderived [GPB07, FMJ11, GPT +
13, HGT + how much? rather than how? .In order to adress the latter question, we will make use of another characterization ofGaussian states: the Wigner function having negative values is actually equivalent to the Husimifunction having zeros, as shown by Lütkenhaus and Barnett [LB95]. Informally, Theorem 2.1.
A pure quantum state is non-Gaussian if and only if its Husimi Q function haszeros.
Since the values of the Q function are the overlaps with coherent states, this result may beunderstood as follows: a pure quantum state is non-Gaussian if and only if it is orthogonal to atleast one coherent state.An interesting point is that for single-mode states, the zeros of the Husimi Q function forma discrete set, as we will show in the next section. The non-Gaussian properties of single-modestates may thus be described by the distribution of these zeros in phase space. Based on thisobservation, we classify single-mode continuous variable quantum states with respect to theirnon-Gaussian properties in the following sections, using the so-called stellar representation, orSegal–Bargmann formalism (see section 1.5), its link with the Husimi Q function and propertiesof holomorphic functions. In what follows, H denotes a single-mode infinite-dimensional Hilbert space. We recall thedefinition of the stellar function [Bar61, SM63] and prove a few important properties. Definition 2.1 (Stellar function) . Let | ψ 〉 = (cid:80) n ≥ ψ n | n 〉 ∈ H be a normalised state. The stellar .1. THE STELLAR FUNCTION function of the state | ψ 〉 is defined as F (cid:63) ψ ( z ) = e | z | 〈 z ∗ | ψ 〉 = (cid:88) n ≥ ψ n (cid:112) n ! z n , (2.1)for all z ∈ (cid:67) , where | z 〉 = e − | z | (cid:80) n ≥ z n (cid:112) n ! | n 〉 ∈ H is the coherent state of amplitude z .We now develop the formalism further, analysing the zeros of the stellar function to characterisestates. The stellar function is a holomorphic function over the complex plane. For any normalisedstate | ψ 〉 ∈ H and all z ∈ (cid:67) , (cid:175)(cid:175)(cid:175) F (cid:63) ψ ( z ) (cid:175)(cid:175)(cid:175) ≤ (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) (cid:88) n ≥ ψ n z n (cid:112) n ! (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ (cid:88) n ≥ | ψ n | (cid:88) n ≥ | z | n n ! = e | z | (2.2)by Cauchy-Schwarz inequality. This implies that the stellar function of a normalised state is offinite order less or equal to 2 and type less or equal to .From the definition of the stellar function, for any state | ψ 〉 ∈ H we may write | ψ 〉 = (cid:88) n ≥ ψ n | n 〉 = F (cid:63) ψ ( ˆ a † ) | 〉 . (2.3)From this equation one may understand the stellar function as an operational recipe for engi-neering a state from the vacuum, using the creation operator ˆ a † . This intuition will be mademore precise in the following sections. An important property is that the stellar representation isunique: Lemma 2.1.
Let | φ 〉 and | ψ 〉 be pure normalised single-mode states such that F (cid:63) φ = F (cid:63) ψ . Then | φ 〉 = | ψ 〉 . Moreover, let | χ 〉 = f ( ˆ a † ) | 〉 be a single-mode normalised pure state, where f is analytic.Then f = F (cid:63) χ . Proof.
With the notations of the Lemma, F (cid:63) φ ( z ) = (cid:80) n ≥ φ n z n (cid:112) n ! and F (cid:63) ψ ( z ) = (cid:80) n ≥ ψ n z n (cid:112) n ! . Thefunctions F (cid:63) φ and F (cid:63) ψ are analytic, so F (cid:63) φ ( z ) = F (cid:63) ψ ( z ) implies that φ n = ψ n for all n ≥
0. Hence | φ 〉 = | ψ 〉 .Now with | χ 〉 = (cid:80) n ≥ χ n | n 〉 = f ( ˆ a † ) | 〉 , let us write f ( z ) = (cid:80) n ≥ f n z n . We obtain | χ 〉 = (cid:88) n ≥ f n ( ˆ a † ) n | 〉= (cid:88) n ≥ f n (cid:112) n ! | n 〉 , (2.4)39 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES so χ n = f n (cid:112) n ! for all n ≥
0. On the other hand, for all z ∈ (cid:67) , F (cid:63) χ ( z ) = e | z | 〈 z ∗ | ψ 〉= (cid:88) n ≥ χ n z n (cid:112) n ! (2.5) = (cid:88) n ≥ f n z n = f ( z ). (cid:4) The stellar function of a state | ψ 〉 ∈ H is related to its Husimi Q function, a smoothed version ofthe Wigner function [CG69a], given by Q ψ ( z ) = π | 〈 z | ψ 〉 | = e −| z | π (cid:175)(cid:175)(cid:175) F (cid:63) ψ ( z ∗ ) (cid:175)(cid:175)(cid:175) , (2.6)for all z ∈ (cid:67) . The zeros of the Husimi Q function are the complex conjugates of the zeros of F (cid:63) ψ .Hence, by Theorem 2.1, a single-mode pure quantum state is non-Gaussian if and only if itsstellar function has zeros. These zeros form a discrete set, as the stellar function is a non-zeroanalytic function. The non-Gaussian properties of a single-mode pure state are then described bythe distribution of the zeros over the complex plane.Figure 2.1: Antistereographic projection of four points onto the sphereUsing anti-stereographic projection [SB07], this amounts to describing the non-Gaussian prop-erties of a pure state with a set of points on the sphere (Fig. 2.1), hence the name stellar40 .1. THE STELLAR FUNCTION representation, where the points on the sphere looked at from the center of the sphere are seenas stars on the celestial vault [TV95, KMW97].In all the chapter we will use for brevity the notations c χ = cosh χ , s χ = sinh χ and t χ = tanh χ , forall χ ∈ (cid:67) . In this section we give the stellar functions of various states and operators.
The displacement operator of amplitude α ∈ (cid:67) is given by ˆ D ( α ) = e α ˆ a † − α ∗ ˆ a . Its action on thevacuum state yields the coherent state | α 〉 . The squeeze operator of parameter ξ = re i θ ∈ (cid:67) isgiven by ˆ S ( ξ ) = e ( ξ ˆ a − ξ ∗ ˆ a † ) . Its action on the vacuum state yields the squeezed state | ξ 〉 . All single-mode Gaussian operations may be decomposed as a squeezing operation and a displacement (seesection 1.3).For any single-mode Gaussian state ˆ S ( ξ ) ˆ D ( α ) | 〉 , where ξ = re i θ , the corresponding stellarfunction is [Vou06]: G (cid:63) ξ , α ( z ) = (1 − | a | ) e − az + bz + c , (2.7)where a : = e − i θ tanh r , b : = α (cid:113) − | a | = α cosh r , c : = a ∗ α − | α | . (2.8)In particular, we obtain G (cid:63) α ( z ) = e α z − | α | , (2.9)for a coherent state of amplitude α ∈ (cid:67) , and G (cid:63) ξ ,0 ( z ) = (cid:112) cosh r e − ( e − i θ tanh r ) z , (2.10)for a squeezed vacuum state with squeezing parameter ξ = re i θ ∈ (cid:67) .For Fock states | n 〉 with n ∈ (cid:78) , the stellar function is simply given by F (cid:63) n ( z ) = z n (cid:112) n ! . (2.11) Let us define for α ∈ (cid:67) the cat + and cat − states: | cat ± α 〉 = (cid:113) N ± α ( | α 〉 ± |− α 〉 ), (2.12)where | α 〉 is a coherent state, and N ± α is a normalisation factor.41 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES
Lemma 2.2.
The stellar functions of cat states are given byF (cid:63) cat + α ( z ) = cosh( α z ) (cid:112) cosh( | α | ) , (2.13) and F (cid:63) cat − α ( z ) = sinh( α z ) (cid:112) sinh( | α | ) , (2.14) for all z , α ∈ (cid:67) . Proof.
The overlap between two coherent states is given by 〈 z | α 〉 = e − ( | z | +| α | − z ∗ α ) , (2.15)for z , α ∈ (cid:67) . Hence, with 〈 cat ± α | cat ± α 〉 = N ± α = ± e − | α | ). (2.16)We then obtain for z , α ∈ (cid:67) , F (cid:63) cat ± α ( z ) = e − | z | 〈 z ∗ | cat ± α 〉= (cid:113) N ± α (cid:179) e − | α | + α z ± e − | α | − α z (cid:180) = (cid:113) (cid:161) e | α | ± e −| α | (cid:162) (cid:161) e α z ± e − α z (cid:162) . (2.17)We finally obtain F (cid:63) cat + α ( z ) = cosh( α z ) (cid:112) cosh( | α | ) , (2.18)and F (cid:63) cat − α ( z ) = sinh( α z ) (cid:112) sinh( | α | ) . (2.19) (cid:4) The set of Gottesman-Kitaev-Preskill (GKP) states have been proposed as a means for encoding aqubit in an oscillator, in a way which is fault-tolerant to small shifts in position and momentum[GKP01]. An example of such states is the simultaneous + e − i (cid:112) π ˆ p and e i (cid:112) π ˆ q . The corresponding encoding may correct for com-parable shifts in ˆ q and ˆ p . An expression for this unphysical state (it has infinite norm) is given42 .1. THE STELLAR FUNCTION by | GKP 〉 = (cid:88) s ∈ (cid:90) e − i (cid:112) π s ˆ p (cid:88) t ∈ (cid:90) e i (cid:112) π t ˆ q | 〉= (cid:88) s , t ∈ (cid:90) ( − st ˆ D (cid:161) (cid:112) π ( s + it ) (cid:162) | 〉= (cid:88) s , t ∈ (cid:90) ( − st | (cid:112) π ( s + it ) 〉 , (2.20)as an infinite superposition of coherent states. The stellar function of this state is then given by F (cid:63) GKP ( z ) = (cid:88) s , t ∈ (cid:90) ( − st F (cid:63) (cid:112) π ( s + it ) ( z ) = (cid:88) s , t ∈ (cid:90) ( − st e − π ( s + t ) e (cid:112) π ( s + it ) z , (2.21)where we used Eq. (2.9) in the second line. This stellar function may be expressed as a Riemanntheta function [Rie57]. Using properties of these functions, we obtain the following result: Lemma 2.3. F (cid:63) GKP has an infinite number of zeros and has exactly zeros counted with multi-plicity in each square region of the complex plane of size (cid:112) π × (cid:112) π . !
Proof.
We first derive a few invariance properties of F (cid:63) GKP and conclude with the argumentprinciple (Theorem 1.1). For all z ∈ (cid:67) , we have F (cid:63) GKP ( iz ) = (cid:88) s , t ∈ (cid:90) ( − st e − π ( s + t ) e (cid:112) π ( − t + is ) z = t →− t (cid:88) s , t ∈ (cid:90) ( − st e − π ( s + t ) e (cid:112) π ( t + is ) z (2.22) = s ↔ t F (cid:63) GKP ( z ).43 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES
Now for all z ∈ (cid:67) , F (cid:63) GKP ( z ) = (cid:88) s , t ∈ (cid:90) ( − st e − π ( s + t ) e (cid:112) π ( s + it ) z = s → s − (cid:88) s , t ∈ (cid:90) ( − ( s + t e − π (( s + + t ) e (cid:112) π (( s + + it ) z = (cid:88) s , t ∈ (cid:90) ( − st e − π ( s + t ) e − π s − π e (cid:112) π ( s + it ) z e (cid:112) π z (2.23) = e (cid:112) π z − π (cid:88) s , t ∈ (cid:90) ( − st e − π ( s + t ) e (cid:112) π ( s + it )( z − (cid:112) π ) = e (cid:112) π z − π F (cid:63) GKP ( z − (cid:112) π )Combining this with Eq. (2.22) we also obtain for all z ∈ (cid:67) , F (cid:63) GKP ( z ) = F (cid:63) GKP ( iz ) = e (cid:112) π iz − π F (cid:63) GKP ( iz − (cid:112) π ) (2.24) = e (cid:112) π iz − π F (cid:63) GKP ( z + i (cid:112) π ).This means that F (cid:63) GKP is quasiperiodic along the horizontal and vertical directions in thecomplex plane, with period 4 (cid:112) π . The functions z (cid:55)→ e (cid:112) π z − π and z (cid:55)→ e (cid:112) π iz − π do notvanish and it is thus sufficient for our purpose to prove that F (cid:63) GKP has at least one zero: thequasiperiodicity implies that F (cid:63) GKP would also vanish on the lattice with square cells of size4 (cid:112) π containing this zero.By Theorem 1.1, the number of zeros of F (cid:63) GKP inside a closed contour C counted withmultiplicity is given by Z C (cid:161) F (cid:63) GKP (cid:162) = i π (cid:73) C ∂ z F (cid:63) GKP ( z ) F (cid:63) GKP ( z ) dz . (2.25)For all ω ∈ (cid:67) , we consider the square contour C ω with corners ω , ω + (cid:112) π , ω + (cid:112) π + i (cid:112) π and ω + i (cid:112) π , parametrised by a ( x ) = ω + (cid:112) π xb ( x ) = ω + (cid:112) π + i (cid:112) π xc ( x ) = ω + (cid:112) π (1 − x ) + i (cid:112) π d ( x ) = ω + i (cid:112) π (1 − x ), (2.26)for x ∈ [0, 1] (Fig. 2.2). We have a (cid:48) ( x ) = (cid:112) π , b (cid:48) ( x ) = i (cid:112) π , c (cid:48) ( x ) = − (cid:112) π and d (cid:48) ( x ) = − i (cid:112) π for all x ∈ [0, 1]. 44 .1. THE STELLAR FUNCTION The quasiperiodicity of F (cid:63) GKP may be rewritten as F (cid:63) GKP ( z + (cid:112) π ) = e π + (cid:112) π z F (cid:63) GKP ( z ), F (cid:63) GKP ( z + i (cid:112) π ) = e π − i (cid:112) π z F (cid:63) GKP ( z ), (2.27)for all z ∈ (cid:67) . Taking the derivative with respect to z we obtain ∂ z F (cid:63) GKP ( z + (cid:112) π ) = e π + (cid:112) π z (cid:163) ∂ z F (cid:63) GKP ( z ) + (cid:112) π F (cid:63) GKP ( z ) (cid:164) , ∂ z F (cid:63) GKP ( z + i (cid:112) π ) = e π − i (cid:112) π z (cid:163) ∂ z F (cid:63) GKP ( z ) − i (cid:112) π F (cid:63) GKP ( z ) (cid:164) , (2.28)and thus ∂ z F (cid:63) GKP ( z + (cid:112) π ) F (cid:63) GKP ( z + (cid:112) π ) = (cid:112) π + ∂ z F (cid:63) GKP ( z ) F (cid:63) GKP ( z ) , ∂ z F (cid:63) GKP ( z + i (cid:112) π ) F (cid:63) GKP ( z + i (cid:112) π ) = − i (cid:112) π + ∂ z F (cid:63) GKP ( z ) F (cid:63) GKP ( z ) . (2.29)With Eq. (2.25), for all ω ∈ (cid:67) , Z C ω (cid:161) F (cid:63) GKP (cid:162) = i π (cid:73) C ω ∂ z F (cid:63) GKP ( z ) F (cid:63) GKP ( z ) dz = i π (cid:90) (cid:34) ∂ z F (cid:63) GKP ( a ( x )) F (cid:63) GKP ( a ( x )) a (cid:48) ( x ) + ∂ z F (cid:63) GKP ( b ( x )) F (cid:63) GKP ( b ( x )) b (cid:48) ( x ) (2.30) + ∂ z F (cid:63) GKP ( c ( x )) F (cid:63) GKP ( c ( x )) c (cid:48) ( x ) + ∂ z F (cid:63) GKP ( d ( x )) F (cid:63) GKP ( d ( x )) d (cid:48) ( x ) (cid:35) dx .Given that c ( x ) = a (1 − x ) + i (cid:112) π , we have (cid:90) ∂ z F (cid:63) GKP ( c ( x )) F (cid:63) GKP ( c ( x )) c (cid:48) ( x ) dx = (cid:90) − (cid:112) π ∂ z F (cid:63) GKP ( a (1 − x ) + i (cid:112) π ) F (cid:63) GKP ( a (1 − x ) + i (cid:112) π ) dx = (cid:90) − (cid:112) π (cid:195) − i (cid:112) π + ∂ z F (cid:63) GKP ( a (1 − x )) F (cid:63) GKP ( a (1 − x )) (cid:33) = i π − (cid:90) ∂ z F (cid:63) GKP ( a (1 − x )) F (cid:63) GKP ( a (1 − x )) a (cid:48) (1 − x ) dx = i π − (cid:90) ∂ z F (cid:63) GKP ( a ( x )) F (cid:63) GKP ( a ( x )) a (cid:48) ( x ) dx , (2.31)where we used Eq. (2.29) in the second line with z = a (1 − x ). Hence,12 i π (cid:90) (cid:34) ∂ z F (cid:63) GKP ( a ( x )) F (cid:63) GKP ( a ( x )) a (cid:48) ( x ) + ∂ z F (cid:63) GKP ( c ( x )) F (cid:63) GKP ( c ( x )) c (cid:48) ( x ) (cid:35) dx =
8. (2.32)Similarly b ( x ) = d (1 − x ) + (cid:112) π gives12 i π (cid:90) (cid:34) ∂ z F (cid:63) GKP ( b ( x )) F (cid:63) GKP ( b ( x )) b (cid:48) ( x ) + ∂ z F (cid:63) GKP ( d ( x )) F (cid:63) GKP ( d ( x )) d (cid:48) ( x ) (cid:35) dx =
8. (2.33)45
HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES
With Eq. (2.30) we finally obtain Z C ω (cid:161) F (cid:63) GKP (cid:162) =
16. (2.34)The function F (cid:63) GKP thus has an infinite number of zeros. Since Eq. (2.34) is independent ofthe choice of ω ∈ (cid:67) , F (cid:63) GKP has exactly 16 zeros (counted with multiplicity) in each squareregion of the complex plane of size 4 (cid:112) π × (cid:112) π . Moreover, with the property of invarianceunder rotation in Eq. (2.22), it has exactly 4 zeros in each square region of size 2 (cid:112) π × (cid:112) π whose corners have coordinates in 2 (cid:112) π (cid:90) (by considering the square region of size 4 (cid:112) π × (cid:112) π centered on the origin). (cid:4) While operators have their own treatment in the Segal–Bargmann formalism [Vou06], it issufficient for our purpose to consider the following correspondences: the creation and annihilationoperators have the stellar representationsˆ a † → z , ˆ a → ∂ z , (2.35)i.e., the operator corresponding to ˆ a † in the stellar representation is the multiplication by z andthe operator in the stellar representation corresponding to ˆ a is the derivative with respect to z . This implies that the stellar function of a photon-added state ˆ a † | ψ 〉 is given by z (cid:55)→ zF (cid:63) ψ ( z ),while the stellar function of a photon-subtracted state ˆ a | ψ 〉 is given by z (cid:55)→ ∂ z F (cid:63) ψ ( z ). In particular,photon-added states are always non-Gaussian, since 0 is a root of their stellar function, whilephoton-subtracted states can be Gaussian (e.g., the Fock state | 〉 , for which ˆ a | 〉 = | 〉 , or thecoherent states | α 〉 , for α ∈ (cid:67) , for which ˆ a | α 〉 = α | α 〉 ).Any operator written as a power series in ˆ a † and ˆ a thus has a stellar representation obtainedby taking the same power series in the operator multiplication by z and the operator derivativewith respect to z , which corresponds to its effect on the stellar function of a state it is acting on.For example, the photon number operator ˆ n = ˆ a † ˆ a acts on the stellar function as F (cid:63) ψ ( z ) (cid:55)→ z ∂ z F (cid:63) ψ ( z ). (2.36)For various operators however, the corresponding stellar representation may be expressed moreconcisely than with a power series. We give a few examples in what follows.The displacement and squeeze operators satisfy the following commutation rules (see section 1.3)ˆ D ( α ) ˆ a † ˆ D † ( α ) = ˆ a † − α ∗ ˆ S ( ξ ) ˆ a † ˆ S † ( ξ ) = c r ˆ a † + s r e i θ ˆ a , (2.37)46 .1. THE STELLAR FUNCTION where α , ξ = re i θ ∈ (cid:67) , with c r = cosh r and s r = sinh r . For all | ψ 〉 = (cid:80) n ≥ ψ n | n 〉 we thus haveˆ D ( α ) | ψ 〉 = ˆ D ( α ) F (cid:63) ψ ( ˆ a † ) | 〉= (cid:88) n ≥ ψ n (cid:112) n ! ˆ D ( α ) ( ˆ a † ) n | 〉= (cid:88) n ≥ ψ n (cid:112) n ! ( ˆ a † − α ∗ ) n ˆ D ( α ) | 〉= F (cid:63) ψ ( ˆ a † − α ∗ ) | α 〉= F (cid:63) ψ ( ˆ a † − α ∗ ) e α ˆ a † − | α | | 〉 , (2.38)where we used Eq. (2.3) in the first line, Eq. (2.1) in the second line, Eq. (2.37) in the third lineand Eq. (2.9) in the last line. Hence, with Lemma 2.1, the displacement operator ˆ D ( α ) acts on thestellar function as F (cid:63) ψ ( z ) (cid:55)→ e α z − | α | F (cid:63) ψ ( z − α ∗ ), (2.39)for all α ∈ (cid:67) . Similarly, for all | ψ 〉 we haveˆ S ( ξ ) | ψ 〉 = ˆ S ( ξ ) F (cid:63) ψ ( ˆ a † ) | 〉= F (cid:63) ψ ( c r ˆ a † + s r e i θ ˆ a ) ˆ S ( ξ ) | 〉= F (cid:63) ψ ( c r ˆ a † + s r e i θ ˆ a ) | ξ 〉= (cid:112) c r F (cid:63) ψ ( c r ˆ a † + s r e i θ ˆ a ) e − e − i θ t r ( ˆ a † ) | 〉 , (2.40)where ξ = re i θ with c r = cosh r , s r = sinh r and t r = tanh r , and where we used Eq. (2.3) in thefirst line, Eq. (2.37) in the second line and Eq. (2.10) in the last line. Hence, with Lemma 2.1, thesqueezing operator ˆ S ( ξ ) acts on the stellar function as F (cid:63) ψ ( z ) (cid:55)→ (cid:112) c r F (cid:63) ψ ( c r z + s r e i θ ∂ z ) e − e − i θ t r z , (2.41)for all ξ = re i θ ∈ (cid:67) .The POVM corresponding to a threshold detection is { | 〉〈 | , (cid:49) − | 〉〈 | } . The projector onto thevacuum acts as | ψ 〉 (cid:55)→ 〈 | ψ 〉 | 〉 , (2.42)so it maps the stellar function of a state | ψ 〉 as F (cid:63) ψ ( z ) (cid:55)→ 〈 | ψ 〉 F (cid:63) | 〉 ( z ). (2.43)We have 〈 | ψ 〉 = F (cid:63) ψ (0) and F (cid:63) | 〉 ( z ) =
1, so the projector | 〉〈 | acts on the stellar function as F (cid:63) ψ ( z ) (cid:55)→ F (cid:63) ψ (0), (2.44)while the projector (cid:49) − | 〉 〈 | acts as F (cid:63) ψ ( z ) (cid:55)→ F (cid:63) ψ ( z ) − F (cid:63) ψ (0). (2.45)47 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES
In particular, the click of a threshold detector projects the measured state onto a non-Gaussianstate for which 0 is a root of the stellar function. This is consistent with the fact that the measuredstate is orthogonal to the vacuum state—a coherent state of amplitude 0—after being projectedonto the support of (cid:49) − | 〉 〈 | .Finally, the parity operator ˆ Π = ( − ˆ a † ˆ a = e i π ˆ n maps the Fock state | n 〉 to ( − n | n 〉 , for all n ∈ (cid:78) .Hence, it acts on the stellar function as F (cid:63) ψ ( z ) (cid:55)→ F (cid:63) ψ ( − z ). (2.46)by Eq. (2.1). The Hilbert space H is naturally partitioned into sets of states whose stellar functions—orequivalently Husimi Q function—have the same number of zeros counted with multiplicity. Weintroduce the following related definition: Definition 2.2 (Stellar rank) . The stellar rank r (cid:63) ( ψ ) of a pure single-mode normalised quantumstate | ψ 〉 ∈ H is defined as the number of zeros of its stellar function F (cid:63) ψ , counted with multiplicity.By analogy with the Schmidt rank in entanglement theory [TH00], we define the stellar rank ofa mixed state ρ as r (cid:63) ( ρ ) : = inf p i , ψ i sup r (cid:63) ( ψ i ), (2.47)where the infimum is over the statistical ensembles { p i , ψ i } such that ρ = (cid:80) i p i | ψ i 〉〈 ψ i | . Inparticular, a mixed quantum state has nonzero rank if and only if it cannot be written as amixture of Gaussian states.We introduce hereafter the notation (cid:78) = (cid:78) ∪ { +∞ } , so that r (cid:63) ( ψ ) ∈ (cid:78) , and extend naturally theordering from (cid:78) to (cid:78) , with the convention N < +∞ ⇔ N ∈ (cid:78) . For N ∈ (cid:78) , we define R N : = { | ψ 〉 ∈ H , r (cid:63) ( ψ ) = N } (2.48)the set of states with stellar rank equal to N . The stellar hierarchy is the hierarchy of statesinduced by the stellar rank (Fig 2.3). The following properties are easily obtained:• By Lemma 2.1, if M (cid:54)= N then R M ∩ R N = ∅ , for all M , N ∈ (cid:78) , so all the ranks in the stellarhierarchy are disjoint.• We have H = (cid:83) N ∈ (cid:78) R N , i.e., the stellar hierarchy covers the whole space of normalisedstates, and the set of states of finite stellar rank is given by (cid:83) N ∈ (cid:78) R N .48 .2. THE STELLAR HIERARCHY ...
Figure 2.3: The stellar hierarchy of single-mode normalised quantum states. Each rank N contains states obtained from the vacuum with N single photon additions and Gaussian unitaryoperations (Theorem 2.2). The states of finite rank are robust, while the states of infinite rankare not (section 2.3).• By Theorem 2.1, the rank zero of the stellar hierarchy R is the set of single-mode nor-malised pure Gaussian states, and non-Gaussian states populate all higher ranks.• For all N ∈ (cid:78) ,the Fock state | N 〉 is of stellar rank N , by Eq. (2.11), while cat states are ofinfinite stellar rank, by Lemma 2.2, so all ranks are non empty.In the following, we investigate further properties of the stellar hierarchy. We prove a first generaldecomposition result for pure states of finite stellar rank: Theorem 2.2.
Let | ψ 〉 ∈ (cid:83) N ∈ (cid:78) R N be a pure state of finite stellar rank. Let { α , . . . , α r (cid:63) ( ψ ) } be theroots of the Husimi Q function of | ψ 〉 , counted with multiplicity. Then, | ψ 〉 = N (cid:34) r (cid:63) ( ψ ) (cid:89) n = ˆ D ( α n ) ˆ a † ˆ D † ( α n ) (cid:35) | G ψ 〉 , (2.49) where ˆ D ( α ) is a displacement operator, | G ψ 〉 is a Gaussian state, and N is a normalisationconstant. Moreover, this decomposition is unique up to reordering of the roots. HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES
Proof.
We consider a state | ψ 〉 of finite stellar rank r (cid:63) ( ψ ) ∈ (cid:78) . Its stellar function is ananalytic function over the complex plane of order less or equal to 2, so by Hadamard-Weierstrass factorization theorem (Theorem 1.2), F (cid:63) ψ ( z ) = z k (cid:34) r (cid:63) ( ψ ) − k (cid:89) n = (cid:181) − zz ∗ n (cid:182) e zz ∗ n + (cid:179) zz ∗ n (cid:180) (cid:35) e g + g z + g z , (2.50)for all z ∈ (cid:67) , where k ∈ (cid:78) is the multiplicity of 0 as a root of F (cid:63) ψ , where the { z n } are the non-zero roots of Q ψ counted with multiplicity (i.e., the { z ∗ n } are the non-zero roots of F (cid:63) ψ countedwith multiplicity), and where g , g , g ∈ (cid:67) . Let us introduce for brevity m = r (cid:63) ( ψ ) − k ∈ (cid:78) .Because the product in the above equation is finite, we need not worry about convergence ofindividual factors, and we may reorder the expression at will. We obtain F (cid:63) ψ ( z ) = z k m (cid:89) n = (cid:181) − zz ∗ n (cid:182) · m (cid:89) n = e zz ∗ n + (cid:179) zz ∗ n (cid:180) · e g + g z + g z = z k m (cid:89) n = (cid:181) − zz ∗ n (cid:182) · e g + (cid:179) g + (cid:80) mn = z ∗ n (cid:180) z + (cid:181) g + (cid:80) mn = z ∗ n )2 (cid:182) z = ( − m (cid:81) mn = z ∗ n (cid:34) z k m (cid:89) n = (cid:161) z − z ∗ n (cid:162)(cid:35) · e g + (cid:179) g + (cid:80) mn = z ∗ n (cid:180) z + (cid:181) g + (cid:80) mn = z ∗ n )2 (cid:182) z . (2.51)With Eqs. (2.3) and (2.37), we obtain, for all α ∈ (cid:67) , | ψ 〉 = F (cid:63) ψ ( ˆ a † ) | 〉= ( − m (cid:81) mn = z ∗ n (cid:34) ( ˆ a † ) k m (cid:89) n = (cid:179) ˆ a † − z ∗ n (cid:180)(cid:35) · e g + (cid:179) g + (cid:80) mn = z ∗ n (cid:180) ˆ a † + (cid:181) g + (cid:80) Mn = z ∗ n )2 (cid:182) ( ˆ a † ) | 〉 (2.52) = ( − m (cid:81) mn = z ∗ n (cid:34) ( ˆ a † ) k m (cid:89) n = ˆ D ( z n ) ˆ a † ˆ D † ( z n ) (cid:35) · e g + (cid:179) g + (cid:80) mn = z ∗ n (cid:180) ˆ a † + (cid:181) g + (cid:80) mn = z ∗ n )2 (cid:182) ( ˆ a † ) | 〉 .Gouping the non-zero roots { z n } and the k zero roots into the set of zeros counted withmultiplicity { α n } , we obtain | ψ 〉 = ( − m (cid:81) mn = z ∗ n (cid:34) r (cid:63) ( ψ ) (cid:89) n = ˆ D ( α n ) ˆ a † ˆ D † ( α n ) (cid:35) · e g + (cid:179) g + (cid:80) mn = z ∗ n (cid:180) ˆ a † + (cid:181) g + (cid:80) mn = z ∗ n )2 (cid:182) ( ˆ a † ) | 〉 . (2.53)The state e g + (cid:179) g + (cid:80) mn = z ∗ n (cid:180) ˆ a † + (cid:181) g + (cid:80) mn = z ∗ n )2 (cid:182) ( ˆ a † ) | 〉 (2.54)is a (non normalised) Gaussian state, by Eq. (2.7) and Lemma 2.1. We finally obtain | ψ 〉 = N (cid:34) r (cid:63) ( ψ ) (cid:89) n = ˆ D ( α n ) ˆ a † ˆ D † ( α n ) (cid:35) | G ψ 〉 , (2.55)50 .2. THE STELLAR HIERARCHY where N is a normalisation constant, and | G ψ 〉 is a Gaussian state. The decomposition isunique by Lemma 2.1 (up to a reordering of the roots). (cid:4) This decomposition implies that any state of finite stellar rank may be obtained from a Gaussianstate by successive applications of the creation operator at different locations in phase space, givenby the zeros of the Husimi Q function. Experimentally, this corresponds to the probabilistic non-Gaussian operation of single-photon addition [ZVB04, MA10, WSPT18]. Using this decomposition,we obtain the following property for the stellar rank: Theorem 2.3.
A unitary operation is Gaussian if and only if it leaves the stellar rank invariant.
Proof.
If a unitary operation leaves the stellar rank invariant, it maps in particular allpure states of stellar rank zero to pure states of stellar rank zero, i.e., all Gaussian states toGaussian states, so it is a Gaussian operation.Reciprocally, let us show that Gaussian unitary operations leave the stellar rank invariant.We first consider finite stellar rank pure states. Let | ψ 〉 be such a state. By Theorem 2, | ψ 〉 = P ψ ( ˆ a † ) | G ψ 〉 , (2.56)where P ψ is a polynomial of degree r (cid:63) ( ψ ) and | G ψ 〉 is a Gaussian state. By Eq. (2.37) and bylinearity we have | ψ α 〉 : = ˆ D ( α ) | ψ 〉 = ˆ P ψ ( ˆ a † − α ∗ ) ˆ D ( α ) | G ψ 〉 , (2.57)and | ψ ξ 〉 : = ˆ S ( ξ ) | ψ 〉 = P ψ ( c r ˆ a † + s r e i θ ˆ a ) ˆ S ( ξ ) | G ψ 〉 , (2.58)where ξ = re i θ . By Eq. (2.39), during a displacement of α , the stellar function of | ψ 〉 ismodified as F (cid:63) ψ ( z ) → F (cid:63) ψ , α ( z ) = e z α − | α | F (cid:63) ψ ( z − α ∗ ) = P ψ ( z − α ∗ ) G (cid:63) α ( z ), (2.59)where G (cid:63) α ( z ) is the Gaussian stellar function corresponding to the Gaussian state ˆ D ( α ) | G ψ 〉 .Moreover, by Eq. (2.41), during a squeezing of ξ , the stellar function of | ψ 〉 is modified as F (cid:63) ψ ( z ) → F (cid:63) ψ , ξ ( z ) = P ψ (cid:179) c r z + s r e i θ ∂ z (cid:180) G (cid:63) ξ ( z ) = Q ψ , r ( z ) G (cid:63) ξ ( z ), (2.60)where G (cid:63) ξ ( z ) is the Gaussian stellar function corresponding to the Gaussian state ˆ S ( ξ ) | G ψ 〉 ,and where Q ψ , ξ ( z ) = G (cid:63) − ξ ( z ) (cid:104) P ψ (cid:161) c r z + s r e i θ ∂ z (cid:162) G (cid:63) ξ ( z ) (cid:105) is a polynomial. Let us compute theleading coefficient of Q ψ , ξ . Writing p the leading coefficient of P ψ , and N = r (cid:63) ( ψ ) its degree51 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES for brevity, the leading coefficient of Q ψ , ξ is given by the leading coefficient of G (cid:63) − ξ ( z ) (cid:183) p (cid:179) c r z + s r e i θ ∂ z (cid:180) N G (cid:63) ξ ( z ) (cid:184) . (2.61)Let us write G (cid:63) ξ ( z ) = e − az + bz + c , as in Eq. (2.7). The leading coefficient of Q ψ , ξ may then beobtained as the leading coefficient of e az (cid:183) p (cid:179) c r z + s r e i θ ∂ z (cid:180) N e − az (cid:184) . (2.62)For all x , λ , we have [Wys17]( x + λ∂ x ) N = (cid:165) N (cid:166) (cid:88) n = N ! λ n ( N − n )! n !2 n N − n (cid:88) k = (cid:195) N − nk (cid:33) x k ∂ N − n − kx , (2.63)so the leading coefficient of Q ψ , ξ is equal to the leading coefficient of: pc Nr (cid:165) N (cid:166) (cid:88) n = z N − n (1 − a ) N − n N ! t nr e in θ ( N − n )! n !2 n , (2.64)where t r = tanh r . Finally, taking the leading coefficient in z of this expression, correspondingto n =
0, gives pc Nr (1 − a ) N . (2.65)It is non-zero unless a =
1, which corresponds to an infinite value for the modulus r of thesqueezing parameter ξ = re i θ by Eq. (2.7). Hence the polynomials P ψ and Q ψ , ξ have the samedegree. This shows that a finite number of zeros is not modified by Gaussian operations.Gaussian operations also map states with infinite number of zeros to states with infinitenumber of zeros. Indeed, assuming there exist a state | φ 〉 with an infinite number of zeroswhich is mapped by a Gaussian operation ˆ G to a state | ψ 〉 with a finite number of zeros, thenˆ G † would map | ψ 〉 to | φ 〉 , thus changing the (finite) number of zeros of F (cid:63) ψ , which would be incontradiction with the previous proof. Hence Gaussian unitary operations leave the stellarrank of pure states invariant.Now by Eq. (2.47), the stellar rank of a mixed state ρ is given by r (cid:63) ( ρ ) = inf p i , ψ i sup r (cid:63) ( ψ i ), (2.66)where the infimum is over the statistical ensembles such that ρ = (cid:80) i p i | ψ i 〉〈 ψ i | . For ˆ G aunitary Gaussian operation, r (cid:63) ( ˆ G ρ ˆ G † ) = inf p i , ψ i sup r (cid:63) ( ˆ G ψ i ) = inf p i , ψ i sup r (cid:63) ( ψ i ) (2.67)52 .2. THE STELLAR HIERARCHY = r (cid:63) ( ρ ),where we used in the second line the fact that Gaussian unitary operations leave the stellarrank of pure states invariant. Hence, Gaussian unitary operations leave the stellar rankinvariant. (cid:4) An interesting consequence is that the number of single-photon additions in the decomposition ofTheorem 2.2 is minimal. Indeed, if a quantum state is obtained from the vacuum by successiveapplications of Gaussian operations and single-photon additions, then its stellar rank is exactlythe number of photon additions, because each single-photon addition increases by one its stellarrank—it adds a zero to the stellar function at zero—while each Gaussian operation leaves thestellar rank invariant by Theorem 2.3. Hence, the stellar rank is a measure of the non-Gaussianproperties of a quantum state which may be interpreted as a minimal non-Gaussian operationalcost, in terms of single-photon additions, for engineering the state from the vacuum.
Now that the first properties of the stellar hierarchy are laid out, we consider the convertibilityof quantum states using Gaussian unitary operations:
Definition 2.3 (Gaussian convertibility) . Two states | φ 〉 and | ψ 〉 are Gaussian-convertible ifthere exists a Gaussian unitary operation ˆ G such that | ψ 〉 = ˆ G | φ 〉 .Note that this notion is different from the notion of Gaussian conversion introduced in [YBT + H . By Theorem 2.3, having thesame stellar rank is a necessary condition for Gaussian convertibility. However, this condition isnot sufficient. In order to derive the equivalence classes for Gaussian convertibility, we introducethe following definition: Definition 2.4 (Core state) . Core states are defined as the single-mode normalised pure quantumstates which have a polynomial stellar function.By Eq. (2.3) and Lemma 2.1, core states are the states with a bounded support over the Fockbasis, i.e., finite superpositions of Fock states. These correspond to the minimal non-Gaussiancore states introduced in [MF09], in the context of non-Gaussian state engineering.With this definition, we obtain our following result.53
HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES
Theorem 2.4.
Let | ψ 〉 ∈ (cid:83) N ∈ (cid:78) R N be a state of finite stellar rank. Then, there exists a unique corestate | C ψ 〉 such that | ψ 〉 and | C ψ 〉 are Gaussian-convertible.By Theorem 2.2, | ψ 〉 = P ψ ( ˆ a † ) | G ψ 〉 , where P ψ is a polynomial of degree r (cid:63) ( ψ ) and | G ψ 〉 = ˆ S ( ξ ) ˆ D ( α ) | 〉 is a Gaussian state, where ˆ D ( α ) = e α ˆ a † − α ∗ ˆ a is a displacement operator, and ˆ S ( ξ ) = e ( ξ ˆ a − ξ ∗ ˆ a † ) is a squeezing operator, with ξ = re i θ . Then, | ψ 〉 = ˆ S ( ξ ) ˆ D ( α ) | C ψ 〉 = ˆ S ( ξ ) ˆ D ( α ) F (cid:63) C ψ ( ˆ a † ) | 〉 , (2.68) where the (polynomial) stellar function of | C ψ 〉 is given byF (cid:63) C ψ ( z ) = P ψ (cid:179) c r z − s r e i θ ∂ z + c r α ∗ − s r e i θ α (cid:180) ·
1, (2.69) for all z ∈ (cid:67) . Proof.
Let | ψ 〉 ∈ (cid:83) N ∈ (cid:78) R N be a state of finite stellar rank. By Theorem 2.2, | ψ 〉 = P ψ ( ˆ a † ) | G ψ 〉 , (2.70)where P ψ is a polynomial of degree r (cid:63) ( ψ ) and | G ψ 〉 = ˆ S ( ξ ) ˆ D ( α ) | 〉 is a Gaussian state,with ξ = re i θ . Let us define | C ψ 〉 = ˆ D † ( α ) ˆ S † ( ξ ) | ψ 〉 . The states | ψ 〉 and | C ψ 〉 are Gaussian-convertible. Moreover, from the commutation relations in Eq. (2.37) and by linearity weobtain | C ψ 〉 = ˆ D † ( α ) ˆ S † ( ξ ) P ψ ( ˆ a † ) | G ψ 〉= ˆ D † ( α ) P ψ (cid:179) c r ˆ a † − s r e i θ ˆ a (cid:180) ˆ S † ( ξ ) | G ψ 〉= P ψ (cid:104) c r ( ˆ a † + α ∗ ) − s r e i θ ( ˆ a + α ) (cid:105) | 〉= P ψ (cid:179) c r ˆ a † − s r e i θ ˆ a + c r α ∗ − s r e i θ α (cid:180) | 〉 , (2.71)where we used Eq. (2.70) in the first line. By Eq. (2.35), the stellar operator correspondingto ˆ a † is the multiplication by z and the stellar operator corresponding to ˆ a is the derivativewith respect to z . Hence, F (cid:63) C ψ ( z ) = P ψ (cid:179) c r z − s r e i θ ∂ z + c r α ∗ − s r e i θ α (cid:180) ·
1, (2.72)for all z ∈ (cid:67) , which is a polynomial function, so the state | C ψ 〉 is a core state.In order to conclude the proof, we need to show that | C ψ 〉 is the unique core state Gaussian-convertible to | ψ 〉 . Let | C 〉 = P C ( ˆ a † ) | 〉 be a core state Gaussian-convertible to | ψ 〉 . The states | C ψ 〉 and | C 〉 are Gaussian-convertible so there exist ξ , α ∈ (cid:67) such that | C ψ 〉 = ˆ S ( ξ ) ˆ D ( α ) | C 〉= ˆ S ( ξ ) ˆ D ( α ) P C ( ˆ a † ) | 〉= P C ( c r ˆ a † + s r e i θ ˆ a − α ∗ ) ˆ S ( ξ ) ˆ D ( α ) | 〉 , (2.73)54 .2. THE STELLAR HIERARCHY where we used Eq. (2.37). Hence, F (cid:63) C ψ ( z ) = P C ( c r z + s r e i θ ∂ z − α ∗ ) G (cid:63) ξ , α ( z ). (2.74)With Eq. (2.7), this function may be expressed as a polynomial multiplied by a Gaussianfunction G (cid:63) ξ , α . On the other hand F (cid:63) C ψ is a polynomial, since | C ψ 〉 is a core state. By compari-son of the speed of convergence, this implies that the Gaussian function G (cid:63) ξ , α is constant, i.e.,that e − i θ tanh r = α (cid:112) − tanh r =
0, (2.75)by Eq. (2.7). This in turn implies ξ = α =
0, and | C 〉 = ˆ S ( ξ ) ˆ D ( α ) | C 〉 = | C ψ 〉 . (cid:4) This result has several consequences:• It implies a second general decomposition result, in addition to Theorem 2.2: by Eq. (2.68),any state of finite stellar rank can be uniquely decomposed as a finite superposition ofequally displaced and equally squeezed number states. This shows that the stellar hierarchymatches the genuine n -photon hierarchy introduced in [LSH + genuine n-photon quantum non-Gaussianity if and only if it has a stellar rank greater orequal to n . Formally, for all N ∈ (cid:78) , the set R N of states of stellar rank equal to N is obtainedby the free action of the group of single-mode Gaussian unitary operations on the set of corestates of stellar rank N , which is isomorphic to the set of normalised complex polynomialsof degree N .• It shows that two different core states are never Gaussian-convertible, while any state offinite stellar rank is always Gaussian-convertible to a unique core state. This implies thatequivalence classes for Gaussian convertibility for states of finite stellar rank correspond tothe orbits of core states under Gaussian operations.• It gives an analytic way to check if two states of finite stellar rank are Gaussian-convertible,given their stellar functions, by checking with Eq. (2.69) if they share the same core state.• It shows that photon-subtracting a state of finite stellar rank, which amounts to derivatingits stellar function, can either decrease its stellar rank by 1, leave it invariant, or increaseit by 1, depending on whether the Gaussian operation which converts the state to itscore state is either the identity, a displacement, or a Gaussian operation with nonzerosqueezing parameter. In particular, this implies that the stellar rank is a lower bound onthe number of photon subtractions necessary to enginer a state from the vacuum, togetherwith Gaussian unitary operations. 55 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES
We consider the following simple example to illustrate the use of Theorem 2.4 for determiningGaussian convertibility: a photon-subtracted squeezed state, a photon-added squeezed state anda single-photon Fock state. We write | φ 〉 = − s ξ ˆ a | ξ 〉 a normalised photon-subtracted squeezedvacuum state and | ψ 〉 = c ξ ˆ a † | ξ 〉 a normalised photon-added squeezed vacuum state, with ξ ∈ (cid:82) ∗ .We write also | χ 〉 = | 〉 a single-photon Fock state. Using Eq. (2.7) and Eq. (2.35), we obtain for all z ∈ (cid:67) F (cid:63) φ ( z ) = − s ξ ∂ z (cid:104) e − t ξ z (cid:105) = zc ξ e − t ξ z , (2.76)and F (cid:63) ψ ( z ) = zc ξ e − t ξ z , (2.77)where c ξ = cosh ξ and t ξ = tanh ξ . Hence F (cid:63) φ = F (cid:63) ψ , so the states | φ 〉 and | ψ 〉 are actually equal.We also have F (cid:63) χ ( z ) = z . With the notations of Theorem 2.4, we have r φ = ξ , r χ = θ φ = θ χ = α φ = α χ =
0, ˆ G φ = ˆ S ( ξ ), ˆ G χ = (cid:49) , P φ ( z ) = zc ξ , and P χ ( z ) = z , so for all z ∈ (cid:67) , P φ (cid:179) c r φ z − s r φ e i θ φ ∂ z + c r φ α ∗ φ − s r φ e i θ φ α φ (cid:180) · = c ξ (cid:161) c ξ z − s ξ ∂ z (cid:162) · = z , (2.78)and P χ (cid:179) c r χ z − s r χ e i θ χ ∂ z + c r χ α ∗ χ − s r χ e i θ χ α χ (cid:180) · = z · = z , (2.79)thus | φ 〉 and | χ 〉 share the same core state. By Theorem 2.4, this means that | φ 〉 and | χ 〉 areGaussian-convertible, and we have | φ 〉 = ˆ G φ ˆ G † χ | χ 〉 , whereˆ G φ ˆ G † χ = ˆ S ( ξ ). (2.80)Using Eq. (2.37) confirms indeed that − s ξ ˆ a | ξ 〉 = c ξ ˆ a † | ξ 〉 = ˆ S ( ξ ) | 〉 . (2.81) The stellar hierarchy provides a ranking of non-Gaussian states, in terms of the minimal numberof photons additions necessary to engineer them. However, for this hierarchy to be relevant inrealistic experimental scenarios, it has to be robust to small deviations. We consider this formallyin what follows and analyse the robustness properties of the stellar hierarchy.56 .3. ROBUSTNESS OF NON-GAUSSIAN STATES
We introduce the following definition:
Definition 2.5 (Stellar robustness) . Let | ψ 〉 ∈ H . The stellar robustness of the state | ψ 〉 is definedas R (cid:63) ( ψ ) : = inf r (cid:63) ( φ ) < r (cid:63) ( ψ ) D ( φ , ψ ), (2.82)where D denotes the trace distance and where the infimum is over all states | φ 〉 ∈ H such that r (cid:63) ( φ ) < r (cid:63) ( ψ ).The stellar robustness quantifies how much one has to deviate from a quantum state in tracedistance to find another quantum state of lower stellar rank: states with a positive stellarrobustness will be referred to as robust . The stellar robustness inherits the property of invarianceunder Gaussian operations of the stellar rank, because the trace distance between two states isinvariant under unitary operations. Because of its operational properties, the choice of the tracedistance is especially relevant in the context of non-Gaussian state engineering and quantumcomputing with non-Gaussian states.A similar notion, though more restricted, is the quantum non-Gaussian depth [SLH + Definition 2.6 ( k -robustness) . Let | ψ 〉 ∈ H . For all k ∈ (cid:78) ∗ , the k-robustness of the state | ψ 〉 isdefined as R (cid:63) k ( ψ ) : = inf r (cid:63) ( φ ) < k D ( φ , ψ ), (2.83)where D denotes the trace distance and where the infimum is over all states | φ 〉 ∈ H such that r (cid:63) ( φ ) < k .For all k ∈ (cid:78) ∗ , the k -robustness quantifies how much one has to deviate from a quantum state intrace distance to find another quantum state which as a stellar rank between 0 and k −
1. Stateswith a positive R (cid:63) k will be referred to as robust with respect to states of stellar rank lower thank . When k = +∞ , the ∞ -robustness quantifies how much one has to deviate from a quantumstate in trace distance to find another quantum state of finite stellar rank. Note that the stellarrobustness satisfies R (cid:63) ( ψ ) = R (cid:63) r (cid:63) ( ψ ) ( ψ ). We introduce the related definition: Definition 2.7 (Robustness profile) . Let | ψ 〉 ∈ H . The robustness profile of the state | ψ 〉 is definedas R ( ψ ) : = (cid:161) R (cid:63) k ( ψ ) (cid:162) k ∈ (cid:78) ∗ . (2.84)The robustness profile is the sequence of k -robustnesses for all k ∈ (cid:78) ∗ . This profile describes howhard a non-Gaussian state is to produce experimentally, using photon additions.57 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES
A dual notion to the robustness is the following:
Definition 2.8 (Smoothed non-Gaussianity of formation) . Let ρ be a single-mode normalisedstate, and let (cid:178) >
0. The (cid:178) -smoothed non-Gaussianity of formation
N G F (cid:178) ( ρ ) is defined as theminimal stellar rank of the states σ that are (cid:178) -close to ρ in trace distance. Formally, N G F (cid:178) ( ρ ) : = inf σ (cid:169) r (cid:63) ( σ ), s.t. D ( ρ , σ ) ≤ (cid:178) (cid:170) , (2.85)where D denotes the trace distance.The infimum in the definition is also a minimum, since the set considered only contains integervalues and is lower bounded by zero. That minimum is not necessarily attained for the energycut-off state (consider, e.g., a Gaussian pure state).The smoothed non-Gaussianity of formation can be obtained directly from the robustnessprofile and gives a smoothed version of the stellar rank, dual to the robustness. By Theorem 2.2,it quantifies the minimal number of single-photon additions that need to be applied to a Gaussianstate in order to obtain a state (cid:178) -close to a target state, and provides an operational cost measurefor non-Gaussian resource states, which is also invariant under Gaussian operations.The robustness is related to the fidelity by the following result: Lemma 2.4.
Let | ψ 〉 ∈ H . For all k ∈ (cid:78) ∗ , sup r (cid:63) ( ρ ) < k F ( ρ , ψ ) = − [ R (cid:63) k ( ψ )] , (2.86) where F is the fidelity. Proof.
For any pure state | ψ 〉 ∈ H , and any set of pure states X , we havesup ρ = (cid:80) p i | φ 〉 i 〈 φ i | (cid:80) p i = φ i ∈ X F ( ρ , ψ ) = sup ρ = (cid:80) p i | φ 〉 i 〈 φ i | (cid:80) p i = φ i ∈ X 〈 ψ | ρ | ψ 〉= sup (cid:80) p i = sup φ i ∈ X (cid:88) p i | 〈 φ i | ψ 〉 | = sup φ ∈ X | 〈 φ | ψ 〉 | = sup φ ∈ X F ( φ , ψ ). (2.87)Hence, for X the set of pure states of stellar rank less than k , R (cid:63) k ( ψ ) = inf r (cid:63) ( φ ) < k D ( φ , ψ ) = inf r (cid:63) ( φ ) < k (cid:113) − | 〈 φ | ψ 〉 | (2.88) = (cid:114) − sup r (cid:63) ( φ ) < k F ( φ , ψ )58 .3. ROBUSTNESS OF NON-GAUSSIAN STATES = (cid:114) − sup r (cid:63) ( ρ ) < k F ( ρ , ψ ) ,where D denotes the trace distance, where we used the definition of the stellar rank formixed states (2.47). We finally obtainsup r (cid:63) ( ρ ) < k F ( ρ , ψ ) = − [ R (cid:63) k ( ψ )] . (2.89) (cid:4) Certifying that a (mixed) state ρ has a fidelity greater than 1 − [ R (cid:63) k ( ψ )] with a given target purestate | ψ 〉 thus ensures that the state ρ has stellar rank greater or equal to k . However, this isonly possible if the two following conditions are met:• The target state | ψ 〉 is robust with respect to states of stellar rank less than k , i.e., R (cid:63) k ( ψ ) > k -robustness R (cid:63) k ( ψ ) is known.We consider these two problems in what follows. First, we determine for all k ∈ (cid:78) ∗ which statesare robust with respect to states of stellar rank less than k . Then, we show how to compute their k -robustness. Determining which states are robust amounts to characterizing the topology of the stellarhierarchy, with respect to the trace norm. Formally, this topology is summarised by the followingresult for states of finite stellar rank:
Theorem 2.5.
For all N ∈ (cid:78) , R N = (cid:91) ≤ K ≤ N R K , (2.90) where X denotes the closure of X for the trace norm in the set of normalised states of H . Proof.
Recall that the set of normalised pure single-mode states is closed for the trace normin the whole Hilbert space, since it is the reciprocal image of { } by the trace norm, which isLipschitz continuous—with Lipschitz constant 1—hence continuous.For the proof, we fix N ∈ (cid:78) . We prove the theorem by showing a double inclusion. Wefirst show that (cid:83) NK = R K ⊂ R N , and then that the set (cid:83) NK = R K is closed in H for the tracenorm. Since the closure of a set X is the smallest closed set containing X , and given that R N ⊂ (cid:83) NK = R K , this will prove the other inclusion and hence the result.We have R N ⊂ R N . Let | ψ 〉 ∈ (cid:83) N − K = R K . There exists K ∈ {
0, . . . , N − } such that r (cid:63) ( ψ ) = K . ByTheorem 4, there exists a core state | C ψ 〉 , with a polynomial stellar function of degree K ,59 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES and a Gaussian operation ˆ G ψ such that | ψ 〉 = ˆ G ψ | C ψ 〉 . We define the sequence of normalisedstates | ψ m 〉 = (cid:115) − m | ψ 〉 + (cid:112) m ˆ G ψ | N 〉 , (2.91)for m ≥
1. We have | ψ m 〉 = ˆ G ψ (cid:195)(cid:115) − m | C ψ 〉 + (cid:112) m | N 〉 (cid:33) , (2.92)and the state (cid:113) − m | C ψ 〉 + (cid:112) m | N 〉 is a normalised core state whose stellar function is apolynomial of degree N , hence | ψ m 〉 ∈ R N . Moreover, { | ψ m 〉 } m ≥ converges to | ψ 〉 in tracenorm. This shows that (cid:83) NK = R K ⊂ R N .We now prove that the set (cid:83) NK = R K is closed in H for the trace norm. For N = + N ≥
0, the sketch of the proof is the following: given a converging sequence in (cid:83) NK = R K , we want to show that its limit has a stellar rank less or equal to N . We firstuse the decomposition result of Theorem 2.4, in order to obtain a sequence of Gaussianoperations acting on a sequence of core states of rank less or equal to N . We make use ofthe compactness of this set of core states to restrict to a unique core state. Then, we showthat the squeezing and the displacement parameters of the sequence of Gaussian operationscannot be unbounded. This allows us to conclude by extracting converging subsequencesfrom these parameters.The trace distance D is induced by the trace norm. Let { | ψ m 〉 } m ∈ (cid:78) ∈ (cid:83) NK = R K be a con-verging sequence for the trace norm, and let | ψ 〉 ∈ H be its limit. By Theorem 2.4, thereexist a sequence of core states { | C m 〉 } m ∈ (cid:78) , with polynomial stellar functions of degrees lessor equal to N , and a sequence of Gaussian operations { ˆ G m } m ∈ (cid:78) such that for all m ∈ (cid:78) , | ψ m 〉 = ˆ G m | C m 〉 .The set of normalised core states with a polynomial stellar function of degree less or equalto N corresponds to the set of normalised states with a support over the Fock basis truncatedat N , and is compact for the trace norm in H (isomorphic to the set of norm 1 vectors in (cid:67) N + ). Hence, the sequence { | C m 〉 } m ∈ (cid:78) admits a converging subsequence { | C m k 〉 } k ∈ (cid:78) . Let thecore state | C 〉 , with a polynomial stellar function of degree less or equal to N , be its limit.Along this subsequence, | ψ m k 〉 = ˆ G m k | C m k 〉 , (2.93)and we have lim k →+∞ D ( | ψ m k 〉 , | ψ 〉 ) = k →+∞ D ( | C m k 〉 , | C 〉 ) =
0. Moreover, for all60 .3. ROBUSTNESS OF NON-GAUSSIAN STATES k ∈ (cid:78) , D ( ˆ G m k | C 〉 , | ψ 〉 ) ≤ D ( ˆ G m k | C 〉 , | ψ m k 〉 ) + D ( | ψ m k 〉 , | ψ 〉 ) = D ( ˆ G m k | C 〉 , ˆ G m k | C m k 〉 ) + D ( | ψ m k 〉 , | ψ 〉 ) = D ( | C 〉 , | C m k 〉 ) + D ( | ψ m k 〉 , | ψ 〉 ), (2.94)where we used the triangular inequality in the first line, Eq. (2.93) in the second line, andthe invariance of the trace distance under unitary transformations in the third line. Hence,the sequence { ˆ G m k | C 〉 } k ∈ (cid:78) converges in trace norm to | ψ 〉 . This shows that we can restrictwithout loss of generality to a unique core state, with a polynomial stellar function of degreeless or equal to N , instead of a sequence of such core states.Let | C 〉 thus be a core state, with a polynomial stellar function of degree K less or equalto N . We write | C 〉 = P C ( ˆ a † ) | 〉 = K (cid:88) n = p n (cid:112) n ! | n 〉 , (2.95)with (cid:80) Kn = | p n | n ! =
1. Let us consider a converging sequence { ˆ G m | C 〉 } m ∈ (cid:78) , where ˆ G m areGaussian operations, and denote | ψ 〉 its limit. There exists two sequences { ξ m } m ∈ (cid:78) and { α m } m ∈ (cid:78) , such that for all m ∈ (cid:78) , ˆ G m = ˆ S ( ξ m ) ˆ D ( α m ). (2.96)We write ξ m = r m e i θ m , with r m ≥
0, for all m ∈ (cid:67) . We may rewrite ˆ G m = ˆ D ( γ m ) ˆ S ( ξ m ), wherefor all m ∈ (cid:78) , γ m = c r m α m + s r m e i θ m α ∗ m , (2.97)where c r m = cosh( r m ) and s r m = sinh( r m ). With these notations, we prove the following result: Lemma 2.5.
The sequences { ξ m } m ∈ (cid:78) and { γ m } m ∈ (cid:78) are bounded. Proof.
We first compute an upper bound for the Q function of the state ˆ G m | C 〉 , which weobtain in Eq. (2.114). For m ∈ (cid:78) , we have: Q ˆ G m | C 〉 ( z ) = Q ˆ D ( γ m ) ˆ S ( ξ m ) | C 〉 ( z ) = Q ˆ S ( ξ m ) | C 〉 ( z − γ m ) = e −| z − γ m | π (cid:175)(cid:175)(cid:175) F (cid:63) ˆ S ( ξ m ) | C 〉 ( z ∗ − γ ∗ m ) (cid:175)(cid:175)(cid:175) , (2.98)for all z ∈ (cid:67) .We have ˆ S ( ξ m ) | C 〉 = ˆ S ( ξ m ) P C ( ˆ a † ) | 〉= P C ( c r m ˆ a † + s r m e i θ m ˆ a ) ˆ S ( ξ m ) | 〉 . (2.99)61 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES
Hence, with Eq. (2.7) and (2.35), F (cid:63) ˆ S ( ξ m ) | C 〉 ( z ) = (1 − | t r m | ) P C ( c r m z + s r m e i θ m ∂ z ) · e − t rm e − i θ m z = (cid:112) c r m K (cid:88) n = p n (cid:112) n ! ( c r m z + s r m e i θ m ∂ z ) n · e − t rm e − i θ m z . (2.100)where t r m = tanh( r m ).The Hermite polynomials [AS65] satisfy the following recurrence relation He n + ( z ) = zHe n ( z ) − ∂ z He n ( z ), (2.101)for all n ≥ z ∈ (cid:67) , and He =
1. Setting f n ( z ) : = e t rm e − i θ m z ( c r m z + s r m e i θ m ∂ z ) n · e − t rm e − i θ m z , (2.102)we obtain f ( z ) =
1, and f n + ( z ) = e t rm e − i θ m z ( c r m z + s r m e i θ m ∂ z ) (cid:104) e − t rm e − i θ m z f n ( z ) (cid:105) = zc r m f n ( z ) + s r m e i θ m ∂ z f n ( z ). (2.103)Hence, with Eq. (2.101), for all n ≥ z ∈ (cid:67) , f n ( z ) = λ n /2 m He n (cid:195) zc r m (cid:112) λ m (cid:33) , (2.104)where we have set λ m = − e i θ m t r m . With Eq. (2.100) we thus obtain F (cid:63) ˆ S ( ξ m ) | C 〉 ( z ) = (cid:112) c r m K (cid:88) n = p n (cid:112) n ! f n ( z ) · e − t rm e − i θ m z = (cid:112) c r m K (cid:88) n = p n λ n /2 m (cid:112) n ! He n (cid:195) zc r m (cid:112) λ m (cid:33) e − t rm e − i θ m z . (2.105)From this and Eq. (2.98) we deduce Q ˆ G m | C 〉 ( z ) = e −| z − γ m | π c r m (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) K (cid:88) n = p n λ n /2 m (cid:112) n ! He n (cid:195) z ∗ − γ ∗ m c r m (cid:112) λ m (cid:33) e − t rm e − i θ m ( z ∗ − γ ∗ m ) (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ e −| z − γ m | π c r m (cid:175)(cid:175)(cid:175) e − t rm e − i θ m ( z ∗ − γ ∗ m ) (cid:175)(cid:175)(cid:175) K (cid:88) n = | p n | n ! · K (cid:88) n = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) λ n /2 m He n (cid:195) z ∗ − γ ∗ m c r m (cid:112) λ m (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) (2.106) = π c r m e −| z − γ m | − t rm [ e i θ m ( z − γ m ) + e − i θ m ( z ∗ − γ ∗ m ) ] K (cid:88) n = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t n /2 r m He n (cid:195) z − γ m c r m (cid:112) λ ∗ m (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ,where we used Cauchy-Schwarz inequality in the second line, | λ m | = t r m and the fact thatthe coefficients of He n are real in the third line.62 .3. ROBUSTNESS OF NON-GAUSSIAN STATES Setting α m ( z ) : = − ie i θ m c r m ( z − γ m ), (2.107)for all m ∈ (cid:78) and for all z ∈ (cid:67) , we obtain Q ˆ G m | C 〉 ( z ) ≤ π c r m e −| z − γ m | − t rm [ e i θ m ( z − γ m ) + e − i θ m ( z ∗ − γ ∗ m ) ] K (cid:88) n = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t n /2 r m He n (cid:195) e − i θ m α m ( z ) (cid:112) t r m (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) = π c r m e − c rm | α m ( z ) | + c rm s rm [ α m ( z ) + α ∗ m ( z )] K (cid:88) n = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t n /2 r m He n (cid:195) e − i θ m α m ( z ) (cid:112) t r m (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) = π c r m e − c rm ( c rm − s rm ) x m ( z ) e − c rm ( c rm + s rm ) y m ( z ) K (cid:88) n = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t n /2 r m He n (cid:195) e − i θ m α m ( z ) (cid:112) t r m (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) , (2.108)where α m ( z ) = x m ( z ) + i y m ( z ). For all r ∈ (cid:82) , c r ( c r − s r ) =
12 (1 + e − r ) >
12 , (2.109)and c r ( c r + s r ) =
12 (1 + e r ) >
12 , (2.110)so with Eq. (2.108) we obtain Q ˆ G m | C 〉 ( z ) ≤ π c r m e − | α m ( z ) | K (cid:88) n = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t n /2 r m He n (cid:195) e − i θ m α m ( z ) (cid:112) t r m (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) . (2.111)Finally, we obtain the following bound for all n ∈ {
0, . . . , K } : (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t n /2 r m He n (cid:195) e − i θ m α m ( z ) (cid:112) t r m (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t n /2 r m (cid:98) n (cid:99) (cid:88) k = ( − k n !2 k k !( n − k )! (cid:195) e − i θ m α m ( z ) (cid:112) t r m (cid:33) n − k (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ (cid:98) n (cid:99) (cid:88) k = n !2 k k !( n − k )! t kr m | α m ( z ) | n − k ≤ (cid:98) n (cid:99) (cid:88) k = n !2 k k !( n − k )! | α m ( z ) | n − k , (2.112)for all m ∈ (cid:78) and all z ∈ (cid:67) . Let us define for brevity the polynomial T ( X ) : = K (cid:88) n = (cid:195) (cid:98) n (cid:99) (cid:88) k = n !2 k k !( n − k )! X n − k (cid:33) . (2.113)Plugging Eq. (2.112) in Eq. (2.111) yields Q ˆ G m | C 〉 ( z ) ≤ π c r m e − | α m ( z ) | T ( | α m ( z ) | ), (2.114)for all m ∈ (cid:78) and all z ∈ (cid:67) . 63 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES
With this bound on the Q function obtained, we may now prove that the sequences { ξ m } m ∈ (cid:78) = { r m e i θ m } m ∈ (cid:78) and { γ m } m ∈ (cid:78) are bounded.Assuming that { r m } m ∈ (cid:78) is unbounded implies that it has a subsequence { r m k } k ∈ (cid:78) goingto infinity. Since the function x (cid:55)→ e − x T ( x ) is bounded, Q ˆ G mk | C 〉 ( z ) → z ∈ (cid:67) when k → +∞ by Eq. (2.114). But Q ˆ G mk | C 〉 ( z ) → Q ψ ( z ) for all z ∈ (cid:67) when k → +∞ , by property ofthe convergence in trace norm. This would imply Q ψ ( z ) = z ∈ (cid:67) , which is impossiblesince | ψ 〉 is normalised. Hence { r m } m ∈ (cid:78) is a bounded sequence, and so is { ξ m } m ∈ (cid:78) .With the same reasoning, if { | α m ( z ) | } m ∈ (cid:78) was unbounded for all z ∈ (cid:67) , this would implyby Eq. (2.114) that Q ψ ( z ) = z ∈ (cid:67) , giving the same contradiction. Hence, there exists z ∈ (cid:67) such that the sequence { | α m ( z ) | } m ∈ (cid:78) is bounded. By Eq. (2.107), this implies that thesequence { γ m } m ∈ (cid:78) is also bounded, since the sequence { r m } m ∈ (cid:78) is bounded. (cid:3) The sequences { ξ m } m ∈ (cid:78) and { γ m } m ∈ (cid:78) being bounded, one can consider simultaneously converg-ing subsequences { ξ m k } k ∈ (cid:78) and { γ m k } k ∈ (cid:78) . We write ξ = re i θ = lim k →∞ ξ m k and γ = lim k →∞ γ m k .On one hand, we have F (cid:63) ˆ G mk | C 〉 ( z ) = F (cid:63) ˆ D ( γ mk ) ˆ S ( ξ mk ) | C 〉 ( z ) = e γ mk z − | γ mk | F (cid:63) ˆ S ( ξ mk ) | C 〉 ( z − γ ∗ m k ) = (cid:112) c r mk K (cid:88) n = p n λ n /2 m k (cid:112) n ! He n (cid:195) z − γ ∗ m k c r mk (cid:112) λ m k (cid:33) e − t rmk e − i θ mk ( z − γ ∗ mk ) + γ mk z − | γ mk | , (2.115)for all k ∈ (cid:78) and all z ∈ (cid:67) , where we have used Eq. (2.59) in the second line, where λ m k =− e i θ mk t r mk , and where we have used Eq. (2.105) in the last line. Setting λ = − e i θ t r , we obtainlim k →∞ F (cid:63) ˆ G mk | C 〉 ( z ) = (cid:112) c r K (cid:88) n = p n λ n /2 (cid:112) n ! He n (cid:181) z − γ ∗ c r (cid:112) λ (cid:182) e − t r e − i θ ( z − γ ∗ ) + γ z − | γ | = F (cid:63) ˆ G | C 〉 ( z ), (2.116)for all z ∈ (cid:67) , where ˆ G = ˆ D ( γ ) ˆ S ( ξ ), and where the second line comes from reversing thecalculations of Eq. (2.115). On the other hand, for all z ∈ (cid:67) ,lim k →∞ F (cid:63) ˆ G mk | C 〉 ( z ) = e | z | lim k →∞ 〈 z ∗ | ˆ G m k | C 〉= e | z | 〈 z ∗ | ψ 〉 (2.117) = F (cid:63) ψ ( z ),by property of the convergence in trace norm. Combining Eq. (2.116) and Eq. (2.117) yields F (cid:63) ψ ( z ) = F (cid:63) ˆ G | C 〉 ( z ), (2.118)64 .3. ROBUSTNESS OF NON-GAUSSIAN STATES for all z ∈ (cid:67) . By Lemma 1, this implies that | ψ 〉 = ˆ G | C 〉 ∈ R K . This shows that (cid:83) NK = R K = (cid:83) NK = R K , so R N ⊂ (cid:83) NK = R K , which concludes the proof. (cid:4) This result implies that the set (cid:83) ≤ K ≤ N R K , containing the states of stellar rank smaller or equalto N , is a closed set in H for the trace norm, for all N ∈ (cid:78) . In particular, since all ranks of thestellar hierarchy are disjoint, for any state of finite rank N , there is no sequence of states ofstrictly lower stellar rank converging to it. Each state of a given finite stellar rank is thus isolatedfrom all the lower stellar ranks, i.e., there is a ball around it in trace norm which only containsstates of equal or higher stellar rank.Moreover, each state of infinite stellar rank is isolated from states of finite stellar rank lowerthan N , for all N ∈ (cid:78) ∗ , i.e., there is a ball around it in trace norm which only contains states ofstellar rank higher than N .On the other hand, with the other inclusion, no state of a given finite stellar rank is isolatedfrom any equal or higher stellar rank, i.e., one can always find a sequence of states of any higherrank converging to this state in trace norm.We also prove the following density result: Lemma 2.6.
The set of states of finite stellar rank is dense for the trace norm in the set ofnormalised pure single-mode states: (cid:91) N ∈ (cid:78) R N = H , (2.119) where X denotes the closure of X for the trace norm in the set of normalised states in H . Proof.
Recall that the set of normalised pure single-mode states is closed for the trace normin the whole Hilbert space, since it is the reciprocal image of { } by the trace norm, which isLipschitz continuous—with Lipschitz constant 1—hence continuous.Let | ψ 〉 ∈ H be a normalised state. We consider the sequence of normalised cut-off states | ψ m 〉 = (cid:112) N m m (cid:88) n = ψ n | n 〉 , (2.120)where N m = (cid:80) mn = | ψ n | is a normalising factor (non-zero for m large enough). All the states | ψ m 〉 have a finite support over the Fock basis, so their stellar function is a polynomial.Hence { | ψ m 〉 } m ∈ (cid:78) ∈ (cid:83) N ∈ (cid:78) R N .Moreover, for all m ∈ (cid:78) , D ( ψ m , ψ ) = (cid:113) − | 〈 ψ m | ψ 〉 | = (cid:115) − m (cid:88) n = | ψ n | (2.121)65 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES = (cid:114) (cid:88) n ≥ m + | ψ n | ,where we used that | ψ 〉 and | ψ m 〉 are pure states in the first line, and the fact that | ψ 〉 isnormalised in the third line. Furthermore, (cid:80) n ≥ m + | ψ n | → m → +∞ , because | ψ 〉 isnormalised. Hence, { | ψ m 〉 } m ∈ (cid:78) converges in trace norm to | ψ 〉 , which concludes the proof. (cid:4) This result implies that states of infinite stellar rank are not isolated from lower stellar ranks,unlike states of finite stellar rank. Given a state of infinite stellar rank, there always exists asequence of states of finite stellar ranks converging to it. However, the ranks of the states in thissequence have to go to infinity, since by Theorem 2.5 states of infinite stellar rank are isolatedfrom states of finite stellar rank lower than N , for all N ∈ (cid:78) ∗ .The consequences of Theorem 2.5 and Lemma 2.6 for the robustness are summarised with thefollowing result: Corollary 2.1.
For all | ψ 〉 ∈ (cid:83) N ∈ (cid:78) R N and for all k ∈ (cid:78) ∗ , (cid:40) R (cid:63) k ( ψ ) > for k ≤ r (cid:63) ( ψ ) , (2.122a) R (cid:63) k ( ψ ) = for k > r (cid:63) ( ψ ) . (2.122b) In particular, states of finite stellar rank are robust: for all states | ψ 〉 ∈ (cid:83) N ∈ (cid:78) R N , we have R (cid:63) ( ψ ) = R (cid:63) r (cid:63) ( ψ ) ( ψ ) > .For all | ψ 〉 ∈ R ∞ and for all k ∈ (cid:78) ∗ , (cid:40) R (cid:63) k ( ψ ) > for k ∈ (cid:78) , (2.123a) R (cid:63) k ( ψ ) = for k = ∞ . (2.123b) In particular, states of infinite stellar rank are not robust: for all states | ψ 〉 ∈ R ∞ , we haveR (cid:63) ( ψ ) = R (cid:63) ∞ ( ψ ) = . Eqs. (2.122a), (2.122b) and (2.123a) are deduced from Theorem 2.5, and Eq. (2.123b) is deducedfrom Lemma 2.6.This result implies that the robust states (i.e., R (cid:63) >
0) are exactly the non-Gaussian states offinite stellar rank. When considering imperfect single-mode non-Gaussian state engineering, onemay thus restrict to states of finite stellar rank, which by Theorem 2.2 are obtained uniquelyby a finite number of single-photon additions to a Gaussian state. Alternatively, one may alsodescribe such states using Theorem 2.4 as finite superpositions of equally displaced and squeezednumber states, or equivalently as Gaussian-convertible to core states. Engineering of such stateshas recently been considered in [SMS19], by photon detection of Gaussian states.Moreover, for k ∈ (cid:78) ∗ , the states that are robust with respect to states of stellar rank lowerthan k (i.e., R (cid:63) k >
0) thus are the states | ψ 〉 such that r (cid:63) ( ψ ) ≥ k .66 .3. ROBUSTNESS OF NON-GAUSSIAN STATES Importantly, the k -robustness is state-dependent and the following result gives a simple expres-sion. Let us define, for all n ∈ (cid:78) , Π n = n (cid:88) m = | m 〉〈 m | (2.124)the projector onto the subspace spanned by the Fock states | 〉 , . . . , | n 〉 . Theorem 2.6.
Let k ∈ (cid:78) ∗ and let | ψ 〉 ∈ H . Then,R (cid:63) k ( ψ ) = (cid:115) − sup ˆ G ∈ G Tr (cid:163) Π k − ˆ G | ψ 〉〈 ψ | ˆ G † (cid:164) , (2.125) where the supremum is over Gaussian unitary operations. Moreover, assuming the optimisationyields a Gaussian operation ˆ G , an optimal approximating state is ˆ G † (cid:195) Π k − ˆ G | ψ 〉 (cid:176)(cid:176) Π k − ˆ G | ψ 〉 (cid:176)(cid:176) (cid:33) . (2.126) Proof.
From Lemma 2.4 and in particular Eq. (2.88) we have R (cid:63) k ( ψ ) = (cid:114) − sup r (cid:63) ( φ ) < k | 〈 φ | ψ 〉 | . (2.127)By Theorem 2.4, for any pure state | φ 〉 such that r (cid:63) ( φ ) < k , there exist a normalised corestate | C φ 〉 of stellar rank lower than k and a Gaussian operation ˆ G φ such that | φ 〉 = ˆ G φ | C φ 〉 . (2.128)We obtain | 〈 φ | ψ 〉 | = | 〈 C φ | ˆ G † φ | ψ 〉 | = | 〈 C φ | Π k − ˆ G † φ | ψ 〉 | ≤ | 〈 C φ | C φ 〉 | | 〈 ψ | ˆ G φ Π k − ˆ G † φ | ψ 〉 | = Tr (cid:104) Π k − ˆ G † φ | ψ 〉〈 ψ | ˆ G φ (cid:105) , (2.129)where we used | C φ 〉 = Π k − | C φ 〉 in the second line, since | C φ 〉 is a core state of stellar ranklower than k (hence its support is contained in the support of Π k − ), Cauchy-Schwarzinequality in the third line and | 〈 C φ | C φ 〉 | = | C φ 〉 = Π k − ˆ G † φ | ψ 〉 (cid:114) Tr (cid:104) Π k − ˆ G † φ | ψ 〉〈 ψ | ˆ G φ (cid:105) , (2.130)which is indeed a normalised core state of stellar rank lower than k .67 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES
With Eqs. (2.127) and (2.129), the robustness of the state | ψ 〉 is then given by R (cid:63) k ( ψ ) = (cid:115) − sup ˆ G φ ∈ G Tr (cid:104) Π k − ˆ G † φ | ψ 〉〈 ψ | ˆ G φ (cid:105) = (cid:115) − sup ˆ G ∈ G Tr (cid:163) Π k − ˆ G | ψ 〉〈 ψ | ˆ G † (cid:164) , (2.131)where the supremum is over Gaussian unitary operations and where we used the fact thatthe set of Gaussian unitary operations is invariant under adjoint in the second line. WithEq. (2.128), assuming the optimisation yields an optimal Gaussian unitary ˆ G , an optimalapproximating state is | φ 〉 = ˆ G φ | C φ 〉 , where ˆ G φ = ˆ G † and where | C φ 〉 is given by Eq. (2.130).Namely, | φ 〉 = ˆ G † (cid:195) Π k − ˆ G | ψ 〉 (cid:176)(cid:176) Π k − ˆ G | ψ 〉 (cid:176)(cid:176) (cid:33) , (2.132)i.e., ˆ G | φ 〉 is the renormalised truncation of ˆ G | ψ 〉 at photon number k − (cid:4) From Theorem 2.6, the robustness profile R ( ψ ) = (cid:161) R (cid:63) k (cid:162) k ∈ (cid:78) ∗ is a non-increasing sequence for anystate | ψ 〉 , and each term in the sequence may be obtained with an optimisation over two complexparameters.In particular, with the Hermite polynomials He m ( z ) = ( − m e z ∂ mz e − z = (cid:98) m (cid:99) (cid:88) p = m !( − p p p !( m − p )! z m − p , (2.133)for all m ∈ (cid:78) and all z ∈ (cid:67) , the robustness of cat states has the following expression: Corollary 2.2.
Let k ∈ (cid:78) ∗ and let α ∈ (cid:67) . Then, writing c x = cosh x, s x = sinh x, and t x = tanh x forbrevity, R (cid:63) k ( cat + α ) = (cid:118)(cid:117)(cid:117)(cid:116) − sup ξ = re i θ , β ∈ (cid:67) e −| β | c r c | α | k − (cid:88) m = t mr m ! (cid:175)(cid:175) u + m ( | α | , ξ , β ) (cid:175)(cid:175) , (2.134) and R (cid:63) k ( cat − α ) = (cid:118)(cid:117)(cid:117)(cid:116) − sup ξ = re i θ , β ∈ (cid:67) e −| β | c r s | α | k − (cid:88) m = t mr m ! (cid:175)(cid:175) u − m ( | α | , ξ , β ) (cid:175)(cid:175) , (2.135) whereu ± m ( | α | , ξ , β ) : = e −| α | β ∗ + t r e i θ ( | α |+ β ) He m (cid:181) | α | + β (cid:112) c r s r e i θ /2 (cid:182) ± e | α | β ∗ + t r e i θ ( β −| α | ) He m (cid:181) β − | α |(cid:112) c r s r e i θ /2 (cid:182) .(2.136)68 .3. ROBUSTNESS OF NON-GAUSSIAN STATES Proof.
Let α ∈ (cid:67) . We have | cat ± α 〉 = (cid:113) N ± α ( | α 〉 ± |− α 〉 ), (2.137)where N ± α = ± e − | α | ). By Theorem 2.6, R (cid:63) k (cat ± α ) = (cid:115) − sup ˆ G ∈ G Tr (cid:163) Π k − ˆ G | cat ± α 〉〈 cat ± α | ˆ G † (cid:164) . (2.138)for all k ∈ (cid:78) ∗ , where Π k − in the projector onto the Fock basis with less the k − (cid:104) Π k − ˆ G | cat ± α 〉〈 cat ± α | ˆ G † (cid:105) = k − (cid:88) m = (cid:175)(cid:175) 〈 m | ˆ G | cat ± α 〉 (cid:175)(cid:175) . (2.139)Setting ˆ G = ˆ S ( ξ ) ˆ D ( β ), for ξ = re i θ , β ∈ (cid:67) , we obtain 〈 m | ˆ G | cat ± α 〉 = (cid:113) N ± α (cid:161) 〈 m | ˆ S ( ξ ) ˆ D ( β ) | α 〉 ± 〈 m | ˆ S ( ξ ) ˆ D ( β ) | − α 〉 (cid:162) = (cid:113) N ± α m ! (cid:179) e ( α ∗ β − αβ ∗ ) 〈 | ˆ a m ˆ S ( ξ ) ˆ D ( α + β ) | 〉 ± e ( αβ ∗ − α ∗ β ) 〈 | ˆ a m ˆ S ( ξ ) ˆ D ( β − α ) | 〉 (cid:180) .(2.140)Switching to the stellar representation we obtain 〈 m | ˆ G | cat ± α 〉 = (cid:113) N ± α m ! (cid:104) ∂ mz e ( α ∗ β − αβ ∗ ) G (cid:63) ξ , α + β ( z ) ± ∂ mz e ( αβ ∗ − α ∗ β ) G (cid:63) ξ , − α + β ( z ) (cid:105) z = = e − ( | α | +| β | ) (cid:113) c r N ± α m ! (cid:183) e − αβ ∗ + t r e i θ ( α + β ) ∂ mz e − e − i θ t r z + α + β cr z ± e αβ ∗ + t r e i θ ( β − α ) ∂ mz e − e − i θ t r z + β − α cr z (cid:184) z = , (2.141)where we used 〈 | ψ 〉 = F (cid:63) ψ (0) and Eq. (2.7). With Eq. (2.133) we have (cid:104) ∂ mz e − az + bz (cid:105) z = = a m /2 He m (cid:181) b (cid:112) a (cid:182) , (2.142)where He m is the m th Hermite polynomial.With Eq. (2.141) we obtain (cid:175)(cid:175) 〈 m | ˆ G | cat ± α 〉 (cid:175)(cid:175) = e − ( | α | +| β | ) t mr c r N ± α m ! (cid:175)(cid:175)(cid:175)(cid:175) e − αβ ∗ + t r e i θ ( α + β ) He m (cid:181) α + β (cid:112) c r s r e i θ /2 (cid:182) ± e αβ ∗ + t r e i θ ( β − α ) He m (cid:181) β − α (cid:112) c r s r e i θ /2 (cid:182) (cid:175)(cid:175)(cid:175)(cid:175) . (2.143)69 HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES
Combining Eqs. (2.138), (2.139) and (2.143) yields R (cid:63) k (cat ± α ) = (cid:118)(cid:117)(cid:117)(cid:116) − sup ξ = re i θ , β ∈ (cid:67) e − ( | α | +| β | ) c r N ± α k − (cid:88) m = t mr m ! (cid:175)(cid:175) u ± m ( α , ξ , β ) (cid:175)(cid:175) , (2.144)where we have set u ± m ( α , ξ , β ) : = e − αβ ∗ + t r e i θ ( α + β ) He m (cid:181) α + β (cid:112) c r s r e i θ /2 (cid:182) ± e αβ ∗ + t r e i θ ( β − α ) He m (cid:181) β − α (cid:112) c r s r e i θ /2 (cid:182) .(2.145)Since the robustness is invariant under Gaussian operations, R (cid:63) k (cat ± α ) does not depend onthe phase of α , since one can map a cat state of amplitude α to a cat state of amplitude e i φ α through a Gaussian rotation (this corresponds to mapping β to e i φ β and θ to θ − φ inthe previous expressions). Hence, we may assume without loss of generality that α ∈ (cid:82) andreplace α by | α | . With c | α | = N + α e −| α | and s | α | = N − α e −| α | , (2.146)this concludes the proof. (cid:4) By Lemma 2.4, the sequence of maximum achievable fidelities for each rank k − F = − R (cid:63) k . We have computed numeri-cally the values of R (cid:63) k (cat + α ) and R (cid:63) k (cat − α ) for different values of k and α and the correspondingachievable fidelities are depicted in Fig. 2.4 and Fig. 2.5. For each rank, if ρ denotes a state forwhich the maximum fidelity is achieved, then any lower fidelity may be obtain by considering thestates ρ p = p |⊥〉 〈⊥| + (1 − p ) ρ , for 0 ≤ p ≤
1, where |⊥〉 is a coherent state orthogonal to the targetstate (which exists by Theorem 2.1, since the target state is non-Gaussian). From the numericsand the obtained profiles of the cat states, we make various observations:• The main difference between low amplitude cat + and cat − states is that the former areeasier to approximate by Gaussian states than the latter: at low amplitude, cat + states arecloser to the vacuum while cat − states are closer to the single photon Fock state.• High amplitude cat states are ‘more non-Gaussian’ than low amplitude cat states, in thesense that one needs more photon additions to approximate them to the same precision.• The maximum achievable fidelity increases more from odd to even ranks (resp. even to oddranks) than from even to odd ranks (resp. odd to even ranks) for cat + states (resp. cat − states). This is due to cat + states (resp. cat − states) having support only on even (resp. odd)Fock states.• For each given amplitude, there is a critical stellar rank after which good approximation ofthe cat state becomes possible. Before that stellar rank, the best Gaussian operation in the70 .3. ROBUSTNESS OF NON-GAUSSIAN STATES | cat +1 i
Let k ∈ (cid:78) ∗ and let | ψ 〉 ∈ H be a non-Gaussian pure state of finite stellar rankr (cid:63) ( ψ ) ≥ k, with core state | C ψ 〉 = (cid:80) r (cid:63) ψ n = C n | n 〉 . Then,R (cid:63) k ( ψ ) = (cid:118)(cid:117)(cid:117)(cid:116) − sup ξ , α ∈ (cid:67) k − (cid:88) m = | u m ( ξ , α ) | , (2.147) where for all m ∈ {
0, . . . , k − } and all ξ = re i θ , α ∈ (cid:67) ,u m ( ξ , α ) = (cid:112) m ! c r r (cid:63) ψ (cid:88) n = C ∗ n (cid:112) n ! (cid:104) ∂ nz ( c r z + s r e i θ ∂ z − α ∗ ) m e − e − i θ t r z + α cr z + e i θ t r α − | α | (cid:105) z = , (2.148) with c r = cosh r, s r = sinh r and t r = tanh r. Moreover, assuming the optimisation yields values ξ , α ∈ (cid:67) , an optimal approximating state is ˆ D † ( α ) ˆ S † ( ξ ) (cid:195) Π k − ˆ S ( ξ ) ˆ D ( α ) | C ψ 〉 (cid:176)(cid:176) Π k − ˆ S ( ξ ) ˆ D ( α ) | C ψ 〉 (cid:176)(cid:176) (cid:33) . (2.149) Proof.
Let | ψ 〉 ∈ H be a pure state of finite stellar rank r (cid:63) ( ψ ) ∈ (cid:78) ∗ with core state | C ψ 〉 = (cid:80) r (cid:63) ψ n = C n | n 〉 . By Theorem 2.4, there exist a Gaussian operation ˆ G ψ such that | ψ 〉 = ˆ G ψ | C ψ 〉 .From Theorem 2.6 we have R (cid:63) k ( ψ ) = (cid:115) − sup ˆ G ∈ G Tr (cid:163) Π k − ˆ G | ψ 〉 〈 ψ | ˆ G † (cid:164) = (cid:115) − sup ˆ G (cid:48) ∈ G Tr (cid:163) Π k − ˆ G (cid:48) | C ψ 〉 〈 C ψ | ˆ G (cid:48) † (cid:164) = (cid:114) − sup ξ , α ∈ (cid:67) Tr (cid:163) Π k − ˆ D † ( α ) ˆ S † ( ξ ) | C ψ 〉 〈 C ψ | ˆ S ( ξ ) ˆ D ( α ) (cid:164) = (cid:118)(cid:117)(cid:117)(cid:116) − sup ξ , α ∈ (cid:67) k − (cid:88) m = (cid:175)(cid:175) 〈 C ψ | ˆ S ( ξ ) ˆ D ( α ) | m 〉 (cid:175)(cid:175) , (2.150)where we used the group structure of the Gaussian unitary operations in the second line andthe fact that any single-mode Gaussian unitary operation may be decomposed as a squeezingand a displacement in the third line. Assuming the optimisation yields values ξ , α ∈ (cid:67) , theoptimal core state used in the approximation is | C 〉 = Π k − ˆ S ( ξ ) ˆ D ( α ) | C ψ 〉 (cid:176)(cid:176) Π k − ˆ S ( ξ ) ˆ D ( α ) | C ψ 〉 (cid:176)(cid:176) . (2.151)72 .3. ROBUSTNESS OF NON-GAUSSIAN STATES Now for all m ∈ {
0, . . . , k − } , 〈 C ψ | ˆ S ( ξ ) ˆ D ( α ) | m 〉 = r (cid:63) ψ (cid:88) n = C ∗ n 〈 n | ˆ S ( ξ ) ˆ D ( α ) | m 〉 , (2.152)and for all n ∈ {
0, . . . , r (cid:63) ψ } , 〈 n | ˆ S ( ξ ) ˆ D ( α ) | m 〉 = (cid:112) m ! n ! 〈 | ˆ a n ˆ S ( ξ ) ˆ D ( α )( ˆ a † ) m | 〉= (cid:112) m ! n ! 〈 | ˆ a n ( c r ˆ a † + s r e i θ ˆ a − α ∗ ) m ˆ S ( ξ ) ˆ D ( α ) | 〉 , (2.153)where we used Eq. (2.37) in the second line, with c r = cosh r , s r = sinh r . Hereafter wealso set t r = tanh r . We have 〈 | χ 〉 = F (cid:63) χ (0) for all states χ , hence switching to the stellarrepresentation Eq. (2.153) rewrites 〈 n | ˆ S ( ξ ) ˆ D ( α ) | m 〉 = (cid:112) m ! n ! c r (cid:104) ∂ nz (cid:179) c r z + s r e i θ ∂ z − α ∗ (cid:180) m e − e − i θ t r z + α cr z + e i θ t r α − | α | (cid:105) z = ,(2.154)where we used Eq. (2.7). Hence, 〈 C ψ | ˆ S ( ξ ) ˆ D ( α ) | m 〉 = r (cid:63) ψ (cid:88) n = C ∗ n (cid:112) m ! n ! c r (cid:104) ∂ nz (cid:179) c r z + s r e i θ ∂ z − α ∗ (cid:180) m e − e − i θ t r z + α cr z + e i θ t r α − | α | (cid:105) z = .(2.155)Setting, for all m ∈ {
0, . . . , k − } , u m ( ξ , α ) = 〈 C ψ | ˆ S ( ξ ) ˆ D ( α ) | m 〉 , thus omitting the dependencyin ψ , we finally obtain with Eq. (2.150), R (cid:63) k ( ψ ) = (cid:118)(cid:117)(cid:117)(cid:116) − sup ξ , α ∈ (cid:67) k − (cid:88) m = | u m ( ξ , α ) | , (2.156)where for all m ∈ {
0, . . . , k − } and all ξ = re i θ , α ∈ (cid:67) , u m ( ξ , α ) = (cid:112) m ! c r r (cid:63) ψ (cid:88) n = C ∗ n (cid:112) n ! (cid:104) ∂ nz ( c r z + s r e i θ ∂ z − α ∗ ) m e − e − i θ t r z + α cr z + e i θ t r α − | α | (cid:105) z = , (2.157)with c r = cosh r , s r = sinh r and t r = tanh r . Moreover, assuming the optimisation yieldsvalues ξ , α ∈ (cid:67) , an optimal approximating state is | φ 〉 = ˆ D † ( α ) ˆ S † ( ξ ) | C 〉 , where | C 〉 isdefined in Eq. (2.151). Namely, | φ 〉 = ˆ D † ( α ) ˆ S † ( ξ ) (cid:195) Π k − ˆ S ( ξ ) ˆ D ( α ) | C ψ 〉 (cid:176)(cid:176) Π k − ˆ S ( ξ ) ˆ D ( α ) | C ψ 〉 (cid:176)(cid:176) (cid:33) , (2.158)i.e., ˆ S ( ξ ) ˆ D ( α ) | φ 〉 is the renormalised truncation of ˆ S ( ξ ) ˆ D ( α ) | C ψ 〉 at photon number k − (cid:4) HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES
From this result, the value of the stellar robustness may be obtained analytically for low stellarrank states and numerically for all finite stellar rank states. In particular, we obtain:
Lemma 2.7.
For the single photon Fock state | 〉 of stellar rank we haveR (cid:63) (1) = (cid:115) − (cid:112) e . (2.159) The corresponding maximum achievable fidelity is given by (cid:112) e ≈ Proof.
The single-photon Fock state | 〉 is a core state of stellar rank 1 (its stellar functionis given by F (cid:63) ( z ) = z for all z ∈ (cid:67) ). Hence, by Corollary 2.3, R (cid:63) (1) = R (cid:63) (1) = (cid:115) − sup ξ = re i θ , α ∈ (cid:67) c r (cid:175)(cid:175)(cid:175)(cid:104) ∂ z e − tr e − i θ z + α cr z + tr e i θ α − | α | (cid:105) z = (cid:175)(cid:175)(cid:175) = (cid:118)(cid:117)(cid:117)(cid:116) − sup ξ = re i θ , α ∈ (cid:67) | α | c r (cid:175)(cid:175)(cid:175) e tr e i θ α − | α | (cid:175)(cid:175)(cid:175) = (cid:118)(cid:117)(cid:117)(cid:116) − sup ξ = re i θ , α ∈ (cid:67) | α | c r e tr ( α e i θ + α ∗ e − i θ ) −| α | . (2.161)Setting γ = x + i y = i α e i θ /2 we obtain | α | c r e tr ( α e i θ + α ∗ e − i θ ) −| α | = | γ | c r e −| γ | − tr ( γ + γ ∗ ) = x + y c r e − (1 + t r ) x e − (1 − t r ) y = (1 − t r ) ( x + y ) e − (1 + t r ) x e − (1 − t r ) y = (1 − t r ) ( x + y ) e − (1 − t r )( x + y ) e − t r x (2.162) ≤ (1 − t r ) ( x + y ) e − (1 − t r )( x + y ) ≤ (1 − t r ) e (1 − t r ) = e (cid:113) (1 − t r )(1 + t r ) ,and this upperbound is attained for x = y = (cid:112) − t r . Finally, we have max u ∈ [0,1] (1 − u )(1 + u ) = .3. ROBUSTNESS OF NON-GAUSSIAN STATES , attained for u = , so we obtain the stellar robustness of a single photon Fock state: R (cid:63) (1) = (cid:115) − (cid:112) e . (2.163) (cid:4) Since the stellar robustness inherits the property of invariance under Gaussian unitary operationsof the stellar rank, Corollary 2.1 implies the same robustness value for states obtained froma single photon Fock state by unitary Gaussian operations, such as photon-added or photon-subtracted squeezed states, by Eq. (2.81).We have computed numerically the stellar robustness for the states cos φ | 〉 + e i χ sin φ | 〉 ,for all φ , χ ∈ [0, 2 π ], which is independent of χ (Fig. 2.6). Setting φ = π yields the single photonFock state, which is thus the most robust state of stellar rank 1, up to Gaussian unitary operations. (cid:1) (cid:1) (cid:1) (cid:1)
0, it is the x -coordinate of the leftmost intersection point of the horizontal line of height1 − (cid:178) with the vertical lines of the profile. For example, let (cid:178) = − (cid:178) = HAPTER 2. STELLAR REPRESENTATION OF NON-GAUSSIAN QUANTUM STATES | i
17, BIS + HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS
We identify the regime for which an efficient classical weak simulation of circuits in this subclasswould imply a collapse of the polynomial hierarchy of complexity classes.This chapter is based on [CMS20, CFGM20, CDM + Depending on the approach used for simulating classically the functioning of quantum devices,several notions of simulability are commonly used. In what follows, we review the ones we will beconsidering in this chapter.
To each quantum computation is associated a probability distribution from which classicaloutcomes are sampled. In the case of continuous variable quantum computations with continuousvariable outcomes, the output probability distribution is replaced by an output probability density.This motivates the following (informal) definition [TD02, PBG20].
Definition 3.1 (Strong simulation) . A quantum computation is strongly simulable if there existsa classical algorithm which evaluates its output probability distribution (density) or any of itsmarginals for any outcome in time polynomial in the size of the quantum computation.Various relaxations of this definition are possible, allowing the classical evaluation to be approx-imate rather than exact, or to abort with a small probability. Hereafter we only consider thedefinition above. When there exists no efficient classical algorithm for strong simulation, we saythat strong simulation is hard .This notion of simulability is referred to as strong because it asks more from the classicalsimulation algorithm than from the quantum computation. Indeed, the quantum computation ismerely sampling from a probability distribution (density), while the classical algorithm has tocompute efficiently probabilities.
A sampling counterpart to the notion of strong simulation is to ask the classical simulationalgorithm to mimic the output of the quantum computation [TD02, PBG20]. Informally:
Definition 3.2 (Weak simulation) . A quantum computation is weakly simulable if there exists aclassical algorithm which outputs samples from its output probability distribution (density) intime polynomial in the size of the quantum computation.Akin to strong simulation, various relaxations of this definition are possible, allowing the classicalsampling to be approximate rather than exact, or to abort with a small probability. Hereafter we80 .1. CLASSICAL SIMULATION OF QUANTUM COMPUTATIONS only consider the definition above. When there exists no efficient classical algorithm for weaksimulation, we say that weak simulation is hard .In the case of continuous variable quantum computations with continuous variable outcomes,a weaker requirement is to ask the classical simulation not to sample from the output probabilitydensity, but rather from a discretised probability distribution obtained from the probabilitydensity by performing an efficient binning of the sample space. Indeed, samples from the outputprobability density yield samples of such a discretised probability distribution with efficientclassical post-processing.Consider a quantum computation of size m yielding discrete classical outcomes from a probabilitydistribution P ( X , . . . , X m ), where X i may take at most M = poly m values for all i ∈ {
1, . . . , m } (thesample space has size M m ). Then, weak simulation is weaker than strong simulation, with thefollowing result [TD02, PBG20]: Lemma 3.1.
An efficient classical algorithm for strong simulation provides an efficient classicalalgorithm for weak simulation (assuming one can efficiently sample from efficiently computableunivariate probability distributions over a polynomial number of samples).
We reproduce the proof below for completeness.
Proof.
Assuming the existence of an efficient classical algorithm for strong simulation ofa quantum computation of size m yielding classical outcomes from a discrete probabilitydistribution P ( X , . . . , X m ), where X i may take at most M = poly m values for all i ∈ {
1, . . . , m } ,one first computes the marginal probabilities P ( X ) for all M possible values of X . Then, onesamples the value x from P ( X ) (which is an efficiently computable univariate probabilitydistribution over a polynomial number of samples). With that sample x , one computes theconditional probability distribution P ( X | x ) = P ( x , X ) P ( x ) , (3.1)for all M possible values of X . Then, one samples the value x from P ( X | x ) (which is alsoan efficiently computable univariate probability distribution over a polynomial number ofsamples). Repeating the same procedure up to P ( X m | x , . . . , x m − ) = P ( x , . . . , x m − , X m ) P ( x , . . . , x m − ) , (3.2)one obtains a sample ( x , . . . , x m ) from P ( X , . . . , X m ) efficiently. (cid:4) For quantum computations yielding continuous variable classical outcomes, the result still holdswith the same proof for binned discretised probability distributions rather than the corresponding81
HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS probability density, as long as the discretised probabilities can be computed efficiently from theprobability density and have support on a polynomial number of bins for each mode.
While the previous two notions of simulation of quantum computations are the most commonlyused, other type of simulation may be useful: if the output samples of a quantum computation areused to compute a quantity which may be computed efficiently classically by other means, it is nolonger necessary to simulate the whole quantum device. We consider two concrete examples whichare prominent for variational quantum algorithms in quantum machine learning: probabilityestimation and overlap estimation [HCT +
19, SK19].
Definition 3.3 (Probability estimation) . Let P be a probability distribution over m outcomes.Given any outcome x in the sample space of P , probability estimation refers to the computationaltask of outputting an estimate ˜ P [ x ] such that P [ x ] − m ≤ ˜ P [ x ] ≤ P [ x ] + m , (3.3)with probability greater than 1 − m .Probability estimation amounts to outputting a polynomially precise additive estimate of theprobability with exponentially small probability of failure. One may use the samples from aquantum computation in order to perform probability estimation for any given outcome: given aquantum device of size m which outputs samples from some probability distribution and a fixedoutcome x in the sample space, one may run the device M = poly m times, recording the value 1whenever the outcome x is obtained and the value 0 otherwise. Then, summing and dividing by M , one obtains the frequency of the outcome x over the M uses of the quantum device, which isa polynomially precise additive estimate of the probability of the outcome x with exponentiallysmall probability of failure, by virtue of Hoeffding inequality [Hoe63].Weak simulation is at least as hard as probability estimation, since by the previous reasoningone may obtain polynomially precise additive estimates of probabilities from samples of theprobability distribution. Moreover, they are some quantum computations for which weak sim-ulation is hard (assuming widely believed conjectures from complexity theory), but probabilityestimation can be done efficiently classically. This is the case for IQP circuits [BJS10, HCT + N is an n bits integer to factor, the period-finding subroutine measuresthe output state 1 N (cid:88) x (cid:88) y e i π xyN | y 〉 | f ( x ) 〉 (3.4)82 .1. CLASSICAL SIMULATION OF QUANTUM COMPUTATIONS in the computational basis, where f is a periodic function over {
0, . . . , N − } which can be evaluatedefficiently. The probability of obtaining an outcome y , f ( x ) is given byPr [ y , f ( x )] = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) N (cid:88) f ( x ) = f ( x ) e i π xy N (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) . (3.5)Now let g x , y : x (cid:55)→ e i π xy N if f ( x ) = f ( x ),0 otherwise. (3.6)The function g x , y can be evaluated efficiently and we havePr [ y , f ( x )] = (cid:175)(cid:175)(cid:175)(cid:175) (cid:69) x ← N [ g x , y ( x )] (cid:175)(cid:175)(cid:175)(cid:175) , (3.7)where (cid:69) x ← N denotes the expected value for x drawn uniformly randomly from {
0, . . . , N − } . Byvirtue of Hoeffding inequality, this quantity may be estimated efficiently (in n the number ofbits of N ) classically by sampling uniformly a polynomial number of values in {
0, . . . , N − } andcomputing the modulus squared of the mean of g x , y for these values.However, note that probability estimation of quantum circuits is a BQP -complete computa-tional task almost by definition, since given a polynomially precise estimate of the probabilityof acceptance of an input x to a quantum circuit, one may determine whether it is acceptedor rejected by the circuit. In particular, unless factoring is in P , probability estimation for thequantum circuit corresponding to Shor’s algorithm as a whole is hard and weak simulation ofthe period-finding subroutine is also hard, since in Shor’s algorithm the output samples fromthe period-finding subroutine are used for a different classical computation than probabilityestimation (essentially obtaining promising candidates for the period).A more general computational task than probability estimation in the context of quantumcomputing is the following: Definition 3.4 (Overlap estimation) . Let | φ 〉 and | ψ 〉 be quantum output states of two quantumcomputations of size m . Overlap estimation refers to the computational task of outputting anestimate ˜ O such that | 〈 φ | ψ 〉 | − m ≤ ˜ O ≤ | 〈 φ | ψ 〉 | + m , (3.8)with probability greater than 1 − m ).The overlap between two quantum states is a measure of their distinguishability [Die88] andoverlap estimation thus is related to quantum state discrimination. Several techniques exist toperform quantumly the overlap estimation of two states | φ 〉 and | ψ 〉 [FRS + + HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS
The complexity of probability estimation and overlap estimation of quantum computations hasbeen well studied in the circuit model [PWB15, BGM19]. In what follows, we consider the case ofpassive linear optical quantum computing with adaptive measurements, which we refer to asadaptive linear optics (Fig. 3.1). We use multi-index notations (see section 1.1.1).Formally, we consider unitary interferometers of size m , described by m × m unitary matrices(see section 1.4). We identify the multimode Fock states with n photons over m modes with theelements of Φ m , n = { s ∈ (cid:78) m , | s | = n } , for all n ∈ (cid:78) . We fix the input state | t 〉 = | n m − n 〉 , with singlephotons in the n first modes, where the superscript indicates the size of the string (0, . . . , 0) or(1, . . . , 1) when there is a possible ambiguity. For p ∈ (cid:78) and p ∈ Φ k , p , let us define U p : = [ (cid:49) k ⊕ U k ( p , . . . , p k )] [ (cid:49) k − ⊕ U k − ( p , . . . , p k − )] . . . [ (cid:49) ⊕ U ( p )] U , (3.9)where (cid:49) j is the identity matrix of size j . The matrices U j depend on the measurement outcomes p , . . . , p j for all j ∈ {
1, . . . , k } . The output state where the adaptive measurement outcome p hasbeen obtained reads Tr k (cid:104) ( | p 〉〈 p | ⊗ (cid:49) m − k ) U p | t 〉〈 t | U p † (cid:105) , (3.10)where the partial trace is over the first k modes and where | p 〉 denotes the k -mode Fock state | p . . . p k 〉 . The matrix U p describes the interferometer in Fig. 3.1, where the adaptive measure-ment outcome p = ( p , . . . , p k ) and the final outcome s = ( s , . . . , s m − k ) have been obtained. ... ... ... . . .. . . . . . {
1, . . . , k } , the unitary interferometer U j , acting on m − j modes, may depend on themeasurement outcomes p , . . . , p j . The adaptive measurement outcomes p , . . . , p k are used todrive the computation, whose final outcome is s , . . . , s m − k .Boson Sampling [AA13] corresponds to the case k = k = O ( m ). We investigate the transition84 .2. ADAPTIVE LINEAR OPTICS between these two cases by giving classical algorithms for probability estimation and overlapestimation and identifying various complexity regimes for different numbers of photons n andadaptive measurements k . For doing probability estimation with a quantum circuit, one samples the circuit O (poly m ) times,obtaining outcomes, for which the frequency gives a polynomially precise additive estimate of theprobability which can be computed efficiently. In the case of a circuit with adaptive measurements,one only looks at the final measurement outcomes and the same holds for adaptive linear opticalcomputations.For doing overlap estimation with unitary quantum circuits, one may run two circuits U and V in parallel and compare their quantum output states, for example with the swap test. Doing so apolynomial number of times provides a polynomially precise estimate of the overlap. Alternatively,one may build the circuit UV † and project the output quantum state onto the input state.In the case of circuits with adaptive measurements, the overlaps are between all possibleoutput states for all possible adaptive measurement results. In particular, if the number ofpossible adaptive measurement outcomes is exponential, then the probability distribution forthese outcomes has to be concentrated on a polynomial number of events for the quantum overlapestimation to be efficient. This is because in order to compute a polynomially precise estimateof the overlap, say, | 〈 φ | ψ 〉 | , the states | φ 〉 and | ψ 〉 , both corresponding to specific adaptivemeasurement results, have to be obtained a polynomial number of times.For adaptive linear optics over m modes with n input photons and k adaptive measurements,the number of possible adaptive measurement outcomes is given by n (cid:88) r = | Φ k , r | = n (cid:88) r = (cid:195) k + r − r (cid:33) = (cid:195) k + nn (cid:33) , (3.11)where the sum is over the total number of photons detected at the stage of the adaptive mea-surements. Hence, either the probability distribution for the adaptive measurements outcomesis concentrated on a polynomial number of outcomes, or (cid:161) n + kn (cid:162) = O (poly m ), which is the case forexample when n = O (1) and k = O ( m ), n = O (log m ) and k = O (log m ), or n = O ( m ) and k = O (1). Inwhat follows, we do not assume concentration of the adaptive measurement outcome probabilitydistribution and consider general interferometers with adaptive measurements. The quantumefficient regime for overlap estimation thus corresponds to (cid:161) n + kn (cid:162) = O (poly m ).Let | φ 〉 and | ψ 〉 be output states of two adaptive linear interferometers over m modes with n input photons and k adaptive measurements. Let p and q denote the outcomes of the adaptivemeasurements for | φ 〉 and | ψ 〉 , respectively. Let U p in Eq. (3.9) be the interferometer for | φ 〉 , with85 HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS input Fock state | t 〉 . We have | 〈 φ | ψ 〉 | = Tr (cid:104) Tr k [( | p 〉〈 p | ⊗ (cid:49) m − k ) U p | t 〉〈 t | U p † ] | ψ 〉〈 ψ | (cid:105) = Tr (cid:104) ( | p 〉〈 p | ⊗ (cid:49) m − k ) U p | t 〉〈 t | U p † ( (cid:49) k ⊗ | ψ 〉〈 ψ | ) (cid:105) = Tr (cid:104) U p | t 〉〈 t | U p † ( | p 〉〈 p | ⊗ | ψ 〉〈 ψ | ) (cid:105) = Tr (cid:104) | t 〉〈 t | U p † (cid:161) | p 〉〈 p | ⊗ | ψ 〉〈 ψ | (cid:162) U p (cid:105) , (3.12)where we used Eq. (3.10) in the first line. Because of the conservation of the total number ofphotons, the overlap between the states | φ 〉 and | ψ 〉 is zero if | p | (cid:54)= | q | . Otherwise, it can beestimated using a polynomial number of copies of the state | ψ 〉 as follows: send the input | p 〉 ⊗ | ψ 〉 into the interferometer with unitary matrix U p † and mesure the photon number in each outputmode. Record the value 1 if the measurement pattern matches the Fock state t and the value 0otherwise. Then, the mean of the obtained values yields a polynomially precise estimate of theoverlap | 〈 φ | ψ 〉 | by Eq. (3.12) and Hoeffding inequality. Note that this overlap estimation requiresthe preparation of the Fock state p . By symmetry, one could estimate the overlap alternativelyusing a polynomial number of copies of the state | φ 〉 and preparing the Fock state | q 〉 . In this section, we obtain a classical algorithm for probability estimation of adaptive linear opticsover m modes with n input photons and k adaptive measurements.We first consider the case k =
0, i.e., Boson Sampling. The probability of the outcome s ∈ Φ m , n for the interferometer U given the input t = ( n , m − n ) ∈ Φ m , n is given by (see section 1.4.5and [AA13]) Pr m , n [ s ] = s ! | Per ( U s , t ) | , (3.13)where U s , t is the n × n matrix obtained from U by repeating s i times its i th row for i ∈ {
1, . . . , m } and removing its j th column for j = { n +
1, . . . , m } , and where the permanent of an n × n squarematrix A = ( a i j )) ≤ i , j ≤ n is given by Per ( A ) = (cid:88) σ ∈ S n n (cid:89) i = a i σ ( i ) , (3.14)where the sum is over the permutations of the set {
1, . . . , n } . When | s | (cid:54)= n however, the probabilityis 0, since t has n photons and the linear interferometer does not change the total number ofphotons. The permanent of a square matrix of size n can be computed exactly in time O ( n n ),thanks to Ryser’s formula [AWH78]. However, polynomially precise estimates of the permanentcan be obtained in polynomial time [Gur05], so the probability estimation can be done classicallyefficiently, which was already noted in [AA13].We now turn to the case k >
0, using notations of Eq. (3.9) and Fig. 3.1. This case is a directextension of the case k =
0. For p ∈ (cid:78) , p ∈ Φ k , p and s ∈ Φ m − k , n − p , the probability of an total86 .2. ADAPTIVE LINEAR OPTICS outcome ( p , s ) ∈ Φ m , n (adaptive measurement and final outcome) is given byPr total m , n [ p , s ] = p ! s ! (cid:175)(cid:175)(cid:175) Per (cid:179) U p ( p , s ), t (cid:180)(cid:175)(cid:175)(cid:175) . (3.15)Let p ∈ {
0, . . . , n } and let s ∈ Φ m − k , n − p . Then, the probability of obtaining the final outcome s afterthe adaptive measurements readsPr final m , n [ s ] = (cid:88) p ∈ Φ k , p Pr total m , n [ p , s ] = s ! (cid:88) p ∈ Φ k , p p ! (cid:175)(cid:175)(cid:175) Per (cid:179) U p ( p , s ), t (cid:180)(cid:175)(cid:175)(cid:175) . (3.16)The sum is taken over the elements of Φ k , p , which has (cid:161) k + p − p (cid:162) ≤ (cid:161) k + n − n (cid:162) elements. This last quan-tity is O (poly m ) when the number of input photons n and the number of adaptive measurements k are small enough compared to m . n k O (1) O (log m ) O ( m ) O (1) O (log m ) O ( m )Table 3.1: Simulability regimes for probability estimation. In blue is the parameter region forwhich the classical algorithm is efficient.The simulability regimes are summarised in Table 3.1, where the regimes are obtained usingStirling equivalent n ! ∼ (cid:112) π n (cid:161) ne (cid:162) n . In particular, as long as both k and n are O (log m ), the outputprobability can be estimated efficiently (and even computed exactly efficiently).The universal quantum computing regime corresponds to n = O ( m ) and k = O ( m ). The timecomplexity of the classical algorithm is O (cid:179)(cid:161) k + n − n (cid:162) poly m (cid:180) , so there is a possibility of subuniversalquantum advantage for probability estimation for n = O (log m ) and k = O ( m ), or n = O ( m ) and k = O (log m ). However, the runtime of the classical algorithm is subexponential in these cases. In this section, we obtain a classical algorithm for overlap estimation of adaptive linear opticsover m modes with n input photons and k adaptive measurements.Once again, we start with k =
0. The output state of an m -mode interferometer U with inputstate t ∈ Φ m , n reads | φ 〉 = (cid:88) s ∈ Φ m , n 〈 s | ˆ U | t 〉 | s 〉= (cid:88) s ∈ Φ m , n Per ( U s , t ) (cid:112) s ! t ! | s 〉 , (3.17)87 HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS where U s , t is the n × n matrix obtained from U by repeating s i times its i th row for i ∈ {
1, . . . , m } and repeating t j times its j th row for j ∈ {
1, . . . , m } . The composition of two interferometers isanother interferometer which unitary representation is the product of the unitary representationsof the composed interferometers. Hence, the inner product of the output states | φ 〉 and | ψ 〉 oftwo m -mode interferometers U and V with the same input state t ∈ Φ m , n , is equal to the matrixelement t , t of ˆ U † ˆ V : 〈 φ | ψ 〉 = (cid:88) u , v ∈ Φ m , n 〈 t | ˆ U † | u 〉 〈 v | ˆ V | t 〉 〈 u | v 〉= (cid:88) s ∈ Φ m , n 〈 t | ˆ U † | s 〉 〈 s | ˆ V | t 〉= 〈 t | ˆ U † ˆ V | t 〉= Per (cid:163) ( U † V ) t , t (cid:164) t ! , (3.18)where we used in the third line t ∈ Φ m , n and the fact that ˆ U † ˆ V conserves the space Φ m , n . Withthe input t = ( n , m − n ) with n photons in m modes, this reduces to 〈 φ | ψ 〉 = Per (cid:104) ( U † V ) n (cid:105) , (3.19)where ( U † V ) n is the n × n top left submatrix of U † V . Hence, the inner product and the over-lap may be approximated to a polynomial precision efficiently, since this is the case for thepermanent [Gur05].We now consider the case k >
0. Let p ∈ (cid:78) and let p ∈ Φ k , p . Writing Pr adap m , n [ p ] the probability ofthe adaptive measurement outcome p , the output state of the interferometer U p with k adaptivemeasurements with input t = ( n , m − n ) in Fig. 3.1, when the adaptive measurement outcome p is obtained, reads 1 (cid:113) Pr adap m , n [ p ] | ψ p 〉 , (3.20)where | ψ p 〉 : = (cid:88) s ∈ Φ m − k , n − p Per (cid:179) U p ( p , s ), t (cid:180)(cid:112) p ! s ! | s 〉 (3.21)and where Pr adap m , n [ p ] = 〈 ψ p | ψ p 〉 . More generally, the inner product of two (not normalised) outputstates | ψ p 〉 and | ψ q 〉 of m -mode interferometers U p and V q with k adaptive measurements thusis zero if | p | (cid:54)= | q | . If r : = | p | = | q | , it is given by 〈 ψ p | ψ q 〉 = (cid:112) p ! q ! (cid:88) s ∈ Φ m − k , n − r s ! Per (cid:179) U p ( p , s ), t (cid:180) ∗ Per (cid:179) V q ( q , s ), t (cid:180) = (cid:112) p ! q ! (cid:88) s ∈ Φ m − k , n − r s ! Per (cid:179) U p † t ,( p , s ) (cid:180) Per (cid:179) V q ( q , s ), t (cid:180) . (3.22)This expression is a sum of | Φ m − k , n − r | terms, which is generally exponential in m whenever n is not constant. It is reminiscent of the permanent composition formula [Per12, Bar16]: for all88 .2. ADAPTIVE LINEAR OPTICS m , n , c ∈ (cid:78) ∗ , all s ∈ (cid:78) , all u ∈ Φ m , s and all v ∈ Φ n , s ,Per (cid:163) ( MN ) u , v (cid:164) = (cid:88) s ∈ Φ c , s s ! Per (cid:161) M u , s (cid:162) Per (cid:161) N s , v (cid:162) (3.23)where M is a m × c matrix and N is a n × c matrix. In what follows, we prove that this expression inEq. (3.22) may be rewritten as a sum over fewer terms using the permanent composition formulain Eq. (3.23). However, this formula is not directly applicable to the expression in Eq. (3.22). Inorder to obtain a suitable expression, we first make use of the Laplace formula for the permanent:we expand the permanent of U p † t ,( p , s ) along the columns that are repeated according to p and weexpand the permanent of V q ( q , s ), t along the rows that are repeated according to q . The generalLaplace column expansion formula for the permanent reads: let n ∈ (cid:78) ∗ , let W be an n × n matrix,and let j ∈ {
0, 1 } n . Then, Per ( W ) = (cid:88) i ∈ { } n | i |=| j | Per (cid:161) W i , j (cid:162) Per (cid:161) W n − i , n − j (cid:162) , (3.24)where W i , j is the matrix obtained from W by keeping only the k th rows and l th columns suchthat i k = j l =
1, respectively, and W n − i , n − j is the matrix obtained from W by keeping onlythe k th rows and l th columns such that i k = j l =
0, respectively. This formula is obtainedby applying the Laplace expansion formula for one column various times, for each column withindex l such that j l =
1, and the same formula holds for rows.
Lemma 3.2.
Let r ∈ (cid:78) . The inner product of two (not normalised) output states | ψ p 〉 and | ψ q 〉 ofm-mode interferometers U p and V q with adaptive measurements outcome p , q ∈ Φ k , r is given by 〈 ψ p | ψ q 〉 = (cid:112) p ! q ! (cid:88) i , j ∈ { } n | i |=| j |= r Per (cid:179) A i (cid:180) Per (cid:179) B j (cid:180) Per (cid:179) C i , j (cid:180) , (3.25) where for all i , j ∈ {
0, 1 } n such that | i | = | j | = r,A i = U p † ( i , m − n ),( p , m − k ) (3.26) is an r × r matrix which can be obtained efficiently from U p ,B j = V q ( q , m − k ),( j , m − n ) (3.27) is an r × r matrix which can be obtained efficiently from V q , andC i , j = U p † ( n − i , m − n ),( k , m − k ) V q ( k , m − k ),( n − j , m − n ) (3.28) is an ( n − r ) × ( n − r ) matrix which can be obtained efficiently from U p and V q . HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS
Proof.
We consider the expression for the inner product obtained in Eq. (3.22): 〈 ψ p | ψ q 〉 = (cid:112) p ! q ! (cid:88) s ∈ Φ m − k , n − r s ! Per (cid:179) U p † t ,( p , s ) (cid:180) Per (cid:179) V q ( q , s ), t (cid:180) . (3.29)We first apply the general column expansion formula in Eq. (3.24) to the matrix U p † t ,( p , s ) with j = ( r , n − r ) ∈ {
0, 1 } n , obtainingPer (cid:179) U p † t ,( p , s ) (cid:180) = (cid:88) i ∈ { } n | i |= r Per (cid:183)(cid:179) U p † t ,( p , s ) (cid:180) i , j (cid:184) Per (cid:183)(cid:179) U p † t ,( p , s ) (cid:180) n − i , n − j (cid:184) . (3.30)Let us consider the matrix (cid:179) U p † t ,( p , s ) (cid:180) i , j appearing in this last expression, for i ∈ {
0, 1 } n . Itsrows are obtained by keeping the first n lines of U p † since t = ( n , m − n ), then by keepingonly the l th rows such that i l =
1. Its columns are obtained by repeating p l times the l th column for l ∈ {
1, . . . , k } and s l times for l ∈ { k +
1, . . . , m } , then by only keeping the first r columns since j = ( r , n − r ). However, since | p | = | j | = r , these are the columes repeatedaccording to p . Hence, (cid:179) U p † t ,( p , s ) (cid:180) i , j = U p † ( i , m − n ),( p , m − k ) , (3.31)where U p † ( i , m − n ),( p , m − k ) is the matrix obtained from U p † by keeping only the l th rows suchthat i l = p l times the l th column for l ∈ {
1, . . . , k } and removing the others. Similarly, with | s | = | n − j | = n − r , (cid:179) U p † t ,( p , s ) (cid:180) n − i , n − j = U p † ( n − i , m − n ),( k , s ) , (3.32)where U p † ( n − i , m − n ),( k , s ) is the matrix obtained from U p † by keeping only the l th rows such that i l = s l times the l th column for l ∈ { k +
1, . . . , m } and removing the others. With Eqs. (3.30), (3.31) and (3.32) we obtainPer (cid:179) U p † t ,( p , s ) (cid:180) = (cid:88) i ∈ { } n | i |= r Per (cid:179) U p † ( i , m − n ),( p , m − k ) (cid:180) Per (cid:179) U p † ( n − i , m − n ),( k , s ) (cid:180) = (cid:88) i ∈ { } n | i |= r Per (cid:179) A i (cid:180) Per (cid:179) U p † ( n − i , m − n ),( k , s ) (cid:180) , (3.33)where we have defined, for all i ∈ {
0, 1 } n such that | i | = r , A i : = U p † ( i , m − n ),( p , m − k ) , (3.34)which is an r × r matrix independent of s that can be obtained efficiently from U p .90 .2. ADAPTIVE LINEAR OPTICS The same reasoning with the general row expansion formula for the matrix V q ( q , s ), t and therows i = ( r , n − r ) givesPer (cid:179) V q ( q , s ), t (cid:180) = (cid:88) j ∈ { } n | j |= r Per (cid:183)(cid:179) V q ( q , s ), t (cid:180) i , j (cid:184) Per (cid:183)(cid:179) V q ( q , s ), t (cid:180) n − i , n − j (cid:184) = (cid:88) j ∈ { } n | j |= r Per (cid:179) V q ( q , m − k ),( j , m − n ) (cid:180) Per (cid:179) V q ( k , s ),( n − j , m − n ) (cid:180) , (3.35)where V q ( q , m − k ),( j , m − n ) is the matrix obtained from V q by repeating q l times the l th row for l ∈ {
1, . . . , k } and removing the others and by keeping only the l th columns such that j l = V q ( k , s ),( n − j , m − n ) is the matrix obtained from V q by repeating s l times the l th rowfor l ∈ { k +
1, . . . , m } and removing the others and by keeping only the l th columns such that j l =
0. Defining, for all j ∈ {
0, 1 } n such that | j | = r , B j : = V q ( q , m − k ),( j , m − n ) , (3.36)the expression in Eq. (3.35) rewritesPer (cid:179) V q ( q , s ), t (cid:180) = (cid:88) j ∈ { } n | j |= r Per (cid:179) B j (cid:180) Per (cid:179) V q ( k , s ),( n − j , m − n ) (cid:180) , (3.37)where B j are r × r matrices independent of s and can be obtained efficiently from V q .Plugging Eqs. (3.33) and (3.37) in Eq. (3.29) we obtain 〈 ψ p | ψ q 〉 = (cid:112) p ! q ! (cid:88) i , j ∈ { } n | i |=| j |= r (cid:183) Per (cid:179) A i (cid:180) Per (cid:179) B j (cid:180) × (cid:88) s ∈ Φ m − k , n − r s ! Per (cid:179) U p † ( n − i , m − n ),( k , s ) (cid:180) Per (cid:179) V q ( k , s ),( n − j , m − n ) (cid:180)(cid:184) . (3.38)The sum appearing in the second line may now be expressed as a single permanent usingthe permanent composition formula: for all i , j ∈ {
0, 1 } n such that | i | = | j | = r , let us definethe ( n − r ) × ( m − k ) matrix ˜ U p , i : = U p † ( n − i , m − n ),( k , m − k ) , (3.39)and the ( m − k ) × ( n − r ) matrix ˜ V q , j : = V q ( k , m − k ),( n − j , m − n ) , (3.40)so that U p † ( n − i , m − n ),( k , s ) = ˜ U p , i n − r , s and V q ( k , s ),( n − j , m − n ) = ˜ V q , js , n − r . (3.41)91 HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS
With the permanent composition formula in Eq. (3.23) we obtain (cid:88) s ∈ Φ m − k , n − r s ! Per (cid:179) U p † ( n − i , m − n ),( k , s ) (cid:180) Per (cid:179) V q ( k , s ),( n − j , m − n ) (cid:180) = Per (cid:183)(cid:179) ˜ U p , i ˜ V q , j (cid:180) n − r , n − r (cid:184) . (3.42)Since ˜ U p , i ˜ V q , j is an ( n − r ) × ( n − r ) matrix we thus have (cid:88) s ∈ Φ m − k , n − r s ! Per (cid:179) U p † ( n − i , m − n ),( k , s ) (cid:180) Per (cid:179) V q ( k , s ),( n − j , m − n ) (cid:180) = Per (cid:179) ˜ U p , i ˜ V q , j (cid:180) . (3.43)Then, Eq. (3.38) rewrites 〈 ψ p | ψ q 〉 = (cid:112) p ! q ! (cid:88) i , j ∈ { } n | i |=| j |= r Per (cid:179) A i (cid:180) Per (cid:179) B j (cid:180) Per (cid:179) C i , j (cid:180) , (3.44)where we have defined C i , j : = ˜ U p , i ˜ V q , j = U p † ( n − i , m − n ),( k , m − k ) V q ( k , m − k ),( n − j , m − n ) , (3.45)is an ( n − r ) × ( n − r ) matrix which can be obtained efficiently from U p and V q . (cid:4) By Lemma 3.2, the overlap is expressed as the modulus squared of a sum over (cid:161) nr (cid:162) products ofthree permanents, of square matrices of sizes | p | = r , | q | = r and ( n − r ), respectively. In the worstcase, when r = n /2, the sum has at most O (4 n ) terms, up to a polynomial factor in n . In particular,when n = O (log m ), the overlap reduces to a sum of a polynomial number of terms, which can allbe computed in time O (poly m ). Moreover, the cost of computing the overlap is independent of thenumber k of adaptive measurements, up to the cost of constructing the matrices with repeatedlines and columns (which is O (poly m )). The overlap of normalised ouput states is given by | 〈 ψ p | ψ q 〉 | 〈 ψ p | ψ p 〉 〈 ψ q | ψ q 〉 , (3.46)which may also be computed efficiently when n = O (log m ). The efficiency of the classical algorithmis summarised as a function of n and k in Table 3.2 and as a function of n and the number ofphotons r detected during the adaptive measurements in Table 3.3, where the regimes areobtained using Stirling equivalent n ! ∼ (cid:112) π n (cid:161) ne (cid:162) n .Since the quantum efficient regime corresponds to (cid:161) k + nn (cid:162) = O (poly m ), there is a possibility ofquantum advantage for overlap estimation when k = O (1) and n = O ( m ).In the case of probability estimation, the possible regimes for quantum advantage do not cor-respond to near-term implementations: k and n must be both greater than log m . However, foroverlap estimation, there is a possiblity of near-term beyond-classical computing with adaptivelinear optics using one adaptive measurement, which requires the preparation of photon number92 .3. THE COMPUTATIONAL POWER OF NON-GAUSSIAN STATES n k O (1) O (log m ) O ( m ) O (1) O (log m ) . . . O ( m ) . . . . . .Table 3.2: Simulability regimes for overlap estimation as a function of n and k . Since the runningtime is independent of k , the columns are the same. In blue is the parameter region for which theclassical algorithm is no longer efficient. The symbol . . . indicates regimes where the quantumalgorithm is not efficient. n r O (1) O (log n ) O ( n ) O (1) O (log m ) O ( m )Table 3.3: Simulability regimes for overlap estimation as a function of n and r . In blue is theparameter region for which the classical algorithm is efficient.states. Note that the interferometer should be concentrating many photons r onto the adap-tive measurement in order to obtain possibly hard to estimate overlaps. Using more adaptivemeasurements does not increase the complexity (apart from polynomial factors in m ).Having characterised these specific simulation regimes, we consider in what follows strongernotions of simulation. In particular, we give classical algorithms for strong simulation of a largeclass of continuous variable quantum computational models. Continuous variable systems are being recognized as a promising alternative to the use ofqubits, as they allow for the deterministic generation of unprecedented large entangled quantumstates, of up to one-million elementary systems [YUA + +
16] and also offer detectiontechniques, such as homodyne and heterodyne, with high efficiency and reliability (see sec-tion 1.4.2). Any given continuous variable quantum circuit is defined by (i) an input state lying inan infinite-dimensional Hilbert space, (ii) an evolution and (iii) measurements (see section 1.1.3).An important theorem [BSBN02, ME12] states that if all these elements are described by positiveWigner functions, then there exists a classical algorithm able to efficiently simulate this circuit.Hence, including a negative Wigner function element is mandatory in order to design a continuous93
HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS variable subuniversal quantum circuit that cannot be efficiently simulated by a classical device.Since Gaussian states and processes have positive Wigner functions, this necessarily correspondsto the use of non-Gaussian resources.Therefore, if one aims at minimal extensions of Gaussian models, three different familiesof non trivial quantum circuits can be defined, depending on whether the element yieldingthe Wigner function negativity is provided by the input state, the unitary evolution, or themeasurement.In what follows, we analyse the computational power of non-Gaussian states and thus focus onthe case where Gaussian circuits and measurements are supplemented with non-Gaussian inputstates as a computational resource. The results obtained have consequences for all three familiesof circuits, since non-Gaussian gates and non-Gaussian measurements can be implemented byGaussian operations together with non-Gaussian ancillary states [GKP01, GS07, SW18].
We first extend a few definitions from the previous chapter to the multimode case, using multi-index notations (see section 1.1.1). First, the stellar function, which provides a representation ofmultimode pure states as multivariate holomorphic functions:
Definition 3.5 (Multimode stellar function) . Let m ∈ (cid:78) ∗ and let | ψ 〉 = (cid:80) n ≥ ψ n | n 〉 ∈ H ⊗ m be anormalised pure state over m modes. The stellar function of the state | ψ 〉 is defined as F (cid:63) ψ ( z ) = e (cid:107) z (cid:107) 〈 z ∗ | ψ 〉 = (cid:88) n ≥ ψ n (cid:112) n ! z n , (3.47)for all z ∈ (cid:67) m , where | z 〉 = e − (cid:107) z (cid:107) (cid:80) n ≥ z n (cid:112) n ! | n 〉 ∈ H ⊗ m is the coherent state of amplitude z .The following definition also extends naturally from the single-mode case: Definition 3.6 (Multimode core state) . Multimode core states are defined as the normalised purequantum states which have a (multivariate) polynomial stellar function.Like in the single-mode case, these are the states with a finite support over the (multimode)Fock basis. For any m ∈ (cid:78) ∗ , the set of multimode core states over m modes is dense in the setof normalised states for the trace norm (by considering renormalised cutoff states). We alsointroduce the following definitions: Definition 3.7 (Degree of a multimode core state) . The degree of a multimode core state isdefined as the degree-sum of its stellar function.
Definition 3.8 (Support of a multimode core state) . The support of a multimode core state is theset of Fock basis elements which have nonzero overlap with the core state.94 .3. THE COMPUTATIONAL POWER OF NON-GAUSSIAN STATES
For example, the 3-mode core state (cid:112) ( | 〉 + | 〉 ) is of degree 3 and has a support of size 2,and its stellar function is given by z z /2 + z / (cid:112) z , z , z ) ∈ (cid:67) .We consider Gaussian circuits with Gaussian measurements, supplemented by non-Gaussianmultimode core states in input, which we refer to as G core circuits. Without loss of generality, aGaussian measurement may be written as a tensor product of single-mode balanced heterodynedetections preceded by a Gaussian unitary (see section 1.4.2). G core circuits are thus describedby two (multidimensional) parameters: a multimode core state | C 〉 in the input and a Gaussianunitary evolution ˆ G (Fig. 3.2). . . . het . . . n
1, . . . , 2 m } , i.e., the partitions of {
1, . . . , 2 m } in subsets of size 2. The hafnian of a matrix of odd size is 0. The hafnian is related to thepermanent by Haf (cid:195) (cid:48) m BB T (cid:48) m (cid:33) = Per ( B ), (3.49)for any m × m matrix B . By convention we set Haf ( (cid:59) ) =
1, where (cid:59) is a square matrix of size 0.The loop hafnian of a square matrix R = ( r i j ) ≤ i , j ≤ r of size r is defined as [BGQ19]lHaf ( R ) : = (cid:88) M ∈ SMP( r ) (cid:89) { i , j } ∈ M r i j , (3.50)95 HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS where the sum is over the single pair matchings of the set {
1, . . . , r } , defined as the set of perfectmatchings of a complete graph with loops with r vertices. This set is isomorphic to the set Π ( {
1, . . . , r } ) of partitions of {
1, . . . , r } in subsets of size 1 and 2 (by mapping a block { k } of size 1of a partition to the matching { k , k } and a block { i , j } of size 2 to the matching { i , j } ). In particular,when R is a matrix whose diagonal entries are all 0, we have lHaf ( R ) = Haf ( R ).We obtain a closed expression for the output probability density of Gaussian circuits with multi-mode core states input in Theorem 3.1, by adapting proof techniques from [HKS +
16, KHS + Lemma 3.3.
Let m ∈ (cid:78) ∗ , let V be a m × m symmetric matrix and let D be a column vector ofsize m. For all p , q ∈ (cid:78) m , there exists a square matrix A p , q ( V , D ) of size | p | + | q | such thatT p , q ( V , D ) : = (cid:90) β ∈ (cid:67) m exp (cid:183)
12 ˜ β T V ˜ β + D T ˜ β (cid:184) (cid:181) ∂∂ β (cid:182) p (cid:181) ∂∂ β ∗ (cid:182) q δ m ( β , β ∗ ) d m β d m β ∗ = ( − | p |+| q | lHaf (cid:163) A p , q ( V , D ) (cid:164) , (3.51) assuming the integral is well defined. The matrix A p , q ( V , D ) is obtained by repeating the entriesof V according to p and q and replacing the diagonal of the matrix obtained by the correspondingelements of D (a detailed example follows the proof). Proof.
Writing p = ( p , . . . , p m ) and q = ( q , . . . , q m ), we first get rid of the integral bysuccessive integration by parts: T p , q ( V , D ) = ( − | p |+| q | (cid:181) ∂∂ β (cid:182) p (cid:181) ∂∂ β ∗ (cid:182) q exp (cid:183)
12 ˜ β T V ˜ β + D T ˜ β (cid:184) (cid:175)(cid:175)(cid:175)(cid:175) ˜ β = = ( − | p |+| q | m (cid:89) j = (cid:181) ∂∂β j (cid:182) p j (cid:195) ∂∂β ∗ j (cid:33) q j exp (cid:183)
12 ˜ β T V ˜ β + D T ˜ β (cid:184) (cid:175)(cid:175)(cid:175)(cid:175) ˜ β = (3.52) = ( − | p |+| q | (cid:89) j ∈ E p , q (cid:195) ∂∂ ˜ β j (cid:33) exp (cid:183)
12 ˜ β T V ˜ β + D T ˜ β (cid:184) (cid:175)(cid:175)(cid:175)(cid:175) ˜ β = ,where the multiset E p , q is defined as the set of size | p | + | q | obtained from {
1, . . . , 2 m } byrepeating p k times the index k and q k times the index m + k , for all k ∈ {
1, . . . , m } .We make use of Faà di Bruno’s formula [Har06] in order to expand the product of partialderivatives and we obtain T p , q ( V , D ) = ( − | p |+| q | (cid:88) π ∈ Π ( E p , q ) (cid:89) B ∈ π (cid:195) ∂ | B | (cid:81) j ∈ B ∂ ˜ β j (cid:33) (cid:183)
12 ˜ β T V ˜ β + D T ˜ β (cid:184) (cid:175)(cid:175)(cid:175)(cid:175) ˜ β = , (3.53)where Π ( E p , q ) denotes the set of all partitions of the multiset E p , q , and where the productruns over the blocks B of the partition π ∈ Π ( E p , q ), with | B | the size of the block. The function˜ β † V ˜ β + D † ˜ β is a sum of a quadratic and a linear functions, so all derivatives of order greater96 .3. THE COMPUTATIONAL POWER OF NON-GAUSSIAN STATES than 2 in the sum vanish. We thus have T p , q ( V , D ) = ( − | p |+| q | (cid:88) π ∈ Π ( E p , q ) (cid:89) B ∈ π (cid:195) ∂ | B | (cid:81) j ∈ B ∂ ˜ β j (cid:33) (cid:183)
12 ˜ β T V ˜ β + D T ˜ β (cid:184) (cid:175)(cid:175)(cid:175)(cid:175) ˜ β = = ( − | p |+| q | (cid:88) π ∈ Π ( E p , q ) (cid:89) { i , j } ∈ π (cid:195) ∂ ∂ ˜ β i ∂ ˜ β j (cid:33) (cid:183)
12 ˜ β T V ˜ β + D T ˜ β (cid:184) (cid:175)(cid:175)(cid:175)(cid:175) ˜ β = × (cid:89) { k } ∈ π (cid:181) ∂∂ ˜ β k (cid:182) (cid:183)
12 ˜ β T V ˜ β + D T ˜ β (cid:184) (cid:175)(cid:175)(cid:175)(cid:175) ˜ β = , (3.54)where Π ( E p , q ) denotes the set of all partitions of the multiset E p , q in subsets of size 1 and2. All derivatives of order 2 of the linear term vanish, and all derivatives of order 1 of thequadratic term vanish when evaluated at ˜ β = . We thus obtain T p , q ( V , D ) = ( − | p |+| q | (cid:88) π ∈ Π ( E p , q ) (cid:89) { i , j } ∈ π (cid:195) ∂ ∂ ˜ β i ∂ ˜ β j (cid:33) (cid:183)
12 ˜ β T V ˜ β (cid:184) (cid:175)(cid:175)(cid:175)(cid:175) ˜ β = (cid:89) { k } ∈ π (cid:181) ∂∂ ˜ β k (cid:182) (cid:104) D T ˜ β (cid:105) (cid:175)(cid:175)(cid:175)(cid:175) ˜ β = .(3.55)Writing V = ( v i j ) ≤ i , j ≤ m , with V = V T , and D = ( d k ) ≤ k ≤ m we obtain T p , q ( V , D ) = ( − | p |+| q | (cid:88) π ∈ Π ( E p , q ) (cid:89) { i , j } ∈ π v i j (cid:89) { k } ∈ π d k . (3.56)We now show that this expression may be rewritten as the loop hafnian of a matrix ofsize | p | + | q | . Define V p , q the ( | p | + | q | ) × ( | p | + | q | ) matrix obtained from V by repeating p k times its k th rows and columns and q k times its ( m + k ) th rows and columns, for k ∈ {
1, . . . , m } . Similarly, define D p , q the column vector of size | p | + | q | obtained from D byrepeating p k times its k th element and q k times its ( m + k ) th element, for k ∈ {
1, . . . , m } .Finally, let A p , q ( V , D ) = ( a i j ) ≤ i , j ≤| p |+| q | be the ( | p | + | q | ) × ( | p | + | q | ) matrix obtained from V p , q by replacing its diagonal with the vector D p , q . Then, Eq. (3.56) rewrites T p , q ( V , D ) = ( − | p |+| q | (cid:88) π ∈ Π ( { | p |+| q | } ) (cid:89) { i , j } ∈ π a i j (cid:89) { k } ∈ π a kk = ( − | p |+| q | (cid:88) M ∈ SMP( | p |+| q | ) (cid:89) { i , j } ∈ M a i j = ( − | p |+| q | lHaf (cid:163) A p , q ( V , D ) (cid:164) , (3.57)where the sum in the first line is over the partitions of {
1, . . . , | p |+| q | } in subsets of size 1 and2, where the sum in the second line is over the single pair matchings of the set {
1, . . . , | p |+| q | } and where the third line comes from the definition of the loop hafnian in Eq. (3.50). (cid:4) Let us illustrate with an example how the matrix A p , q ( V , D ) appearing in Lemma 3.3 is con-structed from the matrix V and the vector D . Let us set m = p = (2, 0) and q = (1, 0). We97 HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS write V = v v v v v v v v v v v v v v v v and D = d d d d . (3.58)We first build the matrix V p , q by repeating p k times the k th rows and columns of V and q k timesthe ( m + k ) th rows and columns. In that case, p = ( p , p ) = (2, 0), so we repeat 2 times the firstrow and column and discard the second row and column, and q = ( q , q ) = (1, 0), so we keep thethird row and column and discard the fourth row and column, obtaining the 3 × V p , q = v v v v v v v v v . (3.59)Similarly, we obtain the vector D p , q by repeating p k times the k th element of D and q k times the( m + k ) th element, as D p , q = d d d . (3.60)Finally, we replace the diagonal of V p , q by D p , q : A p , q ( V , D ) = d v v v d v v v d . (3.61)Note that this construction by repeating rows and columns differ from the one encountered in theprevious section when dealing with the permanent of matrices, for which the first index denoteswhich rows are repeated and the second which columns. Here, we are dealing with hafnians ofmatrices of double size, where the first index denotes which rows and columns are repeated forindices in {
1, . . . , m } , while the second index denotes which rows and columns are repeated forindices in { m +
1, . . . , 2 m } . However, the two constructions coincide when looking at matrices ofthe form (cid:195) (cid:48) m BB T (cid:48) m (cid:33) , (3.62)through the relation in Eq. (3.49): Haf (cid:195) (cid:48) m BB T (cid:48) m (cid:33) = Per ( B ), (3.63)for any m × m square matrix B .Combining Lemma 3.3 with phase space formalism (see section 1.2) and properties of Gaussianstates (see section 1.3), we obtain the following result:98 .3. THE COMPUTATIONAL POWER OF NON-GAUSSIAN STATES Theorem 3.1.
Let m , n ∈ (cid:78) ∗ and let | C 〉 = (cid:88) p ∈ (cid:78) m | p |≤ n c p | p 〉 , (3.64) be an m-mode core state of degree n. Let ˆ G be a Gaussian unitary over m modes. For all α ∈ (cid:67) m , letus write V and ˜ d = ( d , d ∗ ) the covariance matrix and the displacement vector of the Gaussian state ˆ G † | α 〉 . Then, the output probability density for the G core circuit ˆ G with input | C 〉 and heterodynedetection, evaluated at α , is given byPr core [ α ] = κ ( α , ˆ G ) (cid:88) p , q ∈ (cid:78) m | p |≤ n , | q |≤ n ( − | p |+| q | (cid:112) p ! q ! c p c ∗ q lHaf (cid:161) A p , q (cid:162) , (3.65) where A p , q is the square matrix of size | p | + | q | obtained with Lemma 3.3 fromV = (cid:195) (cid:48) m m (cid:49) m (cid:48) m (cid:33) (cid:163) (cid:49) m − ( V + (cid:49) m /2) − (cid:164) and D = (cid:104) ˜ d † ( V + (cid:49) m /2) − (cid:105) T , (3.66) and where κ ( α , ˆ G ) = exp (cid:104) − ˜ d † ( V + (cid:49) m /2) − ˜ d (cid:105) π m (cid:112) Det ( V + (cid:49) m /2) (3.67) is a Gaussian prefactor. Proof.
The Gaussian circuit is composed of a Gaussian unitary ˆ G and balanced heterodynedetection. The output probability density reads, for all α = ( α , . . . , α m ) ∈ (cid:67) m ,Pr core [ α ] = Tr (cid:104) ˆ G | C 〉〈 C | ˆ G † Π α (cid:105) = π m Tr (cid:104) ˆ G † | α 〉〈 α | ˆ G | C 〉〈 C | (cid:105) = (cid:90) β ∈ (cid:67) m Q ˆ G † | α 〉〈 α | ˆ G ( β ) P | C 〉〈 C | ( β ) d m β d m β ∗ , (3.68)where Π α = π m | α 〉〈 α | is the POVM element corresponding to the heterodyne detection of α = ( α , . . . , α m ). The state ˆ G † | α 〉 is a Gaussian state: let V be its covariance matrix and d itsdisplacement vector. For all γ ∈ (cid:67) m , we write ˜ γ = ( γ , . . . , γ m , γ ∗ , . . . , γ ∗ m ). Then, for all β ∈ (cid:67) m , Q ˆ G † | α 〉〈 α | ˆ G ( β ) = π m (cid:112) Det ( V + (cid:49) m /2) exp (cid:183) −
12 ( ˜ β − ˜ d ) † ( V + (cid:49) m /2) − ( ˜ β − ˜ d ) (cid:184) = exp (cid:104) − ˜ d † ( V + (cid:49) m /2) − ˜ d (cid:105) π m (cid:112) Det ( V + (cid:49) m /2) exp (cid:183) −
12 ˜ β † ( V + (cid:49) m /2) − ˜ β + ˜ d † ( V + (cid:49) m /2) − ˜ β (cid:184) ,(3.69)99 HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS i.e., it is a Gaussian function which can be computed efficiently. On the other hand, we have | C 〉〈 C | = (cid:88) p , q ∈ (cid:78) m | p |≤ n , | q |≤ n c p c ∗ q | p 〉〈 q | , (3.70)so that P | C 〉〈 C | ( β ) = (cid:88) p , q ∈ (cid:78) m | p |≤ n , | q |≤ n c p c ∗ q P | p 〉〈 q | ( β ), (3.71)for all β ∈ (cid:67) m . Moreover we have, for all p , q ∈ (cid:78) m and all β ∈ (cid:67) m , P | p 〉〈 q | ( β ) = e (cid:107) β (cid:107) (cid:112) p ! q ! (cid:181) ∂∂ β (cid:182) p (cid:181) ∂∂ β ∗ (cid:182) q δ m ( β , β ∗ ) = e ˜ β † ˜ β (cid:112) p ! q ! (cid:181) ∂∂ β (cid:182) p (cid:181) ∂∂ β ∗ (cid:182) q δ m ( β , β ∗ ), (3.72)where δ m ( β , β ∗ ) = δ ( β ) · · · δ ( β m ) δ ( β ∗ ) · · · δ ( β ∗ m ). Combining Eqs. (3.69), (3.71) and (3.72) withEq. (3.68) we obtainPr core [ α ] = κ ( α , ˆ G ) (cid:88) p , q ∈ (cid:78) m | p |≤ n , | q |≤ n c p c ∗ q (cid:112) p ! q ! (cid:90) β ∈ (cid:67) m (cid:189) exp (cid:183) −
12 ˜ β † ( V + (cid:49) m /2) − ˜ β (cid:184) × exp (cid:104) ˜ d † ( V + (cid:49) m /2) − ˜ β (cid:105) e ˜ β † ˜ β (cid:181) ∂∂ β (cid:182) p (cid:181) ∂∂ β ∗ (cid:182) q δ m ( β , β ∗ ) (cid:190) d m β d m β ∗ , (3.73)where we have set κ ( α , ˆ G ) = exp (cid:104) − ˜ d † ( V + (cid:49) m /2) − ˜ d (cid:105) π m (cid:112) Det ( V + (cid:49) m /2) . (3.74)Given that ˜ β † = ˜ β T (cid:195) (cid:48) m m (cid:49) m (cid:48) m (cid:33) , (3.75)for all β ∈ (cid:67) m , the integral terms in Eq. (3.73) rewrite as (cid:90) β ∈ (cid:67) m exp (cid:183)
12 ˜ β T V ˜ β + D T ˜ β (cid:184) (cid:181) ∂∂ β (cid:182) p (cid:181) ∂∂ β ∗ (cid:182) q δ m ( β , β ∗ ) d m β d m β ∗ , (3.76)for | p | ≤ n and | q | ≤ n , where V = (cid:195) (cid:48) m m (cid:49) m (cid:48) m (cid:33) (cid:163) (cid:49) m − ( V + (cid:49) m /2) − (cid:164) (3.77)is a 2 m × m symmetric matrix, due to the initial structure of the covariance matrix, andwhere D = (cid:104) ˜ d † ( V + (cid:49) m /2) − (cid:105) T (3.78)100 .3. THE COMPUTATIONAL POWER OF NON-GAUSSIAN STATES is a column vector of size 2 m . By Lemma 3.3, the terms in Eq. (3.76) are equal to( − | p |+| q | lHaf (cid:161) A p , q (cid:162) , (3.79)where the square matrices A p , q of size | p | + | q | are obtained from V by repeating its entriesaccording to p and q and replacing the diagonal by the corresponding elements of D (see theexample following Lemma 3.3 for a detailed description of the construction). With Eq. (3.73)we finally obtain Pr core [ α ] = κ ( α , ˆ G ) (cid:88) p , q ∈ (cid:78) m | p |≤ n , | q |≤ n ( − | p |+| q | (cid:112) p ! q ! c p c ∗ q lHaf (cid:161) A p , q (cid:162) , (3.80)where κ ( α , ˆ G ) = exp (cid:104) − ˜ d † ( V + (cid:49) m /2) − ˜ d (cid:105) π m (cid:112) Det ( V + (cid:49) m /2) , (3.81)where V and d are the covariance matrix and the diplacement vector of the Gaussian stateˆ G † | α 〉 , respectively. (cid:4) When the input core state is a multimode Fock state, we refer to the corresponding subclass of G core circuits as G Fock circuits. In that case, the sum in Eq. (3.65) reduces to a single term andwe obtain the following expression:
Corollary 3.1.
Let m , n ∈ (cid:78) ∗ and let p = ( p , . . . , p m ) with | p | = n. Let ˆ G be a Gaussian unitaryover m modes. For all α ∈ (cid:67) m , let us write V and ˜ d = ( d , d ∗ ) the covariance matrix and thedisplacement vector of the Gaussian state ˆ G † | α 〉 . Then, the output probability density for the G Fock circuit ˆ G with Fock state input | p 〉 and heterodyne detection, evaluated at α , is given byPr Fock [ α ] = exp (cid:104) − ˜ d † ( V + (cid:49) m /2) − ˜ d (cid:105) p ! π m (cid:112) Det ( V + (cid:49) m /2) lHaf ( A p , p ), (3.82) where A p , p is the square matrix of size n obtained with Lemma 3.3 fromV = (cid:195) (cid:48) m m (cid:49) m (cid:48) m (cid:33) (cid:163) (cid:49) m − ( V + (cid:49) m /2) − (cid:164) and D = (cid:104) ˜ d † ( V + (cid:49) m /2) − (cid:105) T . (3.83) In this section, we use the expression obtained in Theorem 3.1 in order to study strong simulationof Gaussian circuits with few non-Gaussian elements. The first general result deals with general G core circuits, i.e., Gaussian circuits with multimode core state input.101 HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS
Theorem 3.2.
Let m ∈ (cid:78) ∗ and let | C 〉 be an m-mode core state of support size O (poly m ) anddegree n = O (log m ) . Then, G core circuits over m modes with input | C 〉 and heterodyne detectioncan be strongly simulated efficiently classically. Proof.
By Theorem 3.1, up to an efficiently computable prefactor, the output probabilitydensity is a sum of a polynomial number of loop hafnians, since the support size of the inputcore state is polynomial. The loop hafnian of a matrix of size r may be computed in time O ( r r /2 ) [BGQ19]. For | p | ≤ n and q ≤ n , the matrices A p , q appearing in Eq. (3.65) areefficiently computable square matrices of size | p | + | q | ≤ n , so for n = O (log m ), all the loophafnians may be computed in time O (poly m ). Hence, the output probability density can beevaluated in time O (poly m ).We now consider the evaluations of the marginal probability densities. Let k ∈ {
1, . . . , m − } ,for all α = ( α , . . . , α k ) ∈ (cid:67) k we havePr core [ α ] = Tr (cid:104) ˆ G | C 〉〈 C | ˆ G † ( Π α ⊗ (cid:49) m − k ) (cid:105) = π k Tr (cid:104) ˆ G † ( | α 〉〈 α | ⊗ (cid:49) m − k ) ˆ G | C 〉〈 C | (cid:105) (3.84) = π m − k (cid:90) β ∈ (cid:67) m Q ˆ G † ( | α 〉〈 α |⊗ (cid:49) m − k ) ˆ G ( β ) P | C 〉〈 C | ( β ) d m β d m β ∗ ,where Π α = π k | α , . . . , α k 〉〈 α , . . . , α k | is the POVM element corresponding to the heterodynedetection of ( α , . . . , α k ) over the first k modes. With Lemma 3.3 and the proof of Theorem 3.1,it is sufficient to show that Q ˆ G † ( | α 〉〈 α |⊗ (cid:49) m − k ) ˆ G is an efficiently computable Gaussian function inorder to prove that the marginal probability density can be evaluated efficiently.For all ( α , . . . , α k ) ∈ (cid:67) k and all ( γ , . . . , γ m − k ) ∈ (cid:67) m − k we write α = ( α , . . . , α k , 0, . . . , 0) ∈ (cid:67) m and γ = (0, . . . , 0, γ , . . . , γ m − k ) ∈ (cid:67) m so that α + γ = ( α , . . . , α k , γ , . . . , γ m − k ) ∈ (cid:67) m . Using theovercompleteness of coherent states we obtain, for all ( α , . . . , α k ) ∈ (cid:67) k and for all β ∈ (cid:67) m , π m − k Q ˆ G † ( | α 〉〈 α |⊗ (cid:49) m − k ) ˆ G ( β ) = (cid:90) γ = ( γ ,..., γ m − k ) ∈ (cid:67) m − k Q ˆ G † | α + γ 〉〈 α + γ | ˆ G ( β ) d m − k γ d m − k γ ∗ . (3.85)Let S and ˜ d = ( d , d ∗ ) be the symplectic matrix and the displacement vector associated withthe Gaussian unitary ˆ G † . The Gaussian stateˆ G † | α , . . . , α k , γ , . . . , γ m − k 〉 = ˆ G † | α + γ 〉 (3.86)is described by the covariance matrix V = SS † and the displacement vector S ( ˜ α + ˜ γ ) + ˜ d .Its Q function is thus given by Q ˆ G † | α + γ 〉〈 α + γ | ˆ G ( β ) = exp (cid:163) − ( ˜ β − S ( ˜ α + ˜ γ ) − ˜ d ) † ( V + (cid:49) m /2) − ( ˜ β − S ( ˜ α + ˜ γ ) − ˜ d ) (cid:164) π m (cid:112) Det ( V + (cid:49) m /2) , (3.87)102 .3. THE COMPUTATIONAL POWER OF NON-GAUSSIAN STATES for all ( α , . . . , α k ) ∈ (cid:67) k , for all ( γ , . . . , γ m − k ) ∈ (cid:67) m − k and for all β ∈ (cid:67) m . Let us discard theefficiently computable denominator and expand the product in the exponential. Writing M = ( V + (cid:49) m /2) − , we are left withexp (cid:183) −
12 ( ˜ β − S ˜ α − ˜ d ) † M ( ˜ β − S ˜ α − ˜ d ) (cid:184) · exp (cid:183) −
12 ˜ γ † S † MS ˜ γ + ( ˜ β − S ˜ α − ˜ d ) † MS ˜ γ (cid:184) , (3.88)The first exponential term is an efficiently computable Gaussian function which factors outof the integral in Eq. (3.85). Rewriting Eq. (3.85) up to this efficiently computable Gaussianfunction we are left with (cid:90) γ = (0,...,0, γ ,..., γ m − k ) ∈ (cid:67) m exp (cid:183) −
12 ˜ γ † S † MS ˜ γ + ( ˜ β − S ˜ α − ˜ d ) † MS ˜ γ (cid:184) d m − k γ d m − k γ ∗ = (cid:90) γ = ( γ ,..., γ m − k ) ∈ (cid:67) m − k exp (cid:183) −
12 ˜ γ T V ˜ γ + D T ˜ γ (cid:184) d m − k ) ˜ γ , (3.89)where V is the 2( m − k ) × m − k ) submatrix of (cid:195) (cid:48) m m (cid:49) m (cid:48) m (cid:33) S † MS (3.90)obtained by removing the rows and colums of indices l and m + l for l ∈ {
1, . . . , k } , and where D is the column vector of size 2( m − k ) obtained by removing the elements of (cid:104) ( ˜ β − S ˜ α − ˜ d ) † MS (cid:105) T (3.91)of indices l and m + l for l ∈ {
1, . . . , k } . The matrix V and the vector D are efficiently computable.Moreover, (cid:90) γ = ( γ ,..., γ m − k ) ∈ (cid:67) m − k exp (cid:183) −
12 ˜ γ T V ˜ γ + D T ˜ γ (cid:184) d m − k ) ˜ γ = (2 π ) m − k (cid:112) Det ( V ) exp (cid:183) D T V − D (cid:184) , (3.92)which is an efficiently computable Gaussian function of β .This implies that the value of the marginal probability density Pr [ α , . . . , α k ] may becomputed efficiently. Moreover, it is clear that this does not depent on the choice of k ∈ {
1, . . . , m − } and on the choice of the modes. Hence, all marginal probability densities maybe evaluated in time O (poly m ). (cid:4) This result has consequences for the simulability of various continuous variable quantum comput-ing models, in particular those based on Gaussian operations and photon additions or subtractions.We consider three examples in what follows: Interleaved Photon-Added Gaussian circuits (IPAG),Interleaved Photon-Subtracted Gaussian circuits (IPSG) and Gaussian circuits with input Fockstates ( G Fock ). 103
HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS
The stellar hierarchy of single-mode pure quantum states derived in the previous chapter detailsthe engineering of a single-mode quantum state from vacuum using unitary Gaussian operationsand single photon addition as a non-Gaussian operation. In particular, the states of finite stellarrank, which corresponds to the states that can be obtained from the vacuum using a finite numberof single photon additions or subtractions, are shown to be exactly the states that are obtained byapplying a Gaussian unitary operation to a single-mode core state (Theorem 2.4).As we will see here, the situation is different in the multimode case: we show that the setof states that can be obtained from a multimode core state with a multimode Gaussian unitaryoperation is strictly larger than the set of states that can be obtained from the vacuum usinga finite number of single photon additions and Gaussian unitary operations (Lemma 3.4). Wealso deduce strong simulability results for Gaussian sampling of the latter states. To that end,we consider the family of quantum circuits which sample from states in this set with productunbalanced heterodyne detection, which we refer to as Interleaved Photon-Added Gaussiancircuits (IPAG) due to their structure (Fig. 3.3). . . . | i
1, . . . , m } ,ˆ G ˆ a † k ˆ G † = d k + ( S λ † ) k = d k + m (cid:88) l = s k , l ˆ a † l + s k , m + l ˆ a l , (3.95)where ( S λ † ) k indicates the k th element of the column vector S λ † . Hence, commuting to the rightthe creation operators in Eq. (3.93), starting by the rightmost one, yieldsˆ G n ˆ a † . . . ˆ G ˆ a † ˆ G | 〉 ⊗ m = ˆ G n ˆ a † ˆ G . . . a † ˆ G ˆ G (cid:104) d (0)1 + ( S (0) λ ) (cid:105) | 〉 ⊗ m = . . . = ˆ G n . . . ˆ G (cid:104) d ( n − + ( S ( n − λ ) (cid:105) . . . (cid:104) d (0)1 + ( S (0) λ ) (cid:105) | 〉 ⊗ m , (3.96)where S ( k ) and d ( k ) implement the affine transformation corresponding to the action of ( ˆ G k ˆ G k − . . . ˆ G ) † ,for all k ∈ {
0, . . . , n − } . Writing ˆ G : = ˆ G n ˆ G n − . . . ˆ G , S ( k ) = ( s ( k ) i , j ) ≤ i , j ≤ m , and d ( k ) = ( d ( k )1 , . . . , d ( k ) m )for k ∈ {
0, . . . , n − } , we obtain the output stateˆ G | C IPAG 〉 , (3.97)where the state | C IPAG 〉 : = (cid:195) d ( n − + m (cid:88) l = s ( n − l ˆ a † l + s ( n − m + l ˆ a l (cid:33) . . . (cid:195) d (0)1 + m (cid:88) l = s (0)1, l ˆ a † l + s (0)1, m + l ˆ a l (cid:33) | 〉 ⊗ m (3.98)is a multimode core state of degree n (and not less, by property of symplectic matrices). Usingthis characterisation, we obtain the following result: Lemma 3.4.
The set of output states of IPAG circuits is strictly included in the set of output statesof G core circuits.
HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS
Proof.
The inclusion is immediate with Eq. (3.97). Up to the Gaussian unitary, it is sufficientto consider core states. To prove the strict inclusion, we show that the m -mode core state( | 〉 + | 〉 ) ⊗ | 〉 ⊗ m − (we omit normalisation), which has degree 2, is not a core state of theform of Eq. (3.98).By Eq. (3.98), all m -mode core states of IPAG circuits of degree 2 have the form (cid:195) d (1) + m (cid:88) k = s (1) k ˆ a † k + s (1) m + k ˆ a k (cid:33) (cid:195) d (0) + m (cid:88) l = s (0) l ˆ a † l + s (0)1, m + l ˆ a l (cid:33) | 〉 ⊗ m , (3.99)for some complex numbers d (0) , d (1) , s (0)1 , . . . , s (0)2 m , s (1)1 , . . . , s (1)2 m . This expression rewrites (cid:195) d (1) + m (cid:88) k = s (1) k ˆ a † k + s (1) m + k ˆ a k (cid:33) (cid:195) m (cid:88) l = s (0) l | l 〉 + d (0) | 〉 (cid:33) , (3.100)where for all l ∈ {
1, . . . , m } , we write l = (0, . . . , 0, 1, 0 . . . , 0), with a 1 at the l th position. Wefinally obtain (cid:112) m (cid:88) k = s (0) k s (1) k | k 〉+ m (cid:88) k , l = k (cid:54)= l s (0) k s (1) l | k + l 〉+ m (cid:88) k = (cid:179) d (1) s (0) k + d (0) s (1) k (cid:180) | k 〉+ (cid:195) d (0) d (1) + m (cid:88) k = s (0) k s (1) m + k (cid:33) | 〉 ,(3.101)where for all k ∈ {
1, . . . , m } , we write k = (0, . . . , 0, 2, 0 . . . , 0), with a 2 at the k th position. Onthe other hand we have ( | 〉 + | 〉 ) ⊗ | 〉 ⊗ m − = | 〉 + | 〉 . (3.102)In order for this core state to be of the form of Eq. (3.101) we must have s (0)1 s (1)1 (cid:54)= s (0) k s (1) l =
0, for k (cid:54)= l , (3.103)by considering the first and second terms of Eq. (3.101). This implies s (0) k = s (1) k = k (cid:54)= | 〉 in Eq. (3.101) is equal to 0, while it is nonzero in Eq. (3.102).Therefore the core state described by Eq. (3.102) cannot be generated by an IPAG circuit. (cid:4) In other words, the set of states that can be obtained from a multimode core state with amultimode Gaussian unitary operation is strictly larger than the set of states that can beobtained from the vacuum using a finite number of single photon additions and Gaussian unitaryoperations, unlike in the single mode case, where the two sets coincide.Another consequence of Eqs. (3.97) and (3.98) is the following result:106 .3. THE COMPUTATIONAL POWER OF NON-GAUSSIAN STATES
Lemma 3.5.
IPAG circuits over m modes with n = O (1) photon additions can be strongly simulatedefficiently classically. Proof.
When n = O (1), the support size of the core state | C IPAG 〉 in Eq. (3.98) is O (poly m )and its degree is O (1). Then, the result comes from a direct application of Theorem 3.2. (cid:4) When n = O (log m ) however, the support size of the core state is superpolynomial, so the classicalalgorithm is no longer efficient.Similarly, we can define Interleaved Photon-Subtracted Gaussian circuits (IPSG) by replacingphoton additions by subtractions in the definition of IPAG circuits. With the same reasoning weobtain the following result: Corollary 3.2.
IPSG circuits over m modes with n = O (1) photon subtractions can be stronglysimulated efficiently classically. Proof.
We use again the fact that Gaussian operations induce an affine transformation ofthe vector of annihilation and creation operators. Let ˆ G be an m -mode Gaussian operationwith symplectic matrix S and displacement vector d . Writing λ † = ˆ a † ...ˆ a † m ˆ a ...ˆ a m (3.104)and taking this time the adjoint of Eq. (3.95) we obtainˆ G ˆ a k ˆ G † = d ∗ k + ( S λ † ) † k = d ∗ k + m (cid:88) l = s ∗ k , l ˆ a l + s ∗ k , m + l ˆ a † l , (3.105)for all k ∈ {
1, . . . , m } . The same proof as for IPAG circuits shows that the output state of anIPSG circuit with n photon subtraction and Gaussian evolution ˆ G , . . . , ˆ G n readsˆ G | C IPSG 〉 , (3.106)107 HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS where ˆ G = ˆ G n . . . ˆ G and whereˆ G | C IPSG 〉 : = (cid:195) d ∗ ( n − + m (cid:88) l = s ∗ ( n − l ˆ a l + s ∗ ( n − m + l ˆ a † l (cid:33) . . . (cid:195) d ∗ (0)1 + m (cid:88) l = s ∗ (0)1, l ˆ a l + s ∗ (0)1, m + l ˆ a † l (cid:33) | 〉 ⊗ m ,(3.107)where S ( k ) = ( s ( k ) i , j ) ≤ i , j ≤ m and d ( k ) = ( d ( k )1 , . . . , d ( k ) m ) are the symplectic matrix and the dis-placement vector of ( ˆ G k ˆ G k − . . . ˆ G ) † , for all k ∈ {
0, . . . , n − } . When n = O (1), this core statehas support size O (poly m ) and degree O (1), and Theorem 3.2 concludes the proof. (cid:4) Note that the same reasoning also holds for Gaussian circuits interleaved with both photonadditions and subtractions.A particular subclass of IPAG circuits, where all the photon additions act at the beginning of thecircuit, is the class of G Fock circuits, i.e., Gaussian circuits with Fock state input. In that case,the input is a multimode core state of support size 1. With Corollary 3.1, we obtain the followingresult as an immediate consequence of Theorem 3.2:
Lemma 3.6.
Let m ∈ (cid:78) ∗ and let p ∈ (cid:78) m , such that | p | = O (log m ) . Then, G Fock circuits over mmodes with Fock state input | p 〉 and heterodyne detection can be strongly simulated efficientlyclassically. In other words, sampling with Gaussian measurements over m modes from n = O (log m ) in-distinguishable photons is strongly simulable classically. This contrasts with the case where m = O (poly n ): we show in the next section that strong simulation and even weak simulation ofsampling from n photons in m modes with Gaussian measurements is classically hard in thatcase. In the recent years, there has been an increasing interest in quantum circuits that definesubuniversal models of quantum computation [BJS10, AA13, MFF14, BMS16, FH16, DMK + + + .3. THE COMPUTATIONAL POWER OF NON-GAUSSIAN STATES was proven in [LLRK +
14, HKS + +
17, DMK + SP circuits), single photon-subtracted squeezed vacuum states (CVS PS circuits), or single photon-added squeezed vacuum states (CVS PA circuits), and the measurementis unbalanced heterodyne detection (see section 1.4.2), yielding a continuous variable outcome.These models are analog to the Boson Sampling model [AA13] and the Photon-Added or photon-Subtracted Squeezed Vacuum (PASSV) sampling model [OSM + G core circuits. Their architecture isinspired by recent experiments performed at Laboratoire Kastler Brossel (LKB), where mode-selective single photon subtraction from a collection of multimode squeezed states has beenrecently demonstrated [RJD + + c χ = cosh χ , s χ = sinh χ and t χ = tanh χ , for all χ ∈ (cid:82) . CVS PS circuits are defined formally as follows (see Fig. 3.4, CVS PA and CVS SP are defined analogously bychanging the non-Gaussian input states). Let m be the total number of optical modes. We recallthe definition of the squeezing operator with squeezing parameter ξ ∈ (cid:67) : ˆ S ( ξ ) = e ( ξ ˆ a † − ξ ∗ ˆ a ) . Werestrict to real squeezing parameters in what follows. In that case, ξ < p -squeezingwhile ξ > q -squeezing.The first n modes are single photon-subtracted squeezed vacuum states denoted by ˆ a | ξ 〉 ,where we omit the normalisation factor. The remaining m − n modes are squeezed vacuum states | ξ 〉 . We assume that the real squeezing parameter ξ is uniform over all the modes and does notdepend on the number of modes m . We require that n is even and that m = O (poly n ) ≥ n .The input modes undergo a passive linear evolution ˆ U that is described by an m × m unitarymatrix U of the form U = Oe i φ Σ (3.108)with φ ∈ (cid:82) , O ∈ O ( m ) and Σ ∈ O S ( m ), i.e., O is a real orthogonal matrix, and Σ is a real symmetricorthogonal matrix, and hence satisfies Σ = HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS . . .. . . . . .. . . | i
1, . . . , m } , the j th -mode measurementoutcome ( q j , p j ) falls into the boxes B ( q ) j = (cid:104) b ( q ) j η , ( b ( q ) j + η (cid:105) , B ( p ) j = (cid:104) b ( p ) j η , ( b ( p ) j + η (cid:105) . This prob-ability distribution is related to the real-valued probability density associated with CVS circuits,Pr CVS [ q , p , . . . , q m , p m ], byPr η CVS [ b ] = m (cid:89) j = (cid:90) B ( q ) j (cid:90) B ( p ) j Pr CVS [ q , p , . . . , q m , p m ] dq j d p j , (3.111)where q , p , . . . , q m , p m are the continuously distributed measurement outcomes of the productunbalanced heterodyne detection over m modes. This model of detection is equivalent to perfectheterodyne detection, followed by a binning of the outcome results performed at the stage ofpost-processing. We assume a resolution scaling with the number of modes as η ∼ − poly m .We prove that the probability distribution Pr η CVS [ b ] is hard to sample for a classical computer,both in the worst case scenario—i.e., weak simulation of all CVS circuits is hard—and in theaverage case scenario—i.e., weak simulation of a randomly chosen CVS circuit is hard—underthe assumption that the polynomial hierarchy does not collapse (see section 1.4.5 for a briefreview of the complexity classes appearing in this section). The argument adapts proof techniquesfrom [AA13, HKS +
16, LRKR17, CC17] and follows these lines:• We compute the expression Pr
CVS [ ] of the (continuous) probability density evaluated at = (0, . . . , 0) for a given CVS circuit. 111 HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS • We show that for any real matrix X , one can find a CVS circuit such that the expressionPr CVS [ ] is related to the square of the permanent of X by a multiplicative factor.• We show that a classical machine sampling efficiently from the (discrete) probabilitydistribution Pr η CVS [ b ] associated to this CVS circuit would allow us to approximate multi-plicatively the square of the permanent of X in the third level of the polynomial hierarchy,yielding a contradiction with the widely believed conjecture that the polynomial hierarchydoes not collapse. Lemma 3.7.
We consider a CVS circuit over m modes with n = p non-Gaussian input states,input squeezing ξ ∈ (cid:82) , evolution U = Oe i φ Σ and unbalanced heterodyne detection ζ ∈ (cid:82) . Then,Pr CVS [ ] = κ ( φ , ξ , ζ ) Haf ( Σ n ) , (3.112) where Σ n is the n × n top left submatrix of Σ and where κ ( φ , ξ , ζ ) = m /2 s n ζ sin n (2 φ ) π m (cid:163) + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:164) n + m /2 . (3.113) Proof.
CVS circuits are G Fock circuits. For a CVS circuit over m modes with n = p non-Gaussian input states, input squeezing ξ >
0, evolution U = Oe i φ Σ and unbalanced hetero-dyne detection ζ > | 〉 ⊗ n ⊗ | 〉 ⊗ m − n and thecorresponding Gaussian unitary evolution is given byˆ G = ˆ S † ( ζ ) ⊗ m ˆ U ˆ S ( ξ ) ⊗ m . (3.114)Let V be the covariance matrix of the Gaussian state ˆ G † | 〉 , its displacement vector being .By Corollary 3.1, the output probability density evaluated at (0, . . . , 0) is given byPr CVS [ ] = lHaf ( A n ) π m (cid:112) Det ( V + (cid:49) m /2) , (3.115)where A n is the square matrix of size 2 n obtained with Lemma 3.3 from V = (cid:195) (cid:48) m m (cid:49) m (cid:48) m (cid:33) (cid:163) (cid:49) m − ( V + (cid:49) m /2) − (cid:164) and D =
0, (3.116)by keeping only the k th and ( m + k ) th rows and columns of V for k ∈ {
1, . . . , n } and by replacingits diagonal entries by the corresponding elements of D . Now D =
0, and for a matrix whosediagonal entries are 0, the loop hafnian is equal to the hafnian. Hence,Pr
CVS [ ] = Haf ( A n ) π m (cid:112) Det ( V + (cid:49) m /2) . (3.117)We now derive the expression of the matrix A n in terms of the CVS circuit parameters:112 .3. THE COMPUTATIONAL POWER OF NON-GAUSSIAN STATES Lemma 3.8.
DefineB : = − s ξ c ζ + c ξ s ζ cos(2 φ )1 + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:49) m + i s ζ sin(2 φ )1 + c ξ c ζ − s ξ s ζ cos(2 φ ) Σ . (3.118) Then, A n = (cid:195) B ∗ n (cid:48) n (cid:48) n B n (cid:33) , (3.119) where B n is the n × n top left submatrix of B. Proof.
We show that V = (cid:195) B ∗ (cid:48) m (cid:48) m B (cid:33) . From Eq. (3.116) we have V = (cid:195) (cid:48) m m (cid:49) m (cid:48) m (cid:33) (cid:163) (cid:49) m − ( V + (cid:49) m /2) − (cid:164) , (3.120)where V is the covariance matrix of the Gaussian stateˆ S † ( ξ ) ⊗ m ˆ U † ˆ S ( ζ ) ⊗ m | 〉 〈 | ⊗ m ˆ S † ( ζ ) ⊗ m ˆ U ˆ S ( ξ ) ⊗ m . (3.121)This covariance matrix is given by (see section 1.3.3) V = S − ξ S U † S ζ V vac S † ζ S † U † S † − ξ , (3.122)where V vac = (cid:49) m /2 is the covariance matrix of the vacuum state over m modes, and S − ξ , S U † and S ζ are the symplectic matrices describing the action on the covariance matrixof the operators ˆ S † ( ξ ) ⊗ m , ˆ U † and ˆ S ( ζ ) ⊗ m , respectively. Using the notation c χ = cosh χ and s χ = sinh χ for all χ ∈ (cid:82) , we have S − ξ = (cid:195) c ξ (cid:49) m − s ξ (cid:49) m − s ξ (cid:49) m c ξ (cid:49) m (cid:33) , S U † = (cid:195) U T (cid:48) m (cid:48) m U † (cid:33) , S ζ = (cid:195) c ζ (cid:49) m s ζ (cid:49) m s ζ (cid:49) m c ζ (cid:49) m (cid:33) . (3.123)With Eq. (3.122) we obtain V = (cid:195) c ξ (cid:49) m − s ξ (cid:49) m − s ξ (cid:49) m c ξ (cid:49) m (cid:33) (cid:195) U T (cid:48) m (cid:48) m U † (cid:33) (cid:195) c ζ (cid:49) m s ζ (cid:49) m s ζ (cid:49) m c ζ (cid:49) m (cid:33) (cid:195) c ζ (cid:49) m s ζ (cid:49) m s ζ (cid:49) m c ζ (cid:49) m (cid:33) (cid:195) U ∗ (cid:48) m (cid:48) m U (cid:33) (cid:195) c ξ (cid:49) m − s ξ (cid:49) m − s ξ (cid:49) m c ξ (cid:49) m (cid:33) = (cid:195) c ξ c ζ U T − s ξ s ζ U † c ξ s ζ U T − s ξ c ζ U † − s ξ c ζ U T + c ξ s ζ U † − s ξ s ζ U T + c ξ c ζ U † (cid:33) (cid:195) c ξ c ζ U ∗ − s ξ s ζ U − s ξ c ζ U ∗ + c ξ s ζ Uc ξ s ζ U ∗ − s ξ c ζ U − s ξ s ζ U ∗ + c ξ c ζ U (cid:33) = (cid:195) [ c ξ c ζ − s ξ s ζ cos(2 φ )] (cid:49) m [ − s ξ c ζ + c ξ s ζ cos(2 φ )] (cid:49) m + is ζ sin(2 φ ) Σ [ − s ξ c ζ + c ξ s ζ cos(2 φ )] (cid:49) m − is ζ sin(2 φ ) Σ [ c ξ c ζ − s ξ s ζ cos(2 φ )] (cid:49) m (cid:33) ,(3.124)113 HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS where in the third line we used c χ + s χ = c χ , 2 c χ s χ = s χ , and c χ − s χ =
1, as well as U = Oe i φ Σ with O T O = (cid:49) m and Σ = (cid:49) m , so that U † U = U T U ∗ = (cid:49) m , U T U = cos(2 φ ) (cid:49) m + i sin(2 φ ) Σ and U † U ∗ = cos(2 φ ) (cid:49) m − i sin(2 φ ) Σ . The matrix V + (cid:49) m may thus be expressed as:12 (cid:195) [1 + c ξ c ζ − s ξ s ζ cos(2 φ )] (cid:49) m [ − s ξ c ζ + c ξ s ζ cos(2 φ )] (cid:49) m + is ζ sin(2 φ ) Σ [ − s ξ c ζ + c ξ s ζ cos(2 φ )] (cid:49) m − is ζ sin(2 φ ) Σ [1 + c ξ c ζ − s ξ s ζ cos(2 φ )] (cid:49) m (cid:33) .(3.125)With Eq. (3.120), we simply need to show that the inverse of the above matrix is (cid:49) m − (cid:195) (cid:48) m m (cid:49) m (cid:48) m (cid:33) (cid:195) B ∗ (cid:48) m (cid:48) m B (cid:33) = (cid:195) (cid:49) m − B − B ∗ (cid:49) m (cid:33) . (3.126)A tedious but straightforward matrix multiplication with Eq. (3.125) concludes the proof,using c χ − s χ = Σ = (cid:49) m . (cid:3) With Lemma 3.8 and Eq. (3.117) we havePr
CVS [ ] = π m (cid:112) Det ( V + (cid:49) m /2) Haf (cid:195) B ∗ n (cid:48) n (cid:48) n B n (cid:33) , (3.127)where B n = − s ξ c ζ + c ξ s ζ cos(2 φ )1 + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:49) n + i s ζ sin(2 φ )1 + c ξ c ζ − s ξ s ζ cos(2 φ ) Σ n , (3.128)with Σ n the n × n top left submatrix of Σ . Since the hafnian of a matrix does not depend onits diagonal entries, Eq. (3.127) can be rewritten asPr CVS [ ] = π m (cid:112) Det ( V + (cid:49) m /2) Haf (cid:34) s ζ sin(2 φ )1 + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:195) − i Σ n (cid:48) n (cid:48) n i Σ n (cid:33)(cid:35) = π m (cid:112) Det ( V + (cid:49) m /2) (cid:183) s ζ sin(2 φ )1 + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:184) n Haf (cid:195) − i Σ n (cid:48) n (cid:48) n i Σ n (cid:33) . (3.129)Now Haf ( M ⊕ N ) = Haf ( M ) Haf ( N ), so the previous expression yieldsPr CVS [ ] = π m (cid:112) Det ( V + (cid:49) m /2) (cid:183) s ζ sin(2 φ )1 + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:184) n Haf ( Σ n ) . (3.130)Finally, we compute Det ( V + (cid:49) m /2): Lemma 3.9.
Det ( V + (cid:49) m /2) = m (cid:163) + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:164) m . (3.131)114 .3. THE COMPUTATIONAL POWER OF NON-GAUSSIAN STATES Proof.
From the proof of Lemma 3.8 we have( V + (cid:49) m /2) − = (cid:195) (cid:49) m − B − B ∗ (cid:49) m (cid:33) , (3.132)so that Det ( V + (cid:49) m /2) = (cid:195) (cid:49) m − B − B ∗ (cid:49) m (cid:33) = (cid:49) m − BB ∗ ) . (3.133)Using the expression of the matrix B in Eq. (3.118) we obtain BB ∗ = (cid:181) − s ξ c ζ + c ξ s ζ cos(2 φ )1 + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:49) m + i s ζ sin(2 φ )1 + c ξ c ζ − s ξ s ζ cos(2 φ ) Σ (cid:182) × (cid:181) − s ξ c ζ + c ξ s ζ cos(2 φ )1 + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:49) m − i s ζ sin(2 φ )1 + c ξ c ζ − s ξ s ζ cos(2 φ ) Σ (cid:182) = (cid:163) − s ξ c ζ + c ξ s ζ cos(2 φ ) (cid:164) + s ζ sin (2 φ ) (cid:163) + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:164) (cid:49) m , (3.134)where we used Σ = (cid:49) m . Hence, with Eq. (3.133) we obtainDet ( V + (cid:49) m /2) = (cid:49) m − BB ∗ ) = (cid:183) − [ − s ξ c ζ + c ξ s ζ cos(2 φ ) ] + s ζ sin (2 φ ) [ + c ξ c ζ − s ξ s ζ cos(2 φ ) ] (cid:184) m = (cid:163) + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:164) m (cid:104)(cid:163) + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:164) − (cid:163) − s ξ c ζ + c ξ s ζ cos(2 φ ) (cid:164) − s ζ sin (2 φ ) (cid:105) m (3.135) = m (cid:163) + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:164) m . (cid:3) Combining Eq. (3.130) and Lemma 3.9, we finally obtainPr
CVS [ ] = m /2 s n ζ sin n (2 φ ) π m (cid:163) + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:164) n + m /2 Haf ( Σ n ) , (3.136)where n = p is the number of single photons in the input. (cid:4) Note that the matrix O appearing in the definition of the CVS circuit Eq. (3.108) does not115 HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS contribute to the output probability distribution. It provides additional degrees of freedom thatmay be useful for experimental considerations.Note also that the expression of the prefactor κ ( φ , ξ , ζ ) is left invariant when replacing ξ and ζ by − ξ and − ζ , which corresponds to changing which quadrature is squeezed both in input andoutput.In the case of CVS SP circuits—with single photons as non-Gaussian inputs—the squeezingparameter ξ is equal to 0 and we have the following result, using 1 + c ζ = c ζ and s ζ = c ζ s ζ : Corollary 3.3.
We consider a CVS SP circuit over m modes with n = p non-Gaussian input singlephoton states, evolution U = Oe i φ Σ and unbalanced heterodyne detection ζ ∈ (cid:82) . Then,Pr CVS SP [ ] = κ SP ( φ , ζ ) Haf ( Σ n ) , (3.137) where Σ n is the n × n top left submatrix of Σ and where κ SP ( φ , ζ ) = t n ζ sin n (2 φ ) π m c m ζ , (3.138) with t ζ = tanh ζ and c ζ = cosh ζ . Next, we relate the output probability density evaluated at (0, . . . , 0) of CVS circuits to thepermanent of real matrices. Specifically, we provide an explicit construction holding for any realsquare matrix X . Lemma 3.10.
Let n = p and let X ∈ (cid:82) p × p . For all m ≥ n and ν ≤ || X || there exists a matrix Σ X ∈ O S ( M ) such that its top left n × n submatrix is Σ Xn = ν (cid:195) XX T (cid:33) . (3.139) Proof.
Define Y = ν X . The matrix (cid:49) p − Y T Y is symmetric positive semidefinite since (cid:107) Y (cid:107) ≤ (cid:49) p − Y T Y = Z T Z for some square matrix Z . The columnsof the n × p matrix (cid:195) YZ (cid:33) (3.140)form an orthonormal family that can be completed into an orthonormal basis of (cid:82) n . Thematrix obtained with these columns is orthogonal by construction and reads (cid:195) Y CB T D (cid:33) , (3.141)116 .3. THE COMPUTATIONAL POWER OF NON-GAUSSIAN STATES where B , C , D are p × p matrices. Finally, with the constraint m ≥ n , setting Σ X = Y C Y T B B T D C T D T (cid:49) m − n (3.142)yields an m × m symmetric orthogonal matrix—its columns are orthonormal by construction—which top left n × n submatrix is precisely given by Eq. (3.139). (cid:4) Recall that a specific relation holds between the hafnian and the permanent. Namely, for anysquare matrix X , we have Per ( X ) = Haf (cid:195) XX T (cid:33) . (3.143)Using Lemma 3.7 with the matrix from Lemma 3.10, we get that for any square matrix X thereexists a CVS circuit CVS X which probability density at the origin reads:Pr CVS X [ ] = κ ( φ , ξ , ζ ) Haf (cid:179) Σ Xn (cid:180) = ν n κ ( φ , ξ , ζ ) (cid:34) Haf (cid:195) XX T (cid:33)(cid:35) = ν n κ ( φ , ξ , ζ ) Per ( X ) , (3.144)where ν ≤ (cid:107) X (cid:107) , and where κ ( φ , ξ , ζ ) = m /2 s n ζ sin n (2 φ ) π m (cid:163) + c ξ c ζ − s ξ s ζ cos(2 φ ) (cid:164) n + m /2 . (3.145)By Theorem 28 of [AA13], multiplicative approximation of Per ( X ) is a P -hard problem for realsquare matrices. Formally, for any g ∈ [1, poly n ], the following problem is P -hard: given a realmatrix X ∈ (cid:82) n × n such that 1/ (cid:107) X (cid:107) ≥ − poly(n) , output a nonnegative real number P X such thatPer ( X ) g ≤ P X ≤ g Per ( X ) . (3.146)The multiplying factor ν n κ ( φ , ξ , ζ ) in Eq. (3.144) is finite and non-vanishing for some values of ξ , ζ and φ , so we obtain the following result: Corollary 3.4.
For any g ∈ [1, poly n ] , the following problem is P -hard: given a real matrixX ∈ (cid:82) n × n such that (cid:107) X (cid:107) ≥ − poly(n) , output a nonnegative real number ˜ P X such thatPr CVS X [ ] g ≤ ˜ P X ≤ gPr CVS X [ ]. (3.147)117 HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS
This is because by construction ˜ P X ν n κ ( φ , ξ , ζ ) would then provide a multiplicative approximation ofPer ( X ) . As it turns out, this problem is easier to solve if one can perform weak simulation ofCVS circuits classically: Lemma 3.11.
Given access to a classical oracle which samples from the discretised outputprobability distribution of CVS circuits Pr η CVS of resolution η , for any g ∈ [1, poly n ] , the followingproblem can be solved in the third level of the polynomial hierarchy PH : given a real matrixX ∈ (cid:82) n × n , output a nonnegative real number ˜ P X such thatPr CVS X [ ] g ≤ ˜ P X ≤ gPr CVS X [ ]. (3.148)By classical oracle, we mean here an oracle that takes a uniformly random input string as its onlysource of randomness (it has no built-in randomness as a quantum machine would). Note thatwe consider a classical oracle sampling from the discretised output probability distribution ofCVS circuits Pr η CVS , rather than from the continuous probability density Pr
CVS . This is a strictlyweaker oracle since one may obtain samples from Pr η CVS using samples from Pr
CVS , with efficientclassical post-processing.
Proof.
With Eq. (3.111), the probability distribution for a CVS circuit with a finite resolutionof the heterodyne detection η ∼ − poly m , evaluated at (in a slight abuse of notation wedenote both the outcome and the corresponding discretised box by ), reads:Pr η CVS [ ] = m (cid:89) j = (cid:90) η q j = (cid:90) η p j = dq j d p j Pr CVS [ q , p , . . . , q m , p m ]. (3.149)Performing a Taylor expansion of the multivariate function x (cid:55)→ Pr CVS [ x ] around the value = (0, . . . , 0), we obtain Pr CVS [ x ] = (cid:88) γ ∈ (cid:78) m x γ γ ! ∂ γ Pr CVS [ ]. (3.150)Plugging this expression in Eq. (3.149) and integrating we getPr η CVS [ ] = η m (cid:88) γ ∈ (cid:78) m η | γ | ( γ + γ m + ∂ γ Pr CVS [ ] = η m Pr CVS [ ] + η m + (cid:88) γ ∈ (cid:78) m | γ |> η | γ |− ( γ + γ m + ∂ γ Pr CVS [ ], (3.151)so that Pr η CVS [ ] η m − Pr CVS [ ] = η (cid:88) γ ∈ (cid:78) m | γ |> η | γ |− ( γ + γ m + ∂ γ Pr CVS [ ]. (3.152)If η is small compared to Pr CVS [ ], a multiplicative approximation of Pr η CVS [ ] / η m thusyields a multiplicative approximation of Pr CVS [ ].118 .3. THE COMPUTATIONAL POWER OF NON-GAUSSIAN STATES We have m = poly n and | ζ | = Ω (2 − poly m ), by Eq. (3.109). When considering the cir-cuit CVS X associated to a real matrix X such that 1/ (cid:107) X (cid:107) ≥ − poly m , we have Pr CVS X [ ] = Ω (2 − poly m ) by Eq. (3.144). Hence, with η ∼ − poly m , a multiplicative approximation ofPr η CVS X [ ] / η m is a multiplicative approximation of Pr CVS X [ ].We use Stockmeyer’s approximate counting algorithm [Sto85] in order to conclude theproof: it is a classical algorithm which takes as input the classical description of a circuitsampling from a probability distribution and outputs a multiplicative approximation ofthe probability of a given outcome (see section 1.4.5). This algorithm sits in the third levelof the polynomial hierarchy PH and works as long as the probability to estimate is notsuperexponentially small, i.e., o (2 − poly m ) [LRKR17].We have Pr η CVS [ ] / η m = Ω (2 − poly m ), so with η ∼ − poly m the probability Pr η CVS [ ] isnot superexponentially small. Having at our disposal a classical oracle which samplesfrom the probability distribution Pr η CVS thus allows us to approximate multiplicativelythe probability Pr η CVS [ ] in the third level of the polynomial hierarchy, by making use ofStockmeyer’s algorithm. Dividing the estimate obtained by η m finally yields a multiplicativeapproximation of Pr CVS [ ] in PH (or rather in the class FPH of search problems that maybe solved by a PH machine). (cid:4) This result holds independently of the value of the squeezing parameter ξ , and when the detectionparameter ζ satisfies | ζ | = Ω (2 − poly m ), i.e., even when the detection is very close to a balancedheterodyne detection. When ζ =
0, however, the algorithm fails and the circuit is actually weaklysimulable classically, because the output probability density factorises into products of singlemode output probability densites, due to properties of balanced heterodyne detection. The sameproperty will allow us to derive an efficient verification protocol for Boson Sampling and CVScircuits in the next chapter.Combining Corollary 3.4 and Lemma 3.11 gives the main result of this section:
Theorem 3.3.
Sampling from the discretised output probability distribution of CVS circuits isclassically hard, or the polynomial hierarchy collapses to its third level.
Proof.
Assuming that sampling from the discretised output probability distribution of CVScircuits can be done efficiently classically, Corollary 3.4 and Lemma 3.11 imply P P ⊂ PH (where P P is the class of decision problems that can be solved efficiently using an oracle forthe class of counting problems P ). On the other hand, by Toda’s theorem [Tod91], PH ⊂ P P ,so that PH ⊂ PH , i.e., the polynomial hierarchy collapses to its third level. (cid:4) HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS
Theorem 3.3 implies that using enough non-Gaussian states as computational resources, weaksimulation of Gaussian circuits is no longer classically efficient. This contrast with Theorem 3.2from the previous section, i.e., the fact that strong simulation of Gaussian circuits with fewnon-Gaussian input states is classically efficient.This statement is a worst case statement, i.e., there exists at least one CVS X circuits which ishard to sample classically. In order to obtain an average case statement and identify a fraction ofhard to sample CVS circuits, we define the Real Gaussian Permanent Estimation problem:
Problem 1 (Real Gaussian Permanent Estimation) . Given as input a matrix X ∼ N (0, 1) p × p (cid:82) ofi.i.d. Gaussians together with error bounds (cid:178) , δ > , estimate Per ( X ) to within error ± (cid:178) · | Per ( X ) | ,with probability at least − δ over X , in poly ( p , 1/ (cid:178) , 1/ δ ) time. We can use the construction of Lemma 3.10 for the particular case of i.i.d. Gaussian matrices: forany X ∼ N (0, 1) p × p (cid:82) of i.i.d. Gaussians, we obtain a circuit CVS X such that Eq. ((3.144)) holds.Hence every instance of the RGPE is associated with a specific CVS circuit. In relation to theproblem above, we introduce the Permanent of Real Gaussians
Conjecture:
Conjecture 1 (Permanent of Real Gaussians) . RGPE is P -hard. We also introduce a second conjecture:
Conjecture 2 (Real Permanent Anti-Concentration) . There exists a polynomial P such that forall p and δ > , Pr X ∼ N (0,1) p × p (cid:82) (cid:34) | Per ( X ) | < (cid:112) p ! P ( p , 1/ δ ) (cid:35) < δ . (3.153)This problem and these conjectures are precisely the real version of the Gaussian Permanent Esti-mation problem and the
Permanent-of-Gaussians and
Permanent Anti-Concentration conjecturesintroduced in [AA13]. This leads us to our average case hardness result.
Theorem 3.4.
Assuming Conjecture 2 is true, classical circuits sampling from the (discretised)probability distribution of CVS circuits can be used to solve Real Gaussian Permanent Estimationin the third level of the polynomial hierarchy. Assuming Conjecture 1 is also true, an efficientclassical weak simulation of CVS X circuits, where X ∼ N (0, 1) p × p (cid:82) , would imply a collapse of thepolynomial hierarchy to its third level. Proof.
With the same proof as Lemma 3.11, with η = O (2 − poly m ), classical circuits samplingfrom the (discretised) probability distribution of CVS circuits can be used to obtain a mul-tiplicative approximation of Pr CVS [0, . . . , 0] in the third level of the polynomial hierarchy PH by means of Stockmeyer algorithm. In particular, for X ∼ N (0, 1) p × p (cid:82) a square matrixwhich entries are i.i.d. Gaussians and considering the circuit CVS X , we obtain multiplicativeapproximation of Per ( X ) . 120 .4. DISCUSSION AND OPEN PROBLEMS RGPE however refers to estimating Per ( X ) rather than Per ( X ) . It is easy to see that amultiplicative approximation of Per ( X ) can be turned into a multiplicative approximation | Per ( X ) | by taking the square root of the estimate. Then, in the case of real matrices, onlythe sign of the permanent remains to be determined.A more general version of this question has been addressed in [AA13] where they showedthat (the complex version of) Conjecture 2 allowed one to estimate the phase of Per ( X ) frommultiplicative approximation of | Per ( X ) | , for X i.i.d. complex Gaussian matrix. It impliesin particular that Conjecture 2 allows one to determine the sign of Per ( X ) from Per ( X ) if X is i.i.d. real Gaussian matrix. Hence, assuming Conjecture 2, RGPE can be solved in thethird level of the polynomial hierarchy using a classical circuit sampling from the outputprobability distribution of a CVS circuit as an oracle.Assuming Conjecture 1 is true, RGPE is P -hard. With the above, the existence of anefficient classical algorithm which approximates multiplicatively the output distribution ofCVS X circuits implies the existence of a classical algorithm sitting in the third level of thepolynomial hierarchy able to solve a P -hard problem. This in turn yields a collapse of thepolynomial hierarchy to the third level, thanks to Toda’s theorem [Tod91]. (cid:4) This result is an average case statement, i.e., it implies that a circuit CVS X , where X ∼ N (0, 1) p × p (cid:82) ,is hard to sample with high probability over X , assuming Conjectures 1 and 2 are true. Onceagain, we assumed the existence of a classical oracle sampling from the discretised outputprobability distribution of CVS circuits Pr η CVS , rather than the continuous probability densityPr
CVS . However, one may obtain samples from Pr η CVS using samples from Pr
CVS , with efficientclassical post-processing.
We have considered various notions of classical simulation and have studied the transition fromclassically simulable models to models that are universal for quantum computing for continuousvariables.We have studied the case of adaptive linear optics, an intermediate model between BosonSampling [AA13] and the Knill–Laflamme–Milburn scheme for universal quantum comput-ing [KLM01], obtaining classical algorithms for both probability estimation and overlap esti-mation and analysing their running times. The conclusion to be drawn from our study is thatachieving a quantum advantage for either probability estimation or overlap estimation usinglinear optics, input single photons and adaptive measurements, is challenging.A quantum advantage is not ruled out for probability estimation only if the number of adaptivemeasurements scale at least logarithmically in the size of the interferometer. The challenge121
HAPTER 3. BEYOND-CLASSICAL QUANTUM CONTINUOUS VARIABLE MODELS posed by the implementation of a quantum algorithm with adaptive linear optics for probabilityestimation beyond classical capabilities thus comes from the number of adaptive measurementsneeded.For overlap estimation, a quantum advantage is not ruled out for a constant number adaptivemeasurements, but many overlaps are easy to estimate classically in that case. It is only when asignificant fraction of the input photons is detected at the stage of the adaptive measurementsthat a quantum advantage becomes possible. The challenge posed by the implementation of aquantum algorithm with adaptive linear optics for overlap estimation beyond classical capabilitiesthus comes from the need of photon number-resolving detection and the preparation of manyphoton number states.For strong simulation, we have considered general Gaussian circuits with Gaussian measure-ments and non-Gaussian inputs and we have given sufficient conditions in terms of non-Gaussianresources for an efficient classical strong simulation. We have defined the G core circuits, abroad family of Gaussian circuits supplemented with non-Gaussian input states, where thenon-Gaussian states are multimode core states. We have identified various subclasses of thesecircuits:• The Interleaved Photon-Added Gaussian circuits (IPAG), which are circuits that samplewith Gaussian measurements from states which can be engineered from the vacuum usingmultimode Gaussian unitary operations and a finite number of photon additions.• The G Fock circuits, which are Gaussian circuits supplemented with Fock states in the input.• The CVS
P A /CVS PS /CVS SP circuits, which are specific interferometers with unbalancedheterodyne detection, supplemented with photon-added squeezed states/photon-addedsqueezed states/single photons in the input.The relation between these continuous variable quantum computational models is summarisedas CVS SP ⊂ CVS
P A = CVS PS ⊂ G Fock ⊂ IPAG ⊂ G core , (3.154)from the smallest class of circuits to the largest. The tools developped in this chapter also allowsus to consider Gaussian circuits supplemented with non-Gaussian states and photon counters, bywriting the photon counting POVM element as | n 〉〈 n | = n ! ( ˆ a † ) n | 〉〈 | ˆ a n , for n ∈ (cid:78) and commutingthe creation operators to the input through the Gaussian computation. Classical algorithmssimulating this type of computational model have been derived recently [QA20].For weak simulation, we have proven the computational hardness of a sampling problem thatstems from the family of CVS circuits, relating their discretised output probability density tothe permanent of real matrices. Introducing equivalent conjectures to those of [AA13] for realmatrices, we have extended the hardness result to an average case hardness.122 .4. DISCUSSION AND OPEN PROBLEMS With this collection of results comes various related open problems:One of the main outstanding problems is to prove the hardness of approximately samplingfrom CVS circuits. Following [AA13], this may involve making conjectures about anticoncen-tration and average case hardness of the loop hafnian rather than the permanent, as wellas collecting evidence and ultimately proving these conjectures. These conjectures have al-ready been extended from the permanent to the hafnian for the Gaussian Boson Samplingproposal [HKS +
16, KHS + m or is constant with respect to the number of modes, sincean exponentially small resolution is not experimentally realistic.Comparing more precisely IPAG and IPSG circuit families would give insight on the differ-ences between photon addition and photon subtraction in the multimode case.Whether the set of output states of IPAG circuits is dense in the set of all multimode states(the multimode equivalent of Lemma 2.6 from the previous chapter) is also an interesting question.In other words, is it possible to approximate with arbitrary precision (in trace distance) anymultimode quantum state using only single photon additions and Gaussian unitary operations?Another main open problem, which we solve in the next chapter, is the verification of theoutput of CVS circuits and Boson Sampling, necessary to a proper demonstration of quantumsupremacy with these computational models. 123 H A PT ER C ERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES O ut of the many properties featured by quantum physics, the impossibility to perfectlydetermine an unknown state [DY96] is specially interesting. This property is at the heartof quantum cryptography protocols such as quantum key distribution [BB84a]. On theother hand, it makes certification of the correct functioning of quantum devices a challenge, sincethe output of such devices can only be determined approximately, through repeated measurementsover numerous copies of the output states. The involved configurations spaces have enormousdimensions, a serious burden for any characterization. What is more, certification comes alongwith an ironic twist: it is highly non-trivial in light of the fact that certain quantum computationsare expected to exponentially outperform any attempt at classically solving the same problem.Determining an unknown state is difficult especially for continuous variable quantum states,which are described by possibly infinitely many complex parameters.In this chapter, after introducing known methods for the characterisation of continuousvariable quantum states, we develop new methods using heterodyne measurement in both thetrusted and untrusted settings.Firstly, based on quantum state tomography with heterodyne detection, we introduce areliable method for continuous variable quantum state certification, which directly yields theelements of the density matrix of the state considered with analytical confidence intervals. Thismethod requires neither mathematical reconstruction of the data nor discrete binning of thesample space, and uses a single Gaussian measurement setting, namely heterodyne detection.Secondly, beyond quantum state tomography and without its identical copies assumption,we promote our reliable tomography method to an efficient protocol for verifying single-modecontinuous variable pure quantum states with Gaussian measurements against fully maliciousadversaries, i.e., making no assumptions whatsoever on the state generated by the adversary.125 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES
Thirdly, we generalise the previous protocols to the multimode case and obtain efficientprotocols for verifying a large class of multimode continuous variable quantum states, with andwithout the identical copies assumption. In particular, we show how to efficiently verify theoutput state of a Boson Sampling experiment with a single-mode Gaussian measurement, thusenabling a proper demonstration of quantum supremacy with Boson Sampling.This chapter is based on [EHW +
20, CDG +
20, CRW +
20, CGKM20].
With rapidly developing quantum technologies for communication, simulation, computation andsensing, the ability to assess the correct functioning of quantum devices is of major importance,for near-term systems, the so-called noisy intermediate-scale quantum devices [Pre18], and forthe more sophisticated devices. Depending on the desired level of trust and in particular theassumptions one is ready to make, several methods are available for certifying the output of quan-tum devices [EHW + independent and identically distributed (i.i.d.) over various uses of the device. This implies inparticular that the conclusions drawn from test runs are also valid for future computational runswith the same device.In the following, the task of checking the output state of a quantum device is denoted tomography for state independent methods, when i.i.d. behaviour is assumed, certification for agiven a target state, when i.i.d. behaviour is assumed, and verification for a given target state,with no assumption whatsoever, and in particular without the i.i.d. assumption. Quantum state tomography [DPS03] is an important technique which aims at reconstructing agood approximation of the output state of a quantum device by performing multiple rounds ofmeasurements on several copies of said output states. Given an ensemble of identically preparedsystems, with measurement outcomes from the same observable, one can build up a histogram,from which a probability density can be estimated. According to Born’s rule, this probabilitydensity is the square modulus of the state coefficients, taken in the basis corresponding to themeasurement. However, a single measurement setting cannot yield the full state informationsince the phase of its coefficients are then lost. Many sets of measurements on many subensemblesmust be performed and combined to reconstruct the density matrix of the state. The data do notyield the state directly, but rather indirectly through data analysis. Quantum state tomographycommonly assumes an i.i.d. behaviour for the device, i.e., that the density matrix of the outputstate considered is the same at each round of measurement. This assumption may be relaxedwith a tradeoff in the efficiency of the protocol [CR12].126 .1. BUILDING TRUST FOR A CONTINUOUS VARIABLE QUANTUM STATE
A certification task corresponds to a setting where one wants to benchmark an industrialquantum device, or check the output of a physical experiment. On the other hand, a verificationtask corresponds to a cryptographic scenario, where the device to be tested is untrusted, or thequantum data is given by a potentially malicious party, for example in the context of delegatedquantum computing. In the latter case, the task of quantum verification is to ensure that eitherthe device behaved properly, or the computation aborts with high probability. While delegatedcomputing is a natural platform for the emerging quantum devices, one can provide a physicalinterpretation to this adversarial setting by emphasising that we aim for deriving verificationschemes that make no assumptions whatsoever about the noise model of the underlying systems.Various methods for verification of quantum devices have been investigated, in particular fordiscrete variable quantum information [GKK19], and they provide different efficiencies andsecurity parameters depending on the computational power of the verifier. The common featurefor all these approaches is to utilise some basic obfuscation scheme that allows one to reduce theproblem of dealing with a fully general noise model, or a fully general adversarial deviation ofthe device, to a simple error detection scheme [Vid18].For continuous variable quantum devices, checking that the output state is close to a targetstate may be done with linear optics using optical homodyne tomography [LR09]. This methodallows one to reconstruct the Wigner function of a generic state using only Gaussian measure-ments, namely homodyne detection. Because of the continuous character of its outcomes, onemust proceed to a discrete binning of the sample space, in order to build probability histograms.Then, the state representation in phase space is determined by a mathematical reconstruction.For cases where we have a specific target state, more efficient options are possible. For mul-timode Gaussian states, more efficient certification methods have been derived with Gaussianmeasurements [AGKE15]. These methods involve the computation of a fidelity witness, i.e., alower bound on the fidelity, from the measured samples. The cubic phase state certification proto-col of [LDT +
18] also introduces a fidelity witness, and is an example of certification of a specificnon-Gaussian state with Gaussian measurements, which assumes an i.i.d. state preparation. Theverification protocol for Gaussian continuous variable weighted hypergraph states of [TMM + We address two main issues in what follows. First, existing continuous variable state tomographymethods are not reliable in the sense of [CR12], because errors coming from the reconstructionprocedure are indistinguishable from errors coming from the data. Second, there is no Gaussianverification protocol for non-Gaussian states without i.i.d. assumption.We thus introduce a general receive-and-measure protocol for building trust for single-modecontinuous variable quantum states, using solely Gaussian measurements, namely heterodyne127
HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES
Het
Figure 4.1: A schematic representation of the protocol. The tester (within the dashed rectangle)receives a continuous variable quantum state ρ n over n subsystems. This state could be forexample the outcome of n successive runs of a physical experiment, the output of a commercialquantum device, or directly sent by some untrusted quantum server. The tester measures withheterodyne detection some of the subsystems of ρ n and uses the samples obtained and efficientclassical post-processing to deduce information about the remaining subsystems.detection (see section 1.4.2 and [FOP05, TMJ + heterodyne tomography in what follows. This tomography technique only requires a single fixedmeasurement setting, compared to homodyne tomography. This protocol also provides a means forcertifying single-mode continuous variable quantum states, under the i.i.d. assumption. Finally,the same protocol also allows us to verify single-mode continuous variable quantum states,without the i.i.d. assumption. For these three applications, the measurements performed arethe same. It is only the selection of subsystems to be measured and the classical post-processingperformed that differ from one application to another.The structure of the protocol is depicted in Fig. 6.1: given a quantum state ρ n over n subsys-tems, measure some of the subsystems with balanced heterodyne detection. Then, post-processthe samples obtained to retrieve information about the remaining subsystems. We show in thefollowing sections how this protocol may be used to perform reliable tomography, certification andverification of single-mode continuous variable quantum states, and we detail the correspondingchoices of subsystems and the classical post-processing for each task. In this section, we introduce a generalisation of the optical equivalence theorem for antinormalordering [CG69a], which provides an estimator for the expected value of an operator acting ona state with bounded support over the Fock basis, from samples of heterodyne detection of thestate. From this result, we derive various protocols in the following sections, ranging from statetomography to state verification.We denote by (cid:69) α ← D [ f ( α )] the expected value of a function f for samples drawn from a distribution128 .2. HETERODYNE ESTIMATOR D . Let us introduce for k , l ≥ L k , l ( z ) = e zz ∗ ( − k + l (cid:112) k ! (cid:112) l ! ∂ k + l ∂ z k ∂ z ∗ l e − zz ∗ , (4.1)for z ∈ (cid:67) , which are, up to a normalisation, the Laguerre 2D polynomials, appearing in par-ticular in the expressions of Wigner function of Fock states [Wün98]. For any operator A = (cid:80) +∞ k , l = A kl | k 〉〈 l | and all E ∈ (cid:78) , we define with these polynomials the function f A ( z , η ) = η e (cid:179) − η (cid:180) zz ∗ E (cid:88) k , l = A kl (cid:113) η k + l L k , l (cid:181) z (cid:112) η (cid:182) , (4.2)for all z ∈ (cid:67) , and all 0 < η <
1. We omit the dependency in E for brevity. The function z (cid:55)→ f A ( z , η ),being a polynomial multiplied by a converging Gaussian function, is bounded over (cid:67) . With thesame notations, we also define the following constant: K A = E (cid:88) k , l = | A kl | (cid:112) ( k + l +
1) . (4.3)
Theorem 4.1.
Let E ∈ (cid:78) and let < η < E . Let also A = (cid:80) +∞ k , l = A kl | k 〉〈 l | be an operator and let ρ = (cid:80) Ek , l = ρ kl | k 〉〈 l | be a density operator with bounded support. Then, (cid:175)(cid:175)(cid:175)(cid:175) Tr (cid:161) A ρ (cid:162) − (cid:69) α ← Q ρ [ f A ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175) ≤ η K A , (4.4) where the function f and the constant K are defined in Eqs. (4.2) and (4.3). The function f A defined in Eq. (4.2) is, up to a numerical factor of π , a bounded approximation ofthe Glauber–Sudarshan function P A of the operator A . This approximation is parametrised by aprecision η , and a cutoff value E . The optical equivalence theorem for antinormal ordering reads(see section 1.2 and [CG69a]) Tr ( A ρ ) = π (cid:90) α ∈ (cid:67) Q ρ ( α ) P A ( α ) d α . (4.5)Given that (cid:69) α ← Q ρ [ f A ( α , η )] = (cid:90) α ∈ (cid:67) Q ρ ( α ) f A ( α , η ) d α , (4.6)we would expect that (cid:69) α ← Q ρ [ f A ( α , η )] is an approximation of Tr ( A ρ ) parametrised by η and E .Theorem 4.1 makes this statement more precise. We prove this theorem in what follows. Proof.
With Eq. (4.2) we obtain (cid:175)(cid:175)(cid:175)(cid:175) Tr (cid:161) A ρ (cid:162) − (cid:69) α ← Q ρ [ f A ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) +∞ (cid:88) k , l = A lk Tr (cid:161) | l 〉〈 k | ρ (cid:162) − E (cid:88) k , l = A lk (cid:69) α ← Q ρ [ f | l 〉〈 k | ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) E (cid:88) k , l = A lk (cid:181) Tr (cid:161) | l 〉〈 k | ρ (cid:162) − (cid:69) α ← Q ρ [ f | l 〉〈 k | ( α , η )] (cid:182)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) (4.7) ≤ E (cid:88) k , l = | A lk | (cid:175)(cid:175)(cid:175)(cid:175) Tr (cid:161) | l 〉〈 k | ρ (cid:162) − (cid:69) α ← Q ρ [ f | l 〉〈 k | ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175) ,where we used in the second line the fact that ρ has a bounded support over the Fock basis.This shows that it is sufficient to prove the Theorem for A = | l 〉〈 k | , for all k , l from 0 to E . Wefirst introduce the following result: Lemma 4.1.
For all ≤ k , l ≤ E, (cid:69) α ← Q ρ [ f | l 〉〈 k | ( α , η )] = ρ kl + E (cid:88) m > k , n > lm − n = k − l ρ mn η m + n − k − l (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) mk (cid:33)(cid:195) nl (cid:33) . (4.8) Proof.
Let us fix k , l in 0, . . . , E . By Eqs. (4.1) and (4.2) we have, for all z ∈ (cid:67) , f | l 〉〈 k | ( z ) = (cid:181) η (cid:182) + k + l e (cid:179) − η (cid:180) zz ∗ L l , k (cid:181) z (cid:112) η (cid:182) = (cid:181) η (cid:182) + k + l e zz ∗ ( − k + l (cid:112) k ! (cid:112) l ! ∂ k + l ∂ u ∗ k ∂ u l e − uu ∗ (cid:175)(cid:175)(cid:175)(cid:175) u = z (cid:112) η = η e (cid:179) − η (cid:180) zz ∗ min( k , l ) (cid:88) p = ( − p (cid:112) k ! (cid:112) l ! p !( k − p )!( l − p )! (cid:181) η (cid:182) k + l − p z k − p z ∗ l − p . (4.9)Moreover, for all α ∈ (cid:67) , Q ρ ( α ) = π 〈 α | ρ | α 〉= π E (cid:88) m , n = ρ mn 〈 α | m 〉 〈 n | α 〉= π E (cid:88) m , n = ρ mn α ∗ m α n (cid:112) m ! n ! e −| α | . (4.10)Combining these expressions we obtain (cid:69) α ← Q ρ [ f | l 〉〈 k | ( α , η )] = (cid:90) α ∈ (cid:67) Q ρ ( α ) f | l 〉〈 k | ( α , η ) d α = π E (cid:88) m , n = ρ mn (cid:112) m ! n ! (cid:90) α ∈ (cid:67) α ∗ m α n e −| α | f | l 〉〈 k | ( α , η ) d α (4.11) = πη E (cid:88) m , n = ρ mn (cid:112) k ! (cid:112) l ! (cid:112) m ! (cid:112) n ! min( k , l ) (cid:88) p = ( − p p !( k − p )!( l − p )! (cid:181) η (cid:182) k + l − p (cid:90) α ∈ (cid:67) α k + n − p α ∗ ( l + m − p ) e − η | α | d α .130 .2. HETERODYNE ESTIMATOR Setting α = re i θ , we have d α = rdrd θ and the integral on the last line may be computed as (cid:90) α ∈ (cid:67) α k + n − p α ∗ ( l + m − p ) e − η | α | d α = +∞ (cid:90) r k + l + m + n − p + e − r η dr π (cid:90) e i ( k + n − l − m ) θ d θ = π (cid:179) k + l + m + n − p (cid:180) ! η k + l + m + n − p + for k − l = m − n ,0 for k − l (cid:54)= m − n , (4.12)where we used (cid:82) +∞ r t + e − r η = t ! η t + for t = k + l + m + n − p , which is obtained directly byinduction and integration by parts (note that for k − l = m − n , and p ≤ min( k , l ), we haveindeed t ∈ (cid:78) ). Hence, (cid:69) α ← Q ρ [ f | l 〉〈 k | ( α , η )] = E (cid:88) m , n = m − n = k − l ρ mn (cid:112) k ! (cid:112) l ! (cid:112) m ! (cid:112) n ! min( k , l ) (cid:88) p = ( − p (cid:179) k + l + m + n − p (cid:180) ! p !( k − p )!( l − p )! η m + n − k − l = E (cid:88) m , n = m − n = k − l ρ mn η m + n − k − l (cid:179) k + l + m + n (cid:180) ! (cid:112) m ! (cid:112) n ! (cid:112) k ! (cid:112) l ! min( k , l ) (cid:88) p = ( − p (cid:161) kp (cid:162)(cid:161) lp (cid:162)(cid:161) k + l + m + n p (cid:162) . (4.13)Now for k ≤ l we have, for all q ∈ (cid:78) (see, e.g., result 7.1 of [Gou72]), k (cid:88) p = ( − p (cid:161) kp (cid:162)(cid:161) lp (cid:162)(cid:161) qp (cid:162) = ( q − lk )( qk ) for q ≥ k + l ,0 for q < k + l . (4.14)When k ≤ l , Eq (4.13) thus yields (cid:69) α ← Q ρ [ f | l 〉〈 k | ( α , η )] = E (cid:88) m , n = m − n = k − lm + n ≥ k + l ρ mn η m + n − k − l (cid:179) k + l + m + n (cid:180) ! (cid:112) m ! (cid:112) n ! (cid:112) k ! (cid:112) l ! (cid:161) k + l + m + n − lk (cid:162)(cid:161) k + l + m + n k (cid:162) = E (cid:88) m ≥ k , n ≥ lm − n = k − l ρ mn η m + n − k − l (cid:112) m ! (cid:112) n ! (cid:112) k ! (cid:112) l ! (cid:179) k − l + m + n (cid:180) ! (cid:179) − k + l + m + n (cid:180) ! (cid:179) − k − l + m + n (cid:180) ! = E (cid:88) m ≥ k , n ≥ lm − n = k − l ρ mn η m + n − k − l (cid:112) m ! (cid:112) n ! (cid:112) k ! (cid:112) l ! (cid:112) ( m − k )! (cid:112) ( n − l )! (4.15) = E (cid:88) m ≥ k , n ≥ lm − n = k − l ρ mn η m + n − k − l (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) mk (cid:33)(cid:195) nl (cid:33) ,131 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES where we used that within the summation m − n = k − l . This formula is also valid for l ≤ k ,with the same reasoning. We finally obtain, for any k , l in 0, . . . , E (cid:69) α ← Q ρ [ f | l 〉〈 k | ( α , η )] = E (cid:88) m ≥ k , n ≥ lm − n = k − l ρ mn η m + n − k − l (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) mk (cid:33)(cid:195) nl (cid:33) = ρ kl + E (cid:88) m > k , n > lm − n = k − l ρ mn η m + n − k − l (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) mk (cid:33)(cid:195) nl (cid:33) . (4.16) (cid:3) Using Lemma 4.1, we obtain (cid:175)(cid:175)(cid:175)(cid:175)
Tr ( | l 〉〈 k | ρ ) − (cid:69) α ← Q ρ [ f | l 〉〈 k | ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(cid:175) ρ kl − (cid:69) α ← Q ρ [ f | l 〉〈 k | ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) E (cid:88) m > k , n > lm − n = k − l ρ mn η m + n − k − l (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) mk (cid:33)(cid:195) nl (cid:33) (cid:175)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ E (cid:88) m > k , n > lm − n = k − l | ρ mn | η m + n − k − l (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) mk (cid:33)(cid:195) nl (cid:33) = E − max( k , l ) (cid:88) s = | ρ s + k , s + l | η s (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) s + kk (cid:33)(cid:195) s + ll (cid:33) ≤ E − max( k , l ) (cid:88) s = η s (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) s + kk (cid:33)(cid:195) s + ll (cid:33) (cid:112) ρ s + k , s + k (cid:112) ρ s + l , s + l , (4.17)where we set s = m − k = n − l = m + n − k − l in the third line, and where we used | ρ s + k , s + l | ≤(cid:112) ρ s + k , s + k (cid:112) ρ s + l , s + l in the last line, since ρ is a positive semidefinite matrix. In order to obtainan upper bound independent of ρ , we now show for all s that η s (cid:113)(cid:161) s + kk (cid:162)(cid:161) s + ll (cid:162) ≤ η (cid:112) ( k + l + η ≤ E . For all k , l in 0, . . . , E and for all s in 2, . . . , E − max( k , l ), we have (cid:112) s + k (cid:112) s + ls ≤ E s in 2, . . . , E − max( k , l ) η s (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) s + kk (cid:33)(cid:195) s + ll (cid:33) = η (cid:112) ( s + k )( s + l ) s η s − (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) s − + kk (cid:33)(cid:195) s − + ll (cid:33) ≤ η E η s − (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) s − + kk (cid:33)(cid:195) s − + ll (cid:33) (4.19)132 .2. HETERODYNE ESTIMATOR ≤ η s − (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) s − + kk (cid:33)(cid:195) s − + ll (cid:33) ,since we assumed η ≤ E . Hence by induction, for all s in 2, . . . , E − max( k , l ), η s (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) s + kk (cid:33)(cid:195) s + ll (cid:33) ≤ η (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) + kk (cid:33)(cid:195) + ll (cid:33) = η (cid:112) ( k + l +
1) . (4.20)Combining this with Eq. (4.17) yields (cid:175)(cid:175)(cid:175)(cid:175)
Tr ( | l 〉〈 k | ρ ) − (cid:69) α ← Q ρ [ f | l 〉〈 k | ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175) ≤ η (cid:112) ( k + l + E − max( k , l ) (cid:88) s = (cid:112) ρ s + k , s + k (cid:112) ρ s + l , s + l ≤ η (cid:112) ( k + l + (cid:118)(cid:117)(cid:117)(cid:116) E − max( k , l ) (cid:88) s = ρ s + k , s + kE − max( k , l ) (cid:88) s = ρ s + l , s + l ≤ η (cid:112) ( k + l +
1) , (4.21)for all k , l in 0, . . . , E , where we used Cauchy-Schwarz inequality and the fact that Tr ( ρ ) = E → +∞ . Together with Eq. (4.7) we obtain (cid:175)(cid:175)(cid:175)(cid:175) Tr (cid:161) A ρ (cid:162) − (cid:69) α ← Q ρ [ f A ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175) ≤ η E (cid:88) k , l = | A kl | (cid:112) ( k + l + = η K A , (4.22)by Eq. (4.3). (cid:4) This result provides an estimator for the expected value of any operator A acting on a continuousvariable state ρ with bounded support over the Fock basis. This estimator is the expectedvalue of a bounded function f A over samples drawn from the Husimi Q function of ρ . Thisprobability density corresponds to a Gaussian measurement of ρ , namely heterodyne detection(see section 1.4.2). The right hand side of Eq. (4.4) is an energy bound, which depends on theoperator A and the value E .When the operator A is the density matrix of a continuous variable pure state | ψ 〉 , theprevious estimator approximates the fidelity F ( ψ , ρ ) = 〈 ψ | ρ | ψ 〉 between | ψ 〉〈 ψ | and ρ . With thesame notations: Corollary 4.1.
Let E ∈ (cid:78) and let < η < E . Let also | ψ 〉〈 ψ | = (cid:80) +∞ k , l = ψ k ψ ∗ l | k 〉〈 l | be a normalisedpure state and let ρ = (cid:80) Ek , l = ρ kl | k 〉〈 l | be a density operator with bounded support. Then, (cid:175)(cid:175)(cid:175)(cid:175) F (cid:161) ψ , ρ (cid:162) − (cid:69) α ← Q ρ [ f ψ ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175) ≤ η K ψ ≤ η E + E + where the function f A and the constant K A are defined in Eqs. (4.2) and (4.3), for A = | ψ 〉〈 ψ | . HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES
Proof.
We apply Theorem 4.1 for A = | ψ 〉〈 ψ | a pure state. We obtain (cid:175)(cid:175)(cid:175)(cid:175) 〈 ψ | ρ | ψ 〉 − (cid:69) α ← Q ρ [ f ψ ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175) ≤ η K ψ = η E (cid:88) k , l = | ψ k ψ l | (cid:112) ( k + l + = η (cid:195) E (cid:88) n = | ψ n |(cid:112) n + (cid:33) (4.24) ≤ η E (cid:88) n = | ψ n | E (cid:88) n = ( n + ≤ η E + E + (cid:80) En = | ψ n | ≤ Tr ( | ψ 〉〈 ψ | ) =
1. Since | ψ 〉 is apure state, we have F ( ψ , ρ ) = 〈 ψ | ρ | ψ 〉 , which concludes the proof. (cid:4) This result provides an estimator for the fidelity between any target pure state | ψ 〉 and anycontinuous variable (mixed) state ρ with bounded support over the Fock basis. This estimator isthe expected value of a bounded function f ψ over samples drawn from the probability densitycorresponding to heterodyne detection of ρ . The right hand side of Eq. (4.23) is an energybound, which may be refined depending on the expression of | ψ 〉 . In particular, the second boundis independent of the target state | ψ 〉 . The assumption of bounded support makes sense fortomography, where the energy range of the measured state is known, but not necessarily in amore adversarial setting.Given these results, one may choose a target pure state | ψ 〉 and measure with heterodynedetection various copies of the output (mixed) state ρ of a quantum device with bounded supportover the Fock basis. Then, using the samples obtained, one may estimate the expected value of f ψ , thus obtaining an estimate of the fidelity between the states | ψ 〉〈 ψ | and ρ . Using this result,we introduce a reliable method for performing continuous variable quantum state tomographyusing heterodyne detection. Continuous variable quantum state tomography methods usually make two assumptions: firstlythat the measured states are independent identical copies (i.i.d. assumption, for independentlyand identically distributed ), and secondly that the measured states have a bounded supportover the Fock basis [LR09]. With the same assumptions, we present a reliable method forstate tomography with heterodyne detection which has the advantage of providing analyticalconfidence intervals. Our method directly provides estimates of the elements of the state density134 .3. RELIABLE HETERODYNE TOMOGRAPHY matrix, phase included. As such, neither mathematical reconstruction of the phase, nor binningof the sample space is needed, since the samples are used only to compute expected valuesof bounded functions. Moreover, only a single fixed Gaussian measurement setting is needed,namely heterodyne detection (Fig. 1.4).The law of large numbers ensures that the sample average from independently and identicallydistributed (i.i.d.) random variables converges to the expected value of these random variables,when the number of samples goes to infinity. The following key lemma refines this statement andquantifies the speed of convergence:
Lemma 4.2. (Hoeffding inequality)
Let λ > , let n ≥ , let z , . . . , z n be i.i.d. complex randomvariables from a probability density D over (cid:82) , and let f : (cid:67) (cid:55)→ (cid:82) such that | f ( z ) | ≤ M, for M > and all z ∈ (cid:67) . Then Pr (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) n n (cid:88) i = f ( z i ) − (cid:69) z ← D [ f ( z )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≥ λ (cid:35) ≤ (cid:183) − n λ M (cid:184) . (4.25)This comes directly from Hoeffding inequality [Hoe63] applied to the real bounded i.i.d. randomvariables f ( z ), . . . , f ( z N ). When dealing with complex random variables, we use the followingresult instead: Lemma 4.3. (Hoeffding inequality for complex random variables)
Let λ > , let n ≥ , letz , . . . , z n be i.i.d. complex random variables from a probability density D over (cid:67) , and let f : (cid:67) (cid:55)→ (cid:67) such that | f ( z ) | ≤ M, for M > and all z ∈ (cid:67) . Then Pr (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) n n (cid:88) i = f ( z i ) − (cid:69) z ← D [ f ( z )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≥ λ (cid:35) ≤ (cid:183) − n λ M (cid:184) . (4.26) Proof.
For all a > z ∈ (cid:67) , | z | = (cid:112) ℜ ( z ) + ℑ ( z ) ≥ a implies |ℜ ( z ) | ≥ a / (cid:112) |ℑ ( z ) | ≥ a / (cid:112) | z | ≥ a ] ≤ Pr (cid:183) |ℜ ( z ) | ≥ a (cid:112) (cid:184) + Pr (cid:183) |ℑ ( z ) | ≥ a (cid:112) (cid:184) , (4.27)so applying twice Lemma 4.2 for the real random variables ℜ ( f ( z )) and ℑ ( f ( z )), respectively,yields Lemma 4.3. (cid:4) For tomographic application, all copies of the state are measured. For n ≥
1, let α , . . . , α n ∈ (cid:67) besamples from heterodyne detection of n copies of a quantum state ρ . For (cid:178) > k , l ∈ (cid:78) , wedefine ρ (cid:178) kl = n n (cid:88) i = f | l 〉〈 k | (cid:181) α i , (cid:178) K | l 〉〈 k | (cid:182) , (4.28)135 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES where the function f A and the constant K A are defined in Eqs. (4.2) and (4.3), for A = | l 〉〈 k | , andwhere (cid:178) > ρ (cid:178) kl is the average of the function f | l 〉〈 k | over thesamples α , . . . , α n . The next result shows that this estimator approximates the matrix element k , l of this state with high probability. We use the notations of Theorem 4.1. Theorem 4.2 (Reliable heterodyne tomography) . Let (cid:178) , (cid:178) (cid:48) > , let n ≥ , and let α , . . . , α n besamples obtained by measuring with heterodyne detection n copies of a state ρ = (cid:80) Ek , l = ρ kl | k 〉〈 l | with bounded support, for E ∈ (cid:78) . Then (cid:175)(cid:175) ρ kl − ρ (cid:178) kl (cid:175)(cid:175) ≤ (cid:178) + (cid:178) (cid:48) , (4.29) for all ≤ k , l ≤ E, with probability greater than − (cid:88) ≤ k ≤ l ≤ E exp (cid:183) − n (cid:178) + k + l (cid:178) (cid:48) C kl (cid:184) , (4.30) where the estimate ρ (cid:178) kl is defined in Eq. (4.28), and whereC kl : = [( k + l + + k + l | l − k | (cid:195) max ( k , l )min ( k , l ) (cid:33) (4.31) is a constant independent of ρ . Proof.
In order to prove Theorem 4.2, we apply Lemma 4.3 to the functions z (cid:55)→ f | l 〉〈 k | ( z , η )defined in Eq. (4.2). We first bound these functions: Lemma 4.4.
For all k , l ≥ , defineM kl : = (cid:118)(cid:117)(cid:117)(cid:116) | l − k | (cid:195) max ( k , l )min ( k , l ) (cid:33) . (4.32) Then for all k , l and all z ∈ (cid:67) , (cid:175)(cid:175) f | k 〉〈 l | ( z , η ) (cid:175)(cid:175) ≤ M kl η + k + l . (4.33) Proof.
For k or l > E the inequality is trivial. For all k , l ≤ E and all z ∈ (cid:67) , (cid:175)(cid:175) f | k 〉〈 l | ( z , η ) (cid:175)(cid:175) = (cid:181) η (cid:182) + k + l e (cid:179) − η (cid:180) | z | (cid:175)(cid:175)(cid:175)(cid:175) L k , l (cid:181) z (cid:112) η (cid:182)(cid:175)(cid:175)(cid:175)(cid:175) = η e (cid:179) − η (cid:180) | z | (cid:112) k ! (cid:112) l ! (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) min( k , l ) (cid:88) p = ( − p k ! l ! p !( k − p )!( l − p )! 1 η k + l − p z l − p z ∗ ( k − p ) (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) , (4.34)136 .3. RELIABLE HETERODYNE TOMOGRAPHY where we used Eq. (4.1). Now for all z ∈ (cid:67) ∗ and all a > (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) min( k , l ) (cid:88) p = ( − p k ! l ! p !( k − p )!( l − p )! a k + l − p z l − p z ∗ ( k − p ) (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) = a k l ! | z | k − l (cid:175)(cid:175)(cid:175) L ( k − l ) l (cid:161) a | z | (cid:162)(cid:175)(cid:175)(cid:175) = a l k ! | z | l − k (cid:175)(cid:175)(cid:175) L ( l − k ) k (cid:161) a | z | (cid:162)(cid:175)(cid:175)(cid:175) , (4.35)where L ( α ) n ( x ) = n (cid:88) q = ( − q q ! (cid:195) n + α n − q (cid:33) x q (4.36)are the generalised Laguerre polynomials [AS65], defined for α ∈ (cid:82) and n ∈ (cid:78) . Plugging thisrelation into Eq. (4.34) we obtain (cid:175)(cid:175) f | k 〉〈 l | ( z , η ) (cid:175)(cid:175) = e (cid:179) − η (cid:180) | z | | z | l − k η + l (cid:112) k ! (cid:112) l ! (cid:175)(cid:175)(cid:175)(cid:175) L ( l − k ) k (cid:181) | z | η (cid:182)(cid:175)(cid:175)(cid:175)(cid:175) = e (cid:179) − η (cid:180) | z | | z | k − l η + k (cid:112) l ! (cid:112) k ! (cid:175)(cid:175)(cid:175)(cid:175) L ( k − l ) l (cid:181) | z | η (cid:182)(cid:175)(cid:175)(cid:175)(cid:175) , (4.37)for all z ∈ (cid:67) . The generalised Laguerre polynomials are bounded as [Roo85] (cid:175)(cid:175)(cid:175) L ( α ) n ( x ) (cid:175)(cid:175)(cid:175) ≤ Γ ( n + α + n ! Γ ( α + e x , (4.38)for all x ≥
0, all α ≥ n ∈ (cid:78) , and as (cid:175)(cid:175)(cid:175) L ( α ) n ( x ) (cid:175)(cid:175)(cid:175) ≤ − α e x , (4.39)for all x ≥
0, all α ≤ − and all n ∈ (cid:78) .Let a >
0. Assuming k < l , we have | z | l − k ≤ a l − k for | z | ≤ a , and | z | k − l ≤ a k − l for | z | ≥ a . Thus,the first line of Eq. (4.37), together with Eq. (4.38), give (cid:175)(cid:175) f | k 〉〈 l | ( z , η ) (cid:175)(cid:175) ≤ e (cid:179) − η (cid:180) | z | a l − k η + l (cid:112) k ! (cid:112) l ! l ! k !( l − k )! e | z | η ≤ a l − k η + l (cid:112) l !( l − k )! (cid:112) k ! , (4.40)for | z | ≤ a and k < l . Similarly, the second line of Eq. (4.37), together with Eq. (4.39), give (cid:175)(cid:175) f | k 〉〈 l | ( z , η ) (cid:175)(cid:175) ≤ e (cid:179) − η (cid:180) | z | a k − l η + k (cid:112) l ! (cid:112) k ! 2 l − k e | z | η ≤ a k − l η + k (cid:112) l ! (cid:112) k ! 2 l − k , (4.41)for | z | ≥ a and k < l . 137 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES
These two last bounds in Eqs. (4.40) and (4.41) are equal for a l − k = (2 η ) l − k (cid:112) ( l − k )! , yieldingthe bound (cid:175)(cid:175) f | k 〉〈 l | ( z , η ) (cid:175)(cid:175) ≤ (cid:118)(cid:117)(cid:117)(cid:116) l − k η + k + l (cid:195) lk (cid:33) , (4.42)for all z ∈ (cid:67) and k < l . For l < k the same reasoning gives (cid:175)(cid:175) f | k 〉〈 l | ( z , η ) (cid:175)(cid:175) ≤ (cid:118)(cid:117)(cid:117)(cid:116) k − l η + k + l (cid:195) kl (cid:33) . (4.43)Finally, for k = l the previous bounds also hold, by combining Eqs. (4.37) and (4.38), and thisproves the lemma. (cid:3) Let k , l ≥ n ∈ (cid:78) and (cid:178) (cid:48) >
0. Applying Lemma 4.3 to the function f | l 〉〈 k | , with the boundfrom Lemma 4.4 yieldsPr (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) n n (cid:88) i = f | l 〉〈 k | ( α i , η ) − (cid:69) α ← Q ρ [ f | l 〉〈 k | ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≥ (cid:178) (cid:48) (cid:35) ≤ (cid:34) − n η + k + l (cid:178) (cid:48) M kl (cid:35) . (4.44)Applying Theorem 4.1 for A = | l 〉〈 k | we also obtain (cid:175)(cid:175)(cid:175)(cid:175) ρ kl − (cid:69) α ← Q ρ (cid:163) f | l 〉〈 k | ( α , η ) (cid:164)(cid:175)(cid:175)(cid:175)(cid:175) ≤ η (cid:112) k + (cid:112) l + α , . . . , α n be samples from the Q function of ρ . Combining Eqs. (4.44) and (4.45), weobtain with the triangular inequality (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ρ kl − n n (cid:88) i = f | l 〉〈 k | ( α i , η ) (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ η (cid:112) k + (cid:112) l + + (cid:178) (cid:48) , (4.46)with probability greater than 1 − (cid:34) − n η + k + l (cid:178) (cid:48) M kl (cid:35) . (4.47)We have K | l 〉〈 k | = (cid:112) ( k + l +
1) by Eq. (4.3). Taking η = (cid:178) K | l 〉〈 k | yields (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ρ kl − n n (cid:88) i = f | l 〉〈 k | (cid:181) α i , (cid:178) K | l 〉〈 k | (cid:182)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ (cid:178) + (cid:178) (cid:48) , (4.48)with probability greater than 1 − (cid:183) − n (cid:178) + k + l (cid:178) (cid:48) C kl (cid:184) , (4.49)138 .4. CONTINUOUS VARIABLE QUANTUM STATE CERTIFICATION PROTOCOL where we defined C kl : = [( k + l + + k + l M kl = [( k + l + + k + l | l − k | (cid:195) max ( k , l )min ( k , l ) (cid:33) . (4.50)Now this holds for 0 ≤ k , l ≤ E . Together with the union bound, this proves the theorem. (cid:4) In light of this result, the principle for performing reliable heterodyne tomography is straightfor-ward and as follows: n identical copies ρ ⊗ n of the output quantum state of a physical experimentor quantum device are measured with heterodyne detection, yielding the values α , . . . , α n . Thesevalues are used to compute the estimates ρ (cid:178) kl , defined in Eq. (4.28), for all k , l in the range ofenergy of the experiment. Then, Theorem 4.2 directly provides confidence intervals for all theseestimates of ρ kl , the matrix elements of the density operator ρ , without the need for a binningof the sample space or any additional data reconstruction, using a single measurement setting.For a desired precision (cid:178) and a failure probability δ , the number of samples needed scales as n = poly (1/ (cid:178) , log(1/ δ )).Both homodyne and heterodyne quantum state tomography assume a bounded support overthe Fock basis for the output state considered, i.e., that all matrix elements are equal to zerobeyond a certain value, and that the output quantum states are i.i.d., i.e., that all measuredoutput states are independent and identical. While these assumptions are natural when lookingat the output of a physical experiment, corresponding to a noisy partially trusted quantum devicewith bounded energy, they may be questionable in the context of untrusted devices. We removethese assumptions in what follows: we first drop the bounded support assumption, deriving acertification protocol for continuous variable quantum states of an i.i.d. device with heterodynedetection ; then, we drop both assumptions, deriving a general verification protocol for continuousvariable quantum states against an adversary who can potentially be fully malicious. Given an untrusted source of quantum states, the purpose of state certification and state verifica-tion protocols is to check whether if its output state is close to a given target state, or far from it.To achieve this, a verifier tests the output state of the source. Ideally, one would like to obtainan upper bound on the probability that the state is not close from the target state, given thatit passed a test. However, this is known to be impossible without prior knowledge of the testedstate distribution [GKK19]. Indeed, writing this conditional probabilityPr [incorrect | accept] = Pr [incorrect ∩ accept]Pr [accept] , (4.51)in a situation where the device always produces a bad output state, it is rejected by the verifier’stest most of the time, so the acceptance probability is very small while the conditional probability139 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES is equal to 1. Therefore, the quantity that will always be bounded in certification and verificationprotocols in which one does not have prior knowledge of the device is the joint probability thatthe tested state is not close to the target state and that it passes the test. Equivalently, we obtainlower bounds on the probability that the tested state is close to the target state or that it fails thetest.We first consider the certification of the output of an i.i.d. quantum device, i.e., which output stateis the same at each round. However, we do not assume that the output states of the device havebounded support over the Fock basis anymore. This is instead ensured probabilistically using thesamples from heterodyne detection.Let us define the following operators for E ≥ U = +∞ (cid:88) n = E + | n 〉〈 n | = − Π E , (4.52)where Π E = (cid:80) En = | n 〉〈 n | is the projector onto the Hilbert space ¯ H of states with less than E photons, and T = π (cid:90) | α | ≥ E | α 〉〈 α | d α , (4.53)where | α 〉 is a coherent state. We have the following result, proven in [LGPRC13] by expanding T in the Fock basis: U ≤ T . (4.54)In particular, Tr ( U ρ ) ≤ T ρ ). (4.55)The probability P r that exactly r among n values of | α i | are bigger than E and n − r values arelower, and that the projection Π E of the state ρ onto the Hilbert space ¯ H of states with less than E photons fails is bounded as P r = (cid:195) nr (cid:33) Tr (cid:163) (1 − Π E ) T r (1 − T ) n − r ρ ⊗ n + (cid:164) = (cid:195) nr (cid:33) Tr (cid:163) U T r (1 − T ) n − r ρ ⊗ n + (cid:164) ≤ (cid:195) nr (cid:33) Tr (cid:161) T ρ (cid:162) r + Tr (cid:163) (1 − T ) ρ (cid:164) n − r ≤ (cid:195) nr (cid:33) max p (cid:175)(cid:175) p r + (1 − p ) n − r (cid:175)(cid:175) = (cid:195) nr (cid:33) (cid:181) r + n + (cid:182) r + (cid:181) − r + n + (cid:182) n − r (4.56)140 .4. CONTINUOUS VARIABLE QUANTUM STATE CERTIFICATION PROTOCOL ≤ n r r ! (cid:181) r + n + (cid:182) r + (cid:181) − r + n + (cid:182) n − r ≤ n r r ! ( r + r + n r + exp (cid:183) − ( n − r )( r + n + (cid:184) ≤ n r + (cid:112) π ( r +
1) exp (cid:183) ( r + n + (cid:184) ≤ (cid:112) r + n exp (cid:183) ( r + n + (cid:184) ,where we used Eq. (4.55), 1 − x ≤ e − x and ( r + ≥ (cid:112) π ( r +
1) ( r + r + e − ( r + . For s ∈ (cid:78) , and forall r ≤ s , (cid:112) r + n exp (cid:183) ( r + n + (cid:184) ≤ (cid:112) s + n exp (cid:183) ( s + n + (cid:184) , (4.57)hence the probability that at most s among n values of | α i | are bigger than E , and that theprojection Π E of the state ρ onto the Hilbert space ¯ H of states with less than E photons fails isbounded by P iid support : = ( s + n exp (cid:183) ( s + n + (cid:184) . (4.58)For 1 (cid:191) s (cid:191) n , this implies that either ρ is contained in a lower dimensional subspace, or thescore at the support estimation step is higher than s , with high probability.Our continuous variable quantum state certification protocol is then as follows: let | ψ 〉 be a targetpure state, of which one wants to certify m copies. The values s and E are free parameters of theprotocol. One instructs the i.i.d. device to prepare n + m copies of | ψ 〉 , and the device outputs ani.i.d. (mixed) state ρ ⊗ n + m . One keeps m copies ρ ⊗ m , and measures the n others with heterodynedetection, obtaining the samples α , . . . , α n . One records the number r of samples such that | α i | > E . We refer to this step as support estimation . For a given (cid:178) >
0, one also computes withthe same samples the estimate F ψ ( ρ ) = (cid:34) n n (cid:88) i = f ψ (cid:181) α i , (cid:178) mK ψ (cid:182)(cid:35) m , (4.59)where the function f A and the constant K A are defined in Eqs. (4.2) and (4.3), for A = | ψ 〉〈 ψ | , andwhere (cid:178) > m copies of the output state ρ ⊗ m of the tested device and m copies of the target state | ψ 〉〈 ψ | ⊗ m . Theorem 4.3 (Gaussian certification of continuous variable quantum states) . Let (cid:178) , (cid:178) (cid:48) > , lets ≤ n, and let α , . . . , α n be samples obtained by measuring with heterodyne detection n copies of astate ρ . Let E in (cid:78) , and let r be the number of samples such that | α i | > E. Let also | ψ 〉 be a purestate. Then for all m ∈ (cid:78) ∗ , (cid:175)(cid:175) F ( ψ ⊗ m , ρ ⊗ m ) − F ψ ( ρ ) (cid:175)(cid:175) ≤ (cid:178) + (cid:178) (cid:48) , (4.60)141 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES or r > s, with probability greater than − (cid:179) P iidSupport + P iidHoeffding (cid:180) , (4.61) where P iidSupport = ( s + n exp (cid:183) ( s + n + (cid:184) , (4.62) P iidHoeffding = (cid:34) − n (cid:178) + E (cid:178) (cid:48) m + E C ψ (cid:35) , (4.63) where the estimate F ψ ( ρ ) is defined in Eq. (4.59), and whereC ψ = E (cid:88) k , l = | ψ k ψ l | (cid:179) (cid:178) m (cid:180) E − k + l K + k + l ψ (cid:118)(cid:117)(cid:117)(cid:116) | l − k | (cid:195) max ( k , l )min ( k , l ) (cid:33) (4.64) is a constant independent of ρ , with the constant K defined in Eq. (4.3). In order to prove this theorem we make use of the following simple result:
Lemma 4.5.
Let η > and a , b ∈ [0, 1] such that | a − b | ≤ η . Then for all m ≥ , (cid:175)(cid:175) a m − b m (cid:175)(cid:175) ≤ m | a − b | ≤ m η . (4.65) Proof.
With the notations of the lemma, (cid:175)(cid:175) a m − b m (cid:175)(cid:175) = | a − b | (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) m − (cid:88) j = a j b m − j − (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ m | a − b |≤ m η . (4.66) (cid:4) We first consider the case of m = Proof.
Let us write | ψ 〉 = (cid:80) n ≥ ψ n | n 〉 . For η >
0, the function z (cid:55)→ f ψ ( z , η ) is real-valued,since | ψ 〉〈 ψ | is hermitian. It is bounded as (cid:175)(cid:175) f ψ ( α , η ) (cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) E (cid:88) k , l = ψ k ψ ∗ l f | k 〉〈 l | ( α , η ) (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ E (cid:88) k , l = (cid:175)(cid:175) ψ k ψ ∗ l f | k 〉〈 l | ( α , η ) (cid:175)(cid:175) .4. CONTINUOUS VARIABLE QUANTUM STATE CERTIFICATION PROTOCOL ≤ E (cid:88) k , l = (cid:175)(cid:175) ψ k ψ l (cid:175)(cid:175) M kl η + k + l (4.67) = η + E E (cid:88) k , l = (cid:175)(cid:175) ψ k ψ l (cid:175)(cid:175) η E − ( k + l )/2 M kl = M ψ ( η ) η + E ,where we used Lemma 4.4, and where we defined M ψ ( η ) : = E (cid:88) k , l = (cid:175)(cid:175) ψ k ψ l (cid:175)(cid:175) η E − ( k + l )/2 M kl . (4.68)Applying Lemma 4.2 to the real-valued function z (cid:55)→ f ψ ( z , η ) thus yieldsPr (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) n n (cid:88) i = f ψ ( α i , η ) − (cid:69) α ← Q ρ [ f ψ ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≥ (cid:178) (cid:48) (cid:35) ≤ (cid:34) − n η + E (cid:178) (cid:48) M ψ ( η ) (cid:35) , (4.69)for (cid:178) (cid:48) , η >
0, where the probability is over i.i.d. samples from heterodyne detection of ρ .In what follows, we first assume that ρ ∈ ¯ H . By Corollary 4.1 we have (cid:175)(cid:175)(cid:175)(cid:175) F (cid:161) ψ , ρ (cid:162) − (cid:69) α ← Q ρ [ f ψ ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175) ≤ η K ψ . (4.70)Combining Eqs. (4.69) and (4.70) yields (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) F (cid:161) ψ , ρ (cid:162) − n n (cid:88) i = f ψ ( α i , η ) (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ η K ψ + (cid:178) (cid:48) , (4.71)with probability greater than 1 − (cid:183) − n η + E (cid:178) (cid:48) M ψ ( η ) (cid:184) . Setting η = (cid:178) K ψ yields (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) F (cid:161) ψ , ρ (cid:162) − n n (cid:88) i = f ψ (cid:181) α i , (cid:178) K ψ (cid:182)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ (cid:178) + (cid:178) (cid:48) , (4.72)with probability greater than 1 − (cid:183) − n (cid:178) + E (cid:178) (cid:48) C ψ ,1 ( (cid:178) ) (cid:184) , where we defined C ψ ,1 ( (cid:178) ) : = K + E ψ M ψ (cid:181) (cid:178) K ψ (cid:182) = E (cid:88) k , l = | ψ k ψ l | (cid:178) E − k + l K + k + l ψ (cid:118)(cid:117)(cid:117)(cid:116) | l − k | (cid:195) max ( k , l )min ( k , l ) (cid:33) . (4.73)143 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES
Combining Lemma 4.5 and Eq. (4.72) we obtain (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) F (cid:161) ψ , ρ (cid:162) m − (cid:34) n n (cid:88) i = f ψ (cid:181) α i , (cid:178) K ψ (cid:182)(cid:35) m (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ m ( (cid:178) + (cid:178) (cid:48) ), (4.74)with probability greater than 1 − (cid:183) − n (cid:178) + E (cid:178) (cid:48) C ψ ,1 ( (cid:178) ) (cid:184) . Note that we excluded the pathologicalcase n (cid:80) ni = f ψ ( α i , (cid:178) / K ψ ) >
1: when that is the case we instead set n (cid:80) ni = f ψ ( α i , (cid:178) / K ψ ) =
1. Thetarget state ψ is pure so F ( ψ ⊗ m , ρ ⊗ m ) = F (cid:161) ψ , ρ (cid:162) m . Hence, replacing (cid:178) and (cid:178) (cid:48) by (cid:178) / m and (cid:178) (cid:48) / m ,respectively, gives (cid:175)(cid:175) F (cid:161) ψ , ρ (cid:162) m − F ψ ( ρ ) (cid:175)(cid:175) ≤ (cid:178) + (cid:178) (cid:48) , (4.75)with probability greater than P iid Hoeffding : = − (cid:34) − n (cid:178) + E (cid:178) (cid:48) m + E C ψ (cid:35) , (4.76)where F ψ ( ρ ) = (cid:34) n n (cid:88) i = f ψ (cid:181) α i , (cid:178) mK ψ (cid:182)(cid:35) m , (4.77)and where C ψ : = C ψ ,1 ( (cid:178) / m ) = E (cid:88) k , l = | ψ k ψ l | (cid:179) (cid:178) m (cid:180) E − k + l K + k + l ψ (cid:118)(cid:117)(cid:117)(cid:116) | l − k | (cid:195) max ( k , l )min ( k , l ) (cid:33) . (4.78)Until now we have assumed ρ ∈ ¯ H . By Eq. (4.58), the probability that at most s among n values of | α i | are bigger than E , and that the projection Π E of the state ρ onto the Hilbertspace ¯ H of states with less than E photons fails is bounded by P iid support = ( s + n exp (cid:183) ( s + n + (cid:184) . (4.79)With the union bound we thus obtain (cid:175)(cid:175) F (cid:161) ψ , ρ (cid:162) m − F ψ ( ρ ) (cid:175)(cid:175) ≤ (cid:178) + (cid:178) (cid:48) , (4.80)or r > s , with probability greater than 1 − (cid:179) P iid support + P iid Hoeffding (cid:180) . (cid:4) This result implies that the quantity F ψ ( ρ ) is a good estimate of the fidelity F ( ψ ⊗ m , ρ ⊗ m ), orthe score at the support estimation step is higher than s , with high probability. The values ofthe energy parameters E and s should be chosen to guarantee completeness, i.e., that if thecorrect state | ψ 〉 is sent, then r ≤ s with high probability. This theorem is valid for all continuousvariable target pure states | ψ 〉 , and the failure probability may be greatly reduced depending on144 .5. CONTINUOUS VARIABLE QUANTUM STATE VERIFICATION PROTOCOL the expression of | ψ 〉 . The number of samples needed for certifying a given number of copies m with a precision (cid:178) and a failure probability δ scales as n = poly ( m , 1/ (cid:178) , 1/ δ ).This certification protocol is promoted to a verification protocol for single-mode states in thefollowing section, by removing the i.i.d. assumption. We now consider an adversarial setting, where a verifier delegates the preparation of a continuousvariable quantum state to a potentially malicious party, called the prover . One could see theverifier as the experimentalist in the laboratory and the prover as the noisy device, where we aimnot to make any assumptions about its correct functionality or noise model. Given the absence ofany direct error correction mechanism that permits a fault tolerant run of the device, the aimof verification is to ensure that a wrong outcome is not being accepted. In the context of stateverification, this amounts to making sure that the output state of the tested device is close to anideal target state.The prover is not supposed to have i.i.d. behaviour. In particular, when asked for variouscopies of the same state, the prover may actually send a large state entangled over all subsystems,possibly also entangled with a quantum system on his side. In that case, the certification protocolderived in the previous section is not reliable. With usual tomography measurements, the numberof samples needed for a given precision of the fidelity estimate scales exponentially in the numberof copies to verify. This is an essential limitation of quantum tomography techniques, becausethey check all possible correlations between the different subsystems.However we prove that, because of the symmetry of the protocol, the verifier can assumethat the prover is sending permutation-invariant states, i.e., states that are invariant underany permutation of their subsystems. After a specific support estimation step, reduced states ofpermutation-invariant states are close to mixture almost-i.i.d. states, i.e., states that are i.i.d.on almost all subsystems. At the heart of this reduction is a de Finetti theorem for infinite-dimensional systems [RC09], which allows us to restrict to an almost-i.i.d. prover.
The verification protocol is as follows: the verifier wants to verify m copies of a target pure state | ψ 〉 . The numbers n , k , q , s and E are free parameters of the protocol.• The prover is instructed to prepare n + k copies of | ψ 〉 and send them to the verifier. Wedenote by ρ n + k the state received by the verifier.• The verifier picks k subsystems of the state ρ n + k at random and measures them withheterodyne detection, obtaining the remaining state ρ n and the samples β , . . . , β k . Theverifier records the number r of values | β i | > E (support estimation step).145 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES • The verifier discards 4 q subsystems at random, obtaining the remaining state state ρ n − q ,and measures all the others subsystems but m chosen at random with heterodyne detection,obtaining the remaining state ρ m and the samples α , . . . , α n − q − m .• The verifier computes with these samples the estimate F ψ ( ρ ) = (cid:34) n − q − m n − q − m (cid:88) i = f ψ (cid:181) α i , (cid:178) mK ψ (cid:182)(cid:35) m , (4.81)where the function f A and the constant K A are defined in Eqs. (4.2) and (4.3), for A = | ψ 〉〈 ψ | and where (cid:178) > n by n − q − m . In order to show that this is a good estimate of the fidelity betweenthe remaining state ρ m and m copies of the target state | ψ 〉 , we generalise results from [Ren08,RC08, RC09]. More precisely, we obtain the following results:• Support estimation for permutation-invariant states : with high probability, most of thesubsystems of the permutation-invariant state ρ n − q lie in a lower dimensional subspace,or the score of the state ρ n + k at the support estimation step is high (section 4.5.2).• De Finetti reduction : any permutation-invariant state with most of its subsystems in alower dimensional subspace has a purification in the symmetric subspace that still has mostof its subsystems in a lower dimensional subspace. This purification is well approximatedby a mixture of almost-i.i.d. states (section 4.5.3).•
Hoeffding inequality for almost-i.i.d. states : mixtures of almost-i.i.d. states can be certifiedin a similar fashion as i.i.d. states (section 4.5.4).Using these intermediate results, we obtain the following theorem:
Theorem 4.4 (Gaussian verification of continuous variable quantum states) . Let n ≥ , let s ≤ k,and let ρ n + k be a state over n + k subsystems. Let β , . . . , β k be samples obtained by measuringk subsystems at random with heterodyne detection and let ρ n be the remaining state after themeasurement. Let E in (cid:78) , and let r be the number of samples such that | β i | > E. Let alsoq ≥ m, and let ρ m be the state remaining after discarding q subsystems of ρ n at random, andmeasuring n − q − m other subsystems at random with heterodyne detection, yielding the samples α , . . . , α n − q − m . Let (cid:178) , (cid:178) (cid:48) > and let | ψ 〉 be a target pure state. Then, (cid:175)(cid:175) F (cid:161) ψ ⊗ m , ρ m (cid:162) − F ψ ( ρ ) (cid:175)(cid:175) ≤ (cid:178) + (cid:178) (cid:48) + P deFinetti , (4.82) or r > s, with probability greater than − (cid:161) P support + P deFinetti + P choice + P Hoeffding (cid:162) , (4.83)146 .5. CONTINUOUS VARIABLE QUANTUM STATE VERIFICATION PROTOCOL where P support = k exp (cid:183) − k (cid:181) qn − sk (cid:182) (cid:184) , (4.84) P deFinetti = q ( E + /2 exp (cid:183) − q ( q + n (cid:184) , (4.85) P choice = m (4 q + m − n − q , (4.86) P Hoeffding = (cid:195) n − q q (cid:33) exp (cid:34) − n − q m + E (cid:181) (cid:178) + E (cid:178) (cid:48) C ψ − qm + E n − q − m (cid:182) (cid:35) , (4.87) where the estimate F ψ ( ρ ) is defined in Eq. (4.81), and whereC ψ = E (cid:88) k , l = | ψ k ψ l | (cid:179) (cid:178) m (cid:180) E − k + l K + k + l ψ (cid:118)(cid:117)(cid:117)(cid:116) | l − k | (cid:195) max ( k , l )min ( k , l ) (cid:33) (4.88) is a constant independent of ρ , with the constant K defined in Eq. (4.3). We defer the proof of this result to section 4.5.5. It implies that either the quantity F ψ ( ρ ) is a goodestimate of the fidelity F ( ψ ⊗ m , ρ m ), or the score at the support estimation step is higher than s ,with high probability. Like for the certification protocol, the values of the energy parameters E and s should be chosen by the verifier to guarantee completeness, i.e., that if the prover sends thecorrect state | ψ 〉 , then r ≤ s with high probability.For specific choices of the free parameters of the protocol, detailed in the proof of the theorem,either the estimate F ψ ( ρ ) is polynomially precise in m , or r > s , with polynomial probability in m ,with n , k , q = poly m . In particular, the efficiency of the protocol may be greatly refined by takinginto account the expression of | ψ 〉 in the Fock basis, and optimizing over the free parameters.This verification protocol let the verifier gain confidence about the precision of the estimate ofthe fidelity in Eq. (4.81). If the value of the estimate is close enough to 1, the verifier may decideto use the state to run a computation. Indeed, statements on the fidelity of a state allow oneto infer the correctness of any trusted computation done afterwards using this state. Let β > O be the observable corresponding to the result of the trusted computation performed on ρ m , the reduced state over m subsystems instead of | ψ 〉 ⊗ m , m copies of the target state | ψ 〉 . Inother words, O encodes the resources which the verifier can perform perfectly (ancillary states,evolution and measurements), the imperfections being encoded in ρ . Then, F (cid:161) ψ ⊗ m , ρ m (cid:162) ≥ − β implies the following bound on the total variation distance between the probability densities ofthe computation output of the actual and the target computations: (cid:107) P O ψ ⊗ m − P O ρ m (cid:107) tvd ≤ D ( ψ ⊗ m , ρ m ) ≤ (cid:112) β , (4.89)by standard properties of the trace distance D (see section 1.1.2 and [FVDG99]). What this meansis that the distribution of outcomes for the state ρ m sent by the prover is almost indistinguishablefrom the distribution of outcomes for m copies of the ideal target state | ψ 〉 , when the fidelity isclose enough to one.In what follows, we detail the intermediate steps described above and prove Theorem 4.4.147 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES
We first derive a support estimation step for permutation-invariant states. We will use in thissection the following operators, already introduced in section 4.4: for E ≥ U = +∞ (cid:88) n = E + | n 〉〈 n | = − Π E , (4.90)where Π E = E (cid:88) n = | n 〉〈 n | is the projector onto the Hilbert space ¯ H of states with at most E photons,and T = π (cid:90) | α | ≥ E | α 〉〈 α | d α , (4.91)where | α 〉 is a coherent state. We also recall the following result, from Eq. (4.54), provenin [LGPRC13]: U ≤ T . (4.92)We recall a few notations and results from [RC08]: let A = { A , A } , B = { B , B } be two binaryPOVMs over H . Define for δ > γ A → B ( δ ) = sup ψ (cid:169) Tr ( B ψ ), s.t. Tr ( A ψ ) ≤ δ (cid:170) . (4.93)In particular, γ T → U ( δ ) ≤ δ , (4.94)by Eq. (4.92). We recall the following result (Lemma III.1. of [RC08]): Lemma 4.6.
Let n ≥ k, let δ > , let A = { A , A } and B = { B , B } be two binary POVMs over H , and let x , . . . , x n + k the ( n + k ) -partite classical outcome of the measurement A ⊗ n ⊗ B ⊗ k appliedto any permutation-invariant state ρ n + k . Then Pr (cid:104) x + · · · + x n n > γ B → A (cid:179) x n + + · · · + x n + k k + δ (cid:180) + δ (cid:105) ≤ k e − k δ . (4.95)This result is a refined version of Serfling’s bound [Ser74]. It relates the outcomes of a measure-ment on some subsystems of a symmetric state with the outcomes of a related measurementon the rest of the subsystems. With this technical Lemma, we derive in what follows a supportestimation step for permutation-invariant states using samples from heterodyne detection.Let ρ n + k be a state over n + k subsystems. Applying a random permutation to this state and mea-suring its last k subsystems with heterodyne detection is equivalent to measuring k subsystemsat random. We thus assume in the following that the state ρ n + k is a permutation-invariant state,without loss of generality, and that the verifier measures its last k subsystems with heterodynedetection. 148 .5. CONTINUOUS VARIABLE QUANTUM STATE VERIFICATION PROTOCOL Let T = { − T , T } and U = { − U , U } . Let x , . . . , x n + k the ( n + k )-partite classical outcome ofthe measurement U ⊗ n ⊗ T ⊗ k applied to the permutation-invariant state ρ n + k sent by the prover.A value x i = i ∈
1, . . . , n means that the projection of the i th subsystem onto ¯ H failed, whilea value x j = j ∈ n +
1, . . . , n + k means that the value | β | obtained when measuring the j th subsystem with heterodyne detection was bigger than E . In particular, the number of values β i satisfying | β i | > E , is expressed as x n + + · · · + x n + k . Let T k ≤ s be the event that at most s of the k values β i satisfy | β i | > E , and let F nq be the event that the projection onto ¯ H fails for more than q subsystems of the remaining state ρ n . Then: Lemma 4.7 (Support estimation for permutation-invariant states) . Pr (cid:104) F nq ∩ T k ≤ s (cid:105) ≤ P support . (4.96) where P support = k exp (cid:104) − k (cid:161) qn − sk (cid:162) (cid:105) . Proof.
With Eq. (4.94), we have for all δ > γ T → U (cid:179) x n + + · · · + x n + k k + δ (cid:180) + δ ≤ x n + + · · · + x n + k k + δ . (4.97)Taking δ = (cid:161) qn − sk (cid:162) we obtain γ T → U (cid:179) x n + + · · · + x n + k k + δ (cid:180) + δ ≤ qn + (cid:179) x n + + · · · + x n + k k − sk (cid:180) , (4.98)so if x + · · · + x n > q and x n + + · · · + x n + k ≤ s , then γ T → U (cid:179) x n + + · · · + x n + k k + δ (cid:180) + δ < x + · · · + x n n . (4.99)Hence, Pr (cid:104) F nq ∩ T k ≤ s (cid:105) = Pr [( x + · · · + x n > q ) ∩ ( x n + + · · · + x n + k ≤ s )] ≤ Pr (cid:104)(cid:179) x + · · · + x n n > γ T → U (cid:179) x n + + · · · + x n + k k + δ (cid:180) + δ (cid:180)(cid:105) ≤ k e − k δ = k exp (cid:183) − k (cid:181) qn − sk (cid:182) (cid:184) , (4.100)where we used Lemma 4.6 for A = U and B = T . (cid:4) Recall that ¯ H is the Hilbert space of states with at most E photons, of dimension E +
1. For q ≤ n , let us define the set of permutation-invariant states over n subsystems, with at most q subsystems out of this lower dimensional subspace (introduced in [RC09]): S n ¯ H ⊗ n − q : = span (cid:91) π π (cid:161) ¯ H ⊗ n − q ⊗ H ⊗ q (cid:162) π − , (4.101)149 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES where the union is taken over all permutations. Lemma 4.7 then givesPr (cid:104) F nq ∩ T k ≤ s (cid:105) ≤ P support , (4.102)where F nq is the event that the projection of ρ n (the remaining state after the support estimationstep) onto S n ¯ H ⊗ n − q fails, and where P support = k exp (cid:104) − k (cid:161) qn − sk (cid:162) (cid:105) . For 1 (cid:191) q (cid:191) n and q / n (cid:191) s / k ,this implies that either ρ n has most of its subsystems in a lower dimensional subspace, or thescore at the support estimation step is higher than s , with high probability. We recall in this section two results from [RC09].• The first result says that any permutation-invariant state with most of its subsystems ina lower dimensional subspace has a purification in the symmetric subspace that still hasmost of its subsystems in a lower dimensional subspace. Formally, for n ∈ (cid:78) , and given aHilbert space K , let us write Sym n ( K ) = { φ ∈ K ⊗ n , πφ = φ ( ∀ π ) } the symmetric subspaceof a Hilbert space K ⊗ n , then (Lemma 3 of [RC09]): Lemma 4.8.
For all q ≤ n, any permutation-invariant state ρ n ∈ S n ¯ H ⊗ n − q has a purification ˜ ρ n in Sym n ( H ⊗ H ) (cid:84) S n ( ¯ H ⊗ ¯ H ) ⊗ n − q . The states of the form | v 〉 ⊗ n are the so-called i.i.d. states . For all n , r ≥ | v 〉 ∈ ¯ H ⊗ ¯ H , theset of almost-i.i.d. states along | v 〉 , S nv ⊗ n − r , is defined as the span of all vectors that are, up toreorderings, of the form | v 〉 ⊗ n − r ⊗ | φ 〉 , for an arbitrary φ ∈ ( H ⊗ H ) ⊗ r . In the following, we simplyrefer to these states as almost-i.i.d. states (which becomes relevant when r (cid:191) n ).• The second result is a de Finetti theorem for states in Sym n ( H ⊗ H ) (cid:84) S n ( ¯ H ⊗ ¯ H ) ⊗ n − q , whichsays that reduced states from them are well approximated by mixtures of almost-i.i.d.states. Formally (Theorem 4 of [RC09], applied to K = H ⊗ H and ¯ K = ¯ H ⊗ ¯ H , withdim( ¯ K ) = ( E + ): Theorem 4.5.
Let ˜ ρ n ∈ Sym n ( H ⊗ H ) (cid:84) S n ( ¯ H ⊗ ¯ H ) ⊗ n − q and let ˜ ρ n − q = Tr q ( ˜ ρ n ) . Then, thereexist a finite set V of unit vectors | v 〉 ∈ ¯ H ⊗ ¯ H , a probability distribution { p v } v ∈ V over V , andalmost-i.i.d. states ˜ ρ n − qv ∈ S n − qv ⊗ n − q such thatF (cid:195) ˜ ρ n − q , (cid:88) v ∈ V p v ˜ ρ n − qv (cid:33) > − q ( E + exp (cid:183) − q ( q + n (cid:184) . (4.103)Given a state ρ n ∈ S n ¯ H ⊗ n − q , applying Theorem 4.5 to the purification ˜ ρ n given by Lemma 4.8shows that the reduced state ˜ ρ n − q is close in fidelity to a mixture of states that are i.i.d. on n − q subsystems. 150 .5. CONTINUOUS VARIABLE QUANTUM STATE VERIFICATION PROTOCOL We recall here Lemma 4.2, in the context of a product measurement applied to an i.i.d. state | v 〉〈 v | ⊗ n : Lemma 4.9. (Hoeffding inequality for i.i.d. states)
Let M > ∈ (cid:82) and let f : (cid:67) (cid:55)→ (cid:82) be afunction bounded as | f ( α ) | < M for all α ∈ (cid:67) . Let λ > , let p ∈ (cid:78) ∗ , and let | v 〉 ∈ H . Let M = { M α } α ∈ (cid:67) be a POVM on H and let D | v 〉 be the probability density function of the outcomes of themeasurement M applied to | v 〉〈 v | . Then Pr α (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) p p (cid:88) i = f ( α i ) − (cid:69) β ← D | v 〉 [ f ( β )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≥ λ (cid:35) ≤ (cid:183) − p λ M (cid:184) , (4.104) where the probability is taken over the outcomes α = ( α , . . . , α p ) of the product measurement M ⊗ p applied to | v 〉〈 v | ⊗ p . The next result gives an equivalent statement for almost-i.i.d. states along a state | v 〉 , measuredwith a product measurement. It generalises Theorem 4.5.2 of [Ren08], where the probabilitydistributions over finite sets, corresponding to product measurements with finite number ofoutcomes, are replaced by continuous variable probability densities, corresponding to productmeasurements with continuous variable outcomes. Frequencies estimators are also replacedwith estimators of expected values of bounded functions. We will use this result for the POVMcorresponding to a product heterodyne detection. Lemma 4.10 (Hoeffding inequality for almost-i.i.d. states) . Let M > ∈ (cid:82) and let f : (cid:67) (cid:55)→ (cid:82) be afunction bounded as | f ( α ) | ≤ M for all α ∈ (cid:67) . Let µ > and ≤ m ≤ r < t such that ( t − m ) µ > Mr . (4.105) Let also | v 〉 ∈ ¯ H and | Φ 〉 ∈ S tv ⊗ t − r . Let M = { M α } α ∈ (cid:67) be a POVM on H and let D | v 〉 be the probabilitydensity function of the outcomes of the measurement M applied to | v 〉〈 v | . Then Pr α (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t − m t − m (cid:88) i = f ( α i ) − (cid:69) β ← D | v 〉 [ f ( β )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≥ µ (cid:35) ≤ (cid:195) tr (cid:33) exp (cid:183) − t − r (cid:181) µ M − rt − m (cid:182) (cid:184) , (4.106) where the probability is taken over the outcomes α = ( α , . . . , α t − m ) of the product measurement M ⊗ t − m applied to | Φ 〉〈 Φ | . In essence, this lemma says that a product measurement on all but m subsystems of an almost-i.i.d. state along a state | v 〉 will yield statistics that are similar to the ones that would be obtainedby measuring the i.i.d. state | v 〉 ⊗ t − m . 151 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES
Proof. | Φ 〉 ∈ S tv ⊗ t − r , so by Lemma 4.1.6 of [Ren08], there exist a finite set S of size at most (cid:161) tr (cid:162) , a family of states | ˜ Φ s 〉 ∈ H ⊗ r for s ∈ S , complex amplitudes { γ s } s ∈ S and permutations { π s } s ∈ S over [1, . . . , t ] such that | Φ 〉 : = (cid:88) s ∈ S γ s | Φ s 〉= (cid:88) s ∈ S γ s π s (cid:161) | v 〉 ⊗ t − r ⊗ | ˜ Φ s 〉 (cid:162) . (4.107)With the notations of the Lemma, let us define for µ > Ω µ = (cid:40) α ∈ (cid:67) t − m , (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t − m t − m (cid:88) i = f ( α i ) − (cid:69) β ← D | v 〉 [ f ( β )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) > µ (cid:41) . (4.108)We recall here Lemma of 4.5.1 of [Ren08]: Lemma 4.11.
Let | X | be a finite set and | ψ 〉 = (cid:80) x ∈ X | ψ x 〉 , and let A be a non-negative operator.Then 〈 ψ | A | ψ 〉 ≤ | X | (cid:88) x ∈ X 〈 ψ x | A | ψ x 〉 . (4.109)In particular, using Eq. (4.107) and this lemma when A is a POVM element of the productmeasurement M α ≡ M α ⊗ · · · ⊗ M α t − m , we obtain:Pr α ←| Φ 〉 [ α ∈ Ω µ ] = (cid:90) Ω µ 〈 Φ | M α | Φ 〉 d t − m ) α ≤ (cid:90) Ω µ | S | (cid:88) s ∈ S | γ s | 〈 Φ s | M α | Φ s 〉 d t − m ) α ≤ | S | (cid:88) s ∈ S | γ s | (cid:90) Ω µ 〈 Φ s | M α | Φ s 〉 d t − m ) α = | S | (cid:88) s ∈ S | γ s | Pr α ←| Φ s 〉 [ α ∈ Ω µ ], (4.110)where we write α ← | χ 〉 to indicate that α = ( α , . . . , α t − m ) is distributed according to theoutcomes of the product measurement M ⊗ t − m applied to | χ 〉 .Let α ← | Φ s 〉 . We have | Φ s 〉 = π s ( | v 〉 ⊗ t − r ⊗ | ˜ Φ s 〉 ), and in particular ( α π s (1) , . . . , α π s ( t − r ) ) isdistributed according to the outcomes of the product measurement M ⊗ t − r applied to | v 〉 ⊗ t − r .We also have, for | f | ≤ M , (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t − r t − r (cid:88) i = f ( α π s ( i ) ) − t − m t − m (cid:88) i = f ( α i ) (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t − r t − r (cid:88) i = f ( α π s ( i ) ) − t − m (cid:195) t (cid:88) i = f ( α i ) − t (cid:88) i = t − m + f ( α i ) (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t − r t − r (cid:88) i = f ( α π s ( i ) ) − t − m (cid:195) t (cid:88) i = f ( α π s ( i ) ) − t (cid:88) i = t − m + f ( α i ) (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175)(cid:181) t − r − t − m (cid:182) t − r (cid:88) i = f ( α π s ( i ) ) + t − m (cid:195) t (cid:88) i = t − m + f ( α i ) − t (cid:88) i = t − r + f ( α π s ( i ) ) (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) .5. CONTINUOUS VARIABLE QUANTUM STATE VERIFICATION PROTOCOL ≤ (cid:175)(cid:175)(cid:175)(cid:175) t − r − t − m (cid:175)(cid:175)(cid:175)(cid:175) t − r (cid:88) i = | f ( α π s ( i ) ) | + t − m (cid:195) t (cid:88) i = t − m + | f ( α i ) | + t (cid:88) i = t − r + | f ( α π s ( i ) ) | (cid:33) ≤ | r − m | t − m M + ( m + r ) t − m M = rMt − m , (4.111)where we used r ≥ m . Now for all s ∈ S ,Pr α ←| Φ s 〉 [ α ∈ Ω µ ] = Pr α ←| Φ s 〉 (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t − m t − m (cid:88) i = f ( α i ) − (cid:69) β ← D | v 〉 [ f ( β )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) > µ (cid:35) ≤ Pr α ←| Φ s 〉 (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t − r t − r (cid:88) i = f ( α π s ( i ) ) − (cid:69) β ← D | v 〉 [ f ( β )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) + (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t − m t − m (cid:88) i = f ( α i ) − t − r t − r (cid:88) i = f ( α π s ( i ) ) (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) > µ (cid:35) ≤ Pr α ←| Φ s 〉 (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t − r t − r (cid:88) i = f ( α π s ( i ) ) − (cid:69) β ← D | v 〉 [ f ( β )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) > µ − rMt − m (cid:35) ≤ (cid:183) − t − r (cid:181) µ M − rt − m (cid:182) (cid:184) , (4.112)where we used triangular inequality in the second line, Eq. (4.111) in the third line andLemma 4.9 in the fourth line with p = t − r and λ = µ − rMt − m >
0. Combining this last equationwith Eq. (4.110), and using | S | ≤ (cid:161) tr (cid:162) we finally obtain,Pr α ←| ψ 〉 (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) t − m t − m (cid:88) i = f ( α i ) − (cid:69) β ← D | v 〉 [ f ( β )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≥ µ (cid:35) ≤ (cid:195) tr (cid:33) exp (cid:183) − t − r (cid:181) µ M − rt − m (cid:182) (cid:184) . (4.113) (cid:4) We recall the bound on z (cid:55)→ f ψ ( z , η ) obtained in Eq. (4.68): for all α ∈ (cid:67) , | f ψ ( α , η ) | ≤ M ψ ( η ) η + E , (4.114)where M ψ ( η ) = E (cid:88) k , l = (cid:175)(cid:175) ψ k ψ l (cid:175)(cid:175) η E − ( k + l )/2 (cid:118)(cid:117)(cid:117)(cid:116) | l − k | (cid:195) max ( k , l )min ( k , l ) (cid:33) . (4.115)Let µ , η > E ∈ (cid:78) , let | v 〉 ∈ ¯ H ⊗ ¯ H , and let | Φ v 〉 n − q ∈ S n − qv ⊗ n − q . Applying Lemma 4.10 for thereal-valued function f ψ , for t = n − q , for r = q , for D | v 〉 = Q | v 〉〈 v | , and with the bound fromEq. (4.114), we obtainPr α (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) n − q − m n − q − m (cid:88) i = f ψ ( α i , η ) − (cid:69) β ← Q | v 〉〈 v | [ f ψ ( β , η )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≥ µ (cid:35) ≤ (cid:195) n − q q (cid:33) exp (cid:34) − n − q (cid:181) η + E µ M ψ ( η ) − qn − q − m (cid:182) (cid:35) , (4.116)where the probability is over the outcomes α of a product heterodyne measurement of the first n − q − m subsystems of | Φ v 〉 n − q ∈ S n − qv ⊗ n − q . 153 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES
We introduce the following simple result:
Lemma 4.12.
Let < β < . Let ρ , ρ be two states such that F ( ρ , ρ ) > − β . Let | Φ 〉 be a purestate, then (cid:175)(cid:175) F ( Φ , ρ ) − F ( Φ , ρ ) (cid:175)(cid:175) ≤ D ( ρ , ρ ) ≤ (cid:112) β . (4.117) Proof.
Let us write P Φ ρ and P Φ ρ the probability distributions associated to the binarymeasurement { | Φ 〉〈 Φ | , I − | Φ 〉〈 Φ | } of the states ρ and ρ , respectively. Then, P Φ ρ (0) + P Φ ρ (1) = P Φ ρ (0) + P Φ ρ (1) =
1, and (cid:107) P Φ ρ − P Φ ρ (cid:107) tvd = (cid:179) | P Φ ρ (0) − P Φ ρ (0) | + | P Φ ρ (1) − P Φ ρ (1) | (cid:180) = | P Φ ρ (0) − P Φ ρ (0) | . (4.118)Hence, (cid:175)(cid:175) F ( Φ , ρ ) − F ( Φ , ρ ) (cid:175)(cid:175) = (cid:175)(cid:175) 〈 Φ | ρ | Φ 〉 − 〈 Φ | ρ | Φ 〉 (cid:175)(cid:175) = | P Φ ρ (0) − P Φ ρ (0) |= (cid:107) P Φ ρ − P Φ ρ (cid:107) tvd ≤ D ( ρ , ρ ) ≤ (cid:112) − F ( ρ , ρ ) ≤ (cid:112) β , (4.119)where we used Eqs. (1.11, 1.15). (cid:4) With these intermediate results, we are now in position to prove Theorem 4.4.
Proof.
Let | ψ 〉〈 ψ | be the target pure state, and let ρ n + k be a state sent over n + k subsystems.Let β , . . . , β k be samples obtained by measuring k subsystems at random of ρ n + k withheterodyne detection. Let ρ n be the remaining state after the support estimation step. Inwhat follows, we first assume that ρ n ∈ S n ¯ H ⊗ n − q .Let ρ n − q be the state obtained from ρ n by tracing over the first 4 q subsystems. In thatcase, by section 4.5.3, there exist a finite set V of unit vectors | v 〉 ∈ ¯ H ⊗ ¯ H , a probabilitydistribution { p v } v ∈ V over V , and almost-i.i.d. states ˜ ρ n − qv ∈ S n − qv ⊗ n − q such that F (cid:195) ρ n − q , (cid:88) v ∈ V p v ρ n − qv (cid:33) > − q ( E + exp (cid:183) − q ( q + n (cid:184) , (4.120)154 .5. CONTINUOUS VARIABLE QUANTUM STATE VERIFICATION PROTOCOL where ρ n − qv is the remaining state after tracing over the purifying subsystems, since thefidelity is non-decreasing under quantum operations [BCF + F (cid:195) ρ m , (cid:88) v ∈ V p v ρ mv (cid:33) > − q ( E + exp (cid:183) − q ( q + n (cid:184) , (4.121)where ρ m (resp. ρ mv ) is the remaining state after measuring the first n − q − m subsystemsof ρ n − q (resp. ρ n − qv ) with heterodyne detection.Let α , . . . , α n − q − m be the samples obtained by measuring the first n − q − m subsystemsof ρ n − q with heterodyne detection. The verifier computes the estimate (4.81) F ψ ( ρ ) = (cid:34) n − q − m n − q − m (cid:88) i = f ψ (cid:181) α i , (cid:178) mK ψ (cid:182)(cid:35) m , (4.122)and whenever F ψ ≥ F ψ =
1. Let us define the completely positive map E on H n − q associated to the classical post-processing of the protocol as: σ (cid:55)→ E ( σ ) = (cid:88) e Pr (cid:163) F ψ ( σ ) = e (cid:164) | e 〉〈 e | . (4.123)The sum ranges over the values that the estimate may take. With Eq. (4.120) and Lemma 4.12we obtain D (cid:195) ρ n − q , (cid:88) v ∈ V p v ρ n − qv (cid:33) ≤ q ( E + exp (cid:183) − q ( q + n (cid:184) , (4.124)The trace distance is non-increasing under quantum operations, so Eq. (4.124) implies D (cid:195) E (cid:161) ρ n − q (cid:162) , E (cid:195) (cid:88) v ∈ V p v ρ n − qv (cid:33)(cid:33) ≤ q ( E + exp (cid:183) − q ( q + n (cid:184) . (4.125)Using the definition of the map E , we obtain a bound in total variation distance: (cid:176)(cid:176)(cid:176)(cid:176)(cid:176) P (cid:163) F ψ ( ρ ) (cid:164) − P (cid:34) F ψ (cid:195) (cid:88) v ∈ V p v ρ n − qv (cid:33)(cid:35)(cid:176)(cid:176)(cid:176)(cid:176)(cid:176) tvd ≤ q ( E + exp (cid:183) − q ( q + n (cid:184) , (4.126)where P denotes the probability distributions for the values of the estimates F ψ ( ρ ) and F ψ (cid:179)(cid:80) v ∈ V p v ρ n − qv (cid:180) .In particular, this bound implies that for all λ > (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) Pr (cid:163)(cid:175)(cid:175) F ( ψ ⊗ m , ρ m ) − F ψ ( ρ ) (cid:175)(cid:175) > λ (cid:164) − Pr (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) F ( ψ ⊗ m , ρ m ) − F ψ (cid:195) (cid:88) v ∈ V p v ρ n − qv (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) > λ (cid:35)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ q ( E + exp (cid:183) − q ( q + n (cid:184) , (4.127)155 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES and thusPr (cid:163)(cid:175)(cid:175) F ( ψ ⊗ m , ρ m ) − F ψ ( ρ ) (cid:175)(cid:175) > λ (cid:164) ≤ q ( E + exp (cid:183) − q ( q + n (cid:184) + Pr (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) F ( ψ ⊗ m , ρ m ) − F ψ (cid:195) (cid:88) v ∈ V p v ρ n − qv (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) > λ (cid:35) . (4.128)With Eq. (4.121) and Lemma 4.12 we obtain (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) F (cid:161) ψ ⊗ m , ρ m (cid:162) − F (cid:195) ψ ⊗ m , (cid:88) v ∈ V p v ρ mv (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ q ( E + exp (cid:183) − q ( q + n (cid:184) , (4.129)where ψ ⊗ m is m copies of the target pure state | ψ 〉 . With the triangular inequality, (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) F ( ψ ⊗ m , ρ m ) − F ψ (cid:195) (cid:88) v ∈ V p v ρ n − qv (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) F ( ψ ⊗ m , ρ m ) − F (cid:195) ψ ⊗ m , (cid:88) v ∈ V p v ρ mv (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) + (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) F (cid:195) ψ ⊗ m , (cid:88) v ∈ V p v ρ mv (cid:33) − F ψ (cid:195) (cid:88) v ∈ V p v ρ n − qv (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ q ( E + exp (cid:183) − q ( q + n (cid:184) + (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) F (cid:195) ψ ⊗ m , (cid:88) v ∈ V p v ρ mv (cid:33) − F ψ (cid:195) (cid:88) v ∈ V p v ρ n − qv (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) , (4.130)where we used Eq. (4.129) in the last line. With Eq. (4.128) we obtain, for all λ > (cid:163)(cid:175)(cid:175) F ( ψ ⊗ m , ρ m ) − F ψ ( ρ ) (cid:175)(cid:175) > λ (cid:164) ≤ q ( E + exp (cid:183) − q ( q + n (cid:184) + Pr (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) F (cid:195) ψ ⊗ m , (cid:88) v ∈ V p v ρ mv (cid:33) − F ψ (cid:195) (cid:88) v ∈ V p v ρ n − qv (cid:33)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) > λ − q ( E + e − q ( q + n (cid:35) . (4.131)By linearity of the probabilities, it suffices to bound Pr (cid:163)(cid:175)(cid:175) F ( ψ ⊗ m , Φ m ) − F ψ ( Φ ) (cid:175)(cid:175) > µ (cid:164) , for µ = λ − q ( E + exp (cid:104) − q ( q + n (cid:105) , where | Φ 〉 ∈ S n − qv ⊗ n − q , for | v 〉 ∈ ¯ H ⊗ ¯ H , and where Φ m is thestate obtained from | Φ 〉〈 Φ | by measuring the first n − q − m subsystems with heterodynedetection and tracing over the purifying subsystems. Lemma 4.13.
Let | Φ 〉 ∈ S n − qv ⊗ n − q . For all (cid:178) (cid:48) > , Pr (cid:163)(cid:175)(cid:175) F ( ψ ⊗ m , Φ m ) − F ψ ( Φ ) (cid:175)(cid:175) > (cid:178) + (cid:178) (cid:48) (cid:164) ≤ (cid:195) n − q q (cid:33) exp (cid:34) − n − q m + E (cid:181) (cid:178) + E (cid:178) (cid:48) C ψ − qm + E n − q − m (cid:182) (cid:35) + m (4 q + m − n − q , (4.132)156 .5. CONTINUOUS VARIABLE QUANTUM STATE VERIFICATION PROTOCOL where C ψ = E (cid:88) k , l = | ψ k ψ l | (cid:179) (cid:178) m (cid:180) E − k + l K + k + l ψ (cid:118)(cid:117)(cid:117)(cid:116) | l − k | (cid:195) max ( k , l )min ( k , l ) (cid:33) −→ (cid:178) → | ψ E | K + E ψ . (4.133) Proof.
Let α , . . . , α n − q − m be samples obtained by measuring the first n − q − m subsystemsof | Φ 〉〈 Φ | with heterodyne detection. We have (4.81) F ψ ( Φ ) = (cid:34) n − q − m n − q − m (cid:88) i = f ψ (cid:181) α i , (cid:178) mK ψ (cid:182)(cid:35) m , (4.134)and (cid:175)(cid:175) F ( ψ ⊗ m , Φ m ) − F ψ ( Φ ) (cid:175)(cid:175) ≤ (cid:175)(cid:175) F ( ψ ⊗ m , Φ m ) − F ( ψ ⊗ m , | v 〉〈 v | ⊗ m ) (cid:175)(cid:175) + (cid:175)(cid:175)(cid:175)(cid:175) F ( ψ ⊗ m , | v 〉〈 v | ⊗ m ) − (cid:181) (cid:69) β ← Q | v 〉〈 v | (cid:183) f ψ (cid:181) β , (cid:178) mK ψ (cid:182)(cid:184)(cid:182) m (cid:175)(cid:175)(cid:175)(cid:175) + (cid:175)(cid:175)(cid:175)(cid:175)(cid:181) (cid:69) β ← Q | v 〉〈 v | (cid:183) f ψ (cid:181) β , (cid:178) mK ψ (cid:182)(cid:184)(cid:182) m − F ψ ( Φ ) (cid:175)(cid:175)(cid:175)(cid:175) = (cid:175)(cid:175) F ( ψ ⊗ m , Φ m ) − F ( ψ ⊗ m , | v 〉〈 v | ⊗ m ) (cid:175)(cid:175) + (cid:175)(cid:175)(cid:175)(cid:175) F ( ψ , | v 〉〈 v | ) m − (cid:181) (cid:69) β ← Q | v 〉〈 v | (cid:183) f ψ (cid:181) β , (cid:178) mK ψ (cid:182)(cid:184)(cid:182) m (cid:175)(cid:175)(cid:175)(cid:175) + (cid:175)(cid:175)(cid:175)(cid:175)(cid:175)(cid:181) (cid:69) β ← Q | v 〉〈 v | (cid:183) f ψ (cid:181) β , (cid:178) mK ψ (cid:182)(cid:184)(cid:182) m − (cid:195) n − q − m n − q − m (cid:88) i = f ψ (cid:181) α i , (cid:178) mK ψ (cid:182)(cid:33) m (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ (cid:175)(cid:175) F ( ψ ⊗ m , Φ m ) − F ( ψ ⊗ m , | v 〉〈 v | ⊗ m ) (cid:175)(cid:175) + m (cid:175)(cid:175)(cid:175)(cid:175) F ( ψ , | v 〉〈 v | ) − (cid:69) β ← Q | v 〉〈 v | (cid:183) f ψ (cid:181) β , (cid:178) mK ψ (cid:182)(cid:184)(cid:175)(cid:175)(cid:175)(cid:175) + m (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) (cid:69) β ← Q | v 〉〈 v | (cid:183) f ψ (cid:181) β , (cid:178) mK ψ (cid:182)(cid:184) − n − q − m n − q − m (cid:88) i = f ψ (cid:181) α i , (cid:178) mK ψ (cid:182)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ,(4.135)where we used Lemma 4.5. We bound these three terms in the following.When selecting at random m subsystems from an almost-i.i.d. state over n − q subsystemswhich is i.i.d. on n − q subsystems, the probability that all of the selected states are fromthe n − q i.i.d. subsystems is (cid:161) n − qm (cid:162)(cid:161) n − qm (cid:162) = ( n − q )( n − q −
1) . . . ( n − q − m + n − q )( n − q −
1) . . . ( n − q − m +
1) , (4.136)157
HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES and we have1 − ( n − q )( n − q −
1) . . . ( n − q − m + n − q )( n − q −
1) . . . ( n − q − m + ≤ − ( n − q − m + m ( n − q ) m = − (cid:181) − q + m − n − q (cid:182) m ≤ min (cid:181) m (4 q + m − n − q (cid:182) ≤ m (4 q + m − n − q , (4.137)where we used 1 − (1 − x ) a ≤ ax for all a ≥ x ∈ [0, 1]. In particular, for | Φ 〉 ∈ S n − qv ⊗ n − q , and Φ m its reduced state over m modes chosen at random, we have Φ m = | v 〉 〈 v | ⊗ m , (4.138)with probability greater than 1 − m (4 q + m − n − q , where we used the definition of S n − qv ⊗ n − q , andEq. (4.137). Using Lemma 4.12, the first term in Eq. (4.135) vanishes with probability greaterthan: 1 − m (4 q + m − n − q . (4.139)The bound for the second term is given by Corollary 4.1 applied to the state | v 〉 , for η = (cid:178) mK ψ : m (cid:175)(cid:175)(cid:175)(cid:175) F ( ψ , | v 〉〈 v | ) − (cid:69) β ← Q | v 〉〈 v | (cid:183) f ψ (cid:181) β , (cid:178) mK ψ (cid:182)(cid:184)(cid:175)(cid:175)(cid:175)(cid:175) ≤ (cid:178) . (4.140)The bound for the third term is probabilistic, given by Eq. (4.116), for η = (cid:178) mK ψ and µ = (cid:178) (cid:48) m .For all (cid:178) (cid:48) > α (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) n − q − m n − q − m (cid:88) i = f ψ (cid:181) α i , (cid:178) mK ψ (cid:182) − (cid:69) β ← Q | v 〉〈 v | (cid:183) f ψ (cid:181) β , (cid:178) mK ψ (cid:182)(cid:184)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≥ (cid:178) (cid:48) m (cid:35) ≤ (cid:195) n − q q (cid:33) exp − n − q (cid:178) + E (cid:178) (cid:48) m + E K + E ψ M ψ ( (cid:178) mK ψ ) − qn − q − m . (4.141)We now bring together the previous bounds in order to prove Lemma 4.13. CombiningEqs. (4.135), (4.139), (4.140) and (4.141) yieldsPr (cid:163)(cid:175)(cid:175) F ( ψ ⊗ m , Φ m ) − F ψ ( Φ ) (cid:175)(cid:175) > (cid:178) + (cid:178) (cid:48) (cid:164) ≤ Pr α (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) n − q − m n − q − m (cid:88) i = f ψ (cid:181) α i , (cid:178) mK ψ (cid:182) − (cid:69) β ← Q | v 〉〈 v | (cid:183) f ψ (cid:181) β , (cid:178) mK ψ (cid:182)(cid:184)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≥ (cid:178) (cid:48) m (cid:35) .5. CONTINUOUS VARIABLE QUANTUM STATE VERIFICATION PROTOCOL ≤ (cid:195) n − q q (cid:33) exp − n − q (cid:178) + E (cid:178) (cid:48) m + E K + E ψ M ψ ( (cid:178) mK ψ ) − qn − q − m (4.142) + m (4 q + m − n − q = (cid:195) n − q q (cid:33) exp (cid:34) − n − q m + E (cid:181) (cid:178) + E (cid:178) (cid:48) C ψ − qm + E n − q − m (cid:182) (cid:35) + m (4 q + m − n − q ,where C ψ = K + E ψ M ψ (cid:181) (cid:178) mK ψ (cid:182) = E (cid:88) k , l = | ψ k ψ l | (cid:179) (cid:178) m (cid:180) E − k + l K + k + l ψ (cid:118)(cid:117)(cid:117)(cid:116) | l − k | (cid:195) max ( k , l )min ( k , l ) (cid:33) −→ (cid:178) → | ψ E | K + E ψ . (4.143) (cid:3) Combining Eq. (4.131) and Lemma 4.13, we finally obtainPr (cid:183)(cid:175)(cid:175) F ( ψ ⊗ m , ρ m ) − F ψ ( ρ ) (cid:175)(cid:175) > (cid:178) + (cid:178) (cid:48) + q ( E + e − q ( q + n (cid:184) ≤ q ( E + exp (cid:183) − q ( q + n (cid:184) + (cid:195) n − q q (cid:33) exp (cid:34) − n − q m + E (cid:181) (cid:178) + E (cid:178) (cid:48) C ψ − qm + E n − q − m (cid:182) (cid:35) + m (4 q + m − n − q . (4.144)Setting P Hoeffding = (cid:195) n − q q (cid:33) exp (cid:34) − n − q m + E (cid:181) (cid:178) + E (cid:178) (cid:48) C ψ − qm + E n − q − m (cid:182) (cid:35) , (4.145) P choice = m (4 q + m − n − q , (4.146)and P deFinetti = q ( E + exp (cid:183) − q ( q + n (cid:184) , (4.147)we obtainPr (cid:163)(cid:175)(cid:175) F ( ψ ⊗ m , ρ m ) − F ψ ( ρ ) (cid:175)(cid:175) > (cid:178) + (cid:178) (cid:48) + P deFinetti (cid:164) ≤ P deFinetti + P choice + P Hoeffding . (4.148)Until now we have assumed ρ n ∈ S n ¯ H ⊗ n − q . By section 4.5.2,Pr (cid:104) F nq ∩ T k ≤ s (cid:105) ≤ P support . (4.149)where F nq is the event that the projection of ρ n (the remaining state after the support esti-mation step) onto S n ¯ H ⊗ n − q fails, where T k ≤ s is the event that at most s of the k values β i from159 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES the support estimation step satisfy | β i | > E , and where P support = k exp (cid:104) − k (cid:161) qn − sk (cid:162) (cid:105) .With the union bound we thus obtainPr (cid:104)(cid:161)(cid:175)(cid:175) F ( ψ ⊗ m , ρ m ) − F ψ ( ρ ) (cid:175)(cid:175) > (cid:178) + (cid:178) (cid:48) + P deFinetti (cid:162) ∩ T k ≤ s (cid:105) ≤ P support + P deFinetti + P choice + P Hoeffding ,(4.150)where P support = k exp (cid:183) − k (cid:181) qn − sk (cid:182) (cid:184) , P deFinetti = q ( E + exp (cid:183) − q ( q + n (cid:184) , (4.151) P choice = m (4 q + m − n − q , (4.152) P Hoeffding = (cid:195) n − q q (cid:33) exp (cid:34) − n − q m + E (cid:181) (cid:178) + E (cid:178) (cid:48) C ψ − qm + E n − q − m (cid:182) (cid:35) .The variables (cid:178) , (cid:178) (cid:48) , n , m , q , k , s , E are free parameters of the protocol. Let us fix, e.g., E = O (1), s = O (1), n = O (cid:161) m + E (cid:162) , k = O ( m + E ), q = O (cid:161) m + E (cid:162) , and (cid:178) = (cid:178) (cid:48) = O ( m ). Then, either theestimate F ψ ( ρ ) of the fidelity F ( ψ ⊗ m , ρ m ) is polynomially precise (in m ), or the score at thesupport estimation step is higher than s , with polynomial probability (in m ), by plugging thedifferent scalings in the above expressions. (cid:4) These general single-mode state certification and verification protocols may be used for varioususecases. We present selected applications in the following section, relating to the certification ofnon-Gaussian properties of quantum states.
The stellar hierarchy can be certified with the previous protocol using the estimate of the fidelityobtained as a witness for the stellar rank. We recall a few definitions and results from chapter 2.The stellar rank of a single-mode normalised pure quantum state corresponds to the minimalnumber of photon additions necessary to engineer the state from the vacuum, together withGaussian unitary operations. Moreover, a mixed state which has a stellar rank equal to n cannotbe expressed as a mixture of pure states of ranks strictly lower than n . Given k ∈ (cid:78) ∗ and a targetpure state | ψ 〉 , if a mixed state ρ satisfies F ( ψ , ρ ) > − [ R (cid:63) k ( ψ )] , (4.153)where R (cid:63) k ( ψ ) is the k -robustness of the state | ψ 〉 , then it has a stellar rank greater or equal to k .This in turn can be checked by computing the robustness profile of the state | ψ 〉 .160 .6. CERTIFICATION OF NON-GAUSSIAN PROPERTIES With Theorem 4.3 for m =
1, we obtain the following protocol for certifying the stellar rankunder the i.i.d. assumption, where E , s , (cid:178) and (cid:178) (cid:48) are free parameters:Let | ψ 〉 be a target pure state. First, measure with heterodyne detection n copies of the(mixed) state ρ , obtaining the samples α , . . . , α n . Then, record the number r of samples such that | α i | > E . Compute with the same samples the estimate F ψ ( ρ ) = n n (cid:88) i = f ψ (cid:181) α i , (cid:178) K ψ (cid:182) , (4.154)where the function f A and the constant K A are defined in Eqs. (4.2) and (4.3), for A = | ψ 〉〈 ψ | .Then, (cid:175)(cid:175) F ( ψ , ρ ) − F ψ ( ρ ) (cid:175)(cid:175) ≤ (cid:178) + (cid:178) (cid:48) , (4.155)or r > s , with probability greater than1 − (cid:195) ( s + n exp (cid:183) ( s + n + (cid:184) + (cid:34) − n (cid:178) + E (cid:178) (cid:48) C ψ (cid:35)(cid:33) , (4.156)where C ψ = E (cid:88) k , l = | ψ k ψ l | (cid:178) E − k + l K + k + l ψ (cid:118)(cid:117)(cid:117)(cid:116) | l − k | (cid:195) max ( k , l )min ( k , l ) (cid:33) (4.157)is a constant independent of ρ , with the constant K defined in Eq. (4.3). In particular, if theestimate obtained satisfies F ψ ( ρ ) > − [ R (cid:63) k ( ψ )] + (cid:178) + (cid:178) (cid:48) , (4.158)which can be readily checked from the robustness profile of the target state | ψ 〉 , then either thescore at the support estimation step is high or the state ρ has stellar rank greater or equal to k ,with high probability for a large number of samples. An analogous statement holds for the case ofverification, without the i.i.d. assumption, with Theorem 4.4. In the previous section, we detail how to certify a nonzero stellar rank of any experimental(mixed) state, which implies that this state is non-Gaussian. However, such a mixed state maystill have positive Wigner function. Since processes with positive Wigner functions are classicallysimulable [ME12], negativity of the Wigner function is also a crucial property to look for. Inthis section, we show how our certification protocol with heterodyne detection allows for thecertification of Wigner negativity without the need for a full tomography.The Wigner function of a state ρ evaluated at α ∈ (cid:67) is related to the expected value of theparity operator displaced by α [Roy77]: W ρ ( α ) = π Tr (cid:104) ˆ D ( α ) ˆ Π ˆ D † ( α ) ρ (cid:105) , (4.159)161 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES where ˆ Π = (cid:88) n ≥ ( − n | n 〉〈 n | (4.160)is the parity operator. Hence, we can use the certification protocol to obtain mean value estima-tions of the operator π ˆ D ( α ) ˆ Π ˆ D † ( α ) and retrieve the value of the Wigner function at α . Moreover,since the displacement can be reverted in post-processing by translating the samples by α , we canalternatively obtain mean value estimations of the operator π ˆ Π (which has a simpler expressionin Fock basis) using translated samples.Using either the certification protocol from Theorem 4.3 or the verification protocol fromTheorem 4.4 allows us to witness Wigner negativity under or wihtout the i.i.d. assumption,respectively. The certification and verification protocols described in the previous sections allow us to obtainefficiently estimates of fidelities of any single-mode continuous variable quantum state with anytarget single-mode pure state, with analytical confidence intervals, either with i.i.d. assumptionor with no assumption whatsoever. These protocols also allow us to estimate efficiently fidelitieswith multimode i.i.d. pure states. However, translating them directly to efficient protocols forgeneral multimode states seems hopeless, since verifying a multimode state implies accountingfor all possible correlations between its subsystems, of which there is an exponential number inthe size of the state.On the other hand, we show in what follows that being able to estimate single-mode fidelitieswith heterodyne detection is enough to provide fidelity witnesses for a large class of multimodestates. This result combines the following two observations:• If all the single-mode subsystems ρ i of a multimode quantum state ρ are close enoughto single-mode pure states | ψ i 〉〈 ψ i | , then ρ is close to the tensor product of these purestates (Lemma 4.14). In particular, being able to estimate single-mode fidelities is enoughto provide fidelity witnesses for product of pure states.• Passive linear transformations followed by single-mode Gaussian unitary operations andproduct of single-mode balanced heterodyne detections can be simulated by perform-ing unbalanced heterodyne detections first, then post-processing efficiently the samples(Lemma 4.15). In particular, for such an operation ˆ V , if the multimode state ρ can beefficiently certified using heterodyne detection, then it is also the case for the state ˆ V ρ ˆ V † .This allows us to verify efficiently a large class of multimode continuous variable quantum states,with and without the i.i.d. assumption, including the m -mode states of the form (cid:195) m (cid:79) i = ˆ G i (cid:33) ˆ U (cid:195) m (cid:79) i = | ψ i 〉 (cid:33) , (4.161)162 .7. CERTIFYING MULTIMODE CONTINUOUS VARIABLE QUANTUM STATES where ˆ U is a passive linear transformation (a unitary transformation of the creation and annihi-lation operators of the modes) and where, for all i ∈ {
1, . . . , m } , the state | ψ i 〉 is a single-mode purestate with constant energy (which does not scale with the number of modes m ) and the operationˆ G i is a single-mode Gaussian unitary which may be written as a combination of a single-modedisplacement and a single-mode squeezing (see section 1.3). In particular, these states includesmultimode Gaussian states and the output states of Boson Sampling interferometers and of CVScircuits (see sections 1.4.5 and 3.3.3).The fidelity witnesses presented here extend the work of [AGKE15] in various respects.Their work provides fidelity witnesses for multimode photonic state preparations with Gaussianmeasurements, under the i.i.d. assumption. However, the witnesses are for a more restrictedclass of target states and are efficient for Gaussian pure states only. In particular, the number ofcopies needed to certify with constant precision the output of a Boson Sampling interferometerwith n input photons over m modes with their protocol scales as Ω ( m n + ), which is worse thanexponential in the antibunching regime n = O ( (cid:112) m ), while we show that our protocol providestight fidelity witnesses with constant precision with O ( m log m ) copies. Moreover, we are ableto remove the i.i.d. state preparation assumption, at the cost of an increased—though stillpolynomial—number of measurements needed for the same estimate precision and confidenceinterval.In the following sections, we present the general protocol and detail its application in the case ofBoson Sampling. We present the two versions of the multimode verification protocol, with or without i.i.d. assump-tion. Under the i.i.d. assumption:1. The verifier chooses an m -mode target pure state | τ U , ξ , β 〉 : = ˆ S ( ξ ) ˆ D ( β ) ˆ U ( (cid:78) mi = | ψ i 〉 ), as inEq. (4.161), where for all i ∈ {
1, . . . , m } the state | ψ i 〉 has constant energy, where ˆ U is an m -mode passive linear transformation with m × m unitary matrix U and where ξ , β ∈ (cid:67) m . Theverifier also chooses a precision parameter 0 < η < E , . . . , E m .2. The verifier asks the prover for N = O (poly m ) copies of the target state | τ U , ξ , β 〉 . Let ρ ⊗ N be the ( N × m )-mode (mixed) state sent by the prover, where ρ is an m -mode (mixed) state.3. The verifier measures with unbalanced heterodyne detection with unbalancing parameters ξ all the m subsystems of all the N copies of ρ , obtaining the N vectors of samples γ (1) , . . . , γ ( N ) ∈ (cid:67) m .4. For all k ∈ {
1, . . . , N } , the verifier computes α ( k ) = U † ( γ ( k ) − β ). We write α ( k ) = ( α ( k )1 , . . . , α ( k ) m ).5. For all i ∈ {
1, . . . , m } , the verifier records the number r i of values among α (1) i , . . . , α ( N ) i suchthat | α ( k ) i | > E i ( support estimation ). 163 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES
6. For all i ∈ {
1, . . . , m } , the verifier computes the mean ˜ F i of the function z (cid:55)→ f ψ i ( z , (cid:178) , E i ) overthe same values α (1) i , . . . , α ( N ) i , where the function f is defined in Eq.( 4.2).7. The verifier computes ˜ W = − (cid:80) mi = (1 − ˜ F i ).The cutoff values E , . . . , E m should be chosen by the verifier to guarantee completeness for theestimation of the single-mode fidelities, i.e., that if the prover is sending a near-ideal state it isaccepted with high probability. For a sufficiently large number of copies N = O (poly m ), we showin what follows that ˜ W is a tight lower bound on the fidelity with inverse polynomial precision, orone of the scores r , . . . , r m is high, with high probability.Without i.i.d. assumption, an equivalent protocol is obtained by using the version of theprotocol which does not assume i.i.d. state preparation for estimating the single-mode fidelitiesin Theorem 4.4. In that case, the final protocol is nearly identical, up to slight differences forthe classical post-processing: a small fraction of the measured subsystems have to be discardedat random and the samples used for the support estimation step must be randomly chosenand cannot be used to compute the fidelity estimates. This comes at the cost of an increasednumber of measurements necessary for the same witness precision and confidence interval, whichcorresponds however to a polynomial overhead in m .For both protocols, note that the efficiency may be greatly refined by taking into accountthe expression of the single-mode target pure states | ψ i 〉 in Fock basis. We give an exampleof such optimisation in the next section, in the case of Boson Sampling output states, whenthe single-mode target pure states are either single-photon Fock states or vacuum states. Inparticular, for the protocol under i.i.d. assumption, if the single-mode target states have a finitesupport over the Fock basis then the support estimation step is no longer needed.We now show that the estimate ˜ W is a tight fidelity witness for a number of samples O (poly m ).We first prove the following result: Lemma 4.14.
Let ρ be an m-mode state. For all i ∈ {
1, . . . , m } , we denote by ρ i the single-modereduced state of ρ over the i th mode. Let | ψ 〉 , . . . , | ψ m 〉 be single-mode pure states. For all i ∈ {
1, . . . , m } , we write F ( ρ i , ψ i ) = − (cid:178) i , where F is the fidelity. Then, − m (cid:88) i = (cid:178) i ≤ F ( ρ , ψ ⊗ · · · ⊗ ψ m ) ≤ m (cid:89) i = (1 − (cid:178) i ). (4.162) In particular, when (cid:178) = · · · = (cid:178) m = (cid:178) , − m (cid:178) ≤ F ( ρ , ψ ⊗ · · · ⊗ ψ m ) ≤ (1 − (cid:178) ) m . (4.163) Proof.
Since | ψ 〉 , . . . , | ψ m 〉 are pure states, F ( ρ , ψ ⊗ · · · ⊗ ψ m ) = Tr [ ρ | ψ 〉〈 ψ | ⊗ · · · ⊗ | ψ m 〉〈 ψ m | ], (4.164)164 .7. CERTIFYING MULTIMODE CONTINUOUS VARIABLE QUANTUM STATES and F ( ρ i , ψ i ) = Tr[ ρ i | ψ i 〉〈 ψ i | ] = Tr[ ρ (cid:49) i − ⊗ | ψ i 〉〈 ψ i | ⊗ (cid:49) m − i ] (4.165)for all i ∈ {
1, . . . , m } . The left hand side of Eq. (4.162) is obtained by writing F ( ρ , ψ ⊗ · · · ⊗ ψ m )as a telescopic sum:Tr [ ρ | ψ 〉〈 ψ | ⊗ · · · ⊗ | ψ m 〉〈 ψ m | ] = Tr [ ρ (cid:49) m ] − Tr [ ρ ( (cid:49) − | ψ 〉〈 ψ | ) ⊗ (cid:49) m − ] − Tr [ ρ | ψ 〉〈 ψ | ⊗ ( (cid:49) − | ψ 〉〈 ψ | ) ⊗ (cid:49) m − ] − Tr [ ρ | ψ 〉〈 ψ | ⊗ | ψ 〉〈 ψ | ⊗ ( (cid:49) − | ψ 〉〈 ψ | ) ⊗ (cid:49) m − ] − . . . − Tr [ ρ | ψ 〉〈 ψ | ⊗ | ψ 〉〈 ψ | ⊗ · · · ⊗ ( (cid:49) − | ψ m 〉〈 ψ m | )] ≥ − m (cid:88) i = (cid:161) − Tr[ ρ (cid:49) i − ⊗ | ψ i 〉〈 ψ i | ⊗ (cid:49) m − i ] (cid:162) , (4.166)by linearity of the trace, where we used Tr ( ρ ) =
1. This gives F ( ρ , ψ ⊗ · · · ⊗ ψ m ) ≥ − m (cid:88) i = (cid:161) − F ( ρ i , ψ i ) (cid:162) , (4.167)with Eqs. (4.164) and (4.165).The right hand side of Eq. (4.162) is obtained by Cauchy-Schwarz inequality and a simpleinduction:Tr [ ρ | ψ 〉〈 ψ | ⊗ · · · ⊗ | ψ m 〉〈 ψ m | ] = Tr (cid:163)(cid:161) (cid:112) ρ | ψ 〉〈 ψ | ⊗ (cid:49) m − (cid:162) (cid:161) (cid:49) ⊗ | ψ 〉〈 ψ | ⊗ · · · ⊗ | ψ m 〉〈 ψ m | (cid:112) ρ (cid:162)(cid:164) ≤ Tr (cid:163) ρ | ψ 〉〈 ψ | ⊗ (cid:49) m − (cid:164) Tr (cid:163) ρ (cid:49) ⊗ | ψ 〉〈 ψ | ⊗ · · · ⊗ | ψ m 〉〈 ψ m | (cid:164) ≤ . . . ≤ m (cid:89) i = Tr [ ρ (cid:49) i − ⊗ | ψ i 〉〈 ψ i | ⊗ (cid:49) m − i ], (4.168)where we used the cyclicity of the trace and the fact that | ψ 〉 , . . . , | ψ m 〉 are pure states. Thisgives F ( ρ , ψ ⊗ · · · ⊗ ψ m ) ≤ m (cid:89) i = F ( ρ i , ψ i ), (4.169)with Eqs. (4.164) and (4.165).Writing F ( ρ i , ψ i ) = − (cid:178) i , we obtain, with Eqs. (4.167) and (4.169),1 − m (cid:88) i = (cid:178) i ≤ F ( ρ , ψ ⊗ · · · ⊗ ψ m ) ≤ m (cid:89) i = (1 − (cid:178) i ), (4.170)165 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES which concludes the proof. Additionnally, by the inequality of arithmetic and geometricmeans, m (cid:89) i = (1 − (cid:178) i ) ≤ (cid:195) − m m (cid:88) i = (cid:178) i (cid:33) m (4.171) ≤ exp (cid:195) − m (cid:88) i = (cid:178) i (cid:33) ,which gives a looser bound in terms of the total single-mode deviation (cid:80) mi = (cid:178) i :1 − m (cid:88) i = (cid:178) i ≤ F ( ρ , ψ ⊗ · · · ⊗ ψ m ) ≤ exp (cid:195) − m (cid:88) i = (cid:178) i (cid:33) . (4.172) (cid:4) Note that Eq. (4.163) is tight for small (cid:178) , since its right hand side is then equivalent to 1 − m (cid:178) .Lemma 4.14 implies that if the fidelities of single-mode subsystems of an m -mode quantum statewith some target pure states are higher than 1 − λ m , for some λ >
0, then the m -mode state hasfidelity at least 1 − λ with the target m -mode product state.Together with the union bound and the single-mode certification and verification protocolsfrom Theorems 4.3 and 4.4, this provides a means for obtaining efficiently tight fidelity witnesseswith any target tensor product of single-mode pure states with analytical confidence intervals,with and without i.i.d. assumption.At this point, we can obtain fidelity witnesses only for pure product states, with no entanglement,using a fidelity estimation protocol for each of the single-mode subsystems in parallel. We makeuse of the properties of heterodyne detection in order to extend the class of target states for whichfidelity witnesses can be efficiently obtained, from pure product states to the multimode statesthat are obtained from a pure product state with a passive linear transformation followed bysingle-mode Gaussian unitary operations, as in Eq. (4.161).The POVM elements of product single-mode unbalanced heterodyne detection over m modeswith unbalancing parameters ξ ∈ (cid:67) m are given by (see section 1.4.2) Π ξα = π m | α , ξ 〉〈 α , ξ | , (4.173)for all α = ( α , . . . , α m ) ∈ (cid:67) m , where | α , ξ 〉 = (cid:78) mi = | α i , ξ i 〉 is a product of squeezed coherent statesˆ S ( ξ i ) ˆ D ( α i ) | 〉 .The POVM elements of product single-mode balanced heterodyne detection are given by Π α ,for all α ∈ (cid:67) m , and we have Π ξα = ˆ S ( ξ ) Π α ˆ S † ( ξ ). In particular, a single-mode squeezing followedby a single-mode balanced heterodyne detection can be simulated by performing directly anunbalanced heterodyne detection according to the squeezing parameter. One retrieves balancedheterodyne detection by setting the unbalancing parameter to 0 and homodyne detection byletting the modulus of the unbalancing parameter go to infinity.166 .7. CERTIFYING MULTIMODE CONTINUOUS VARIABLE QUANTUM STATES Passive linear transformations correspond to unitary transformations of the creation andannihilation operators of the modes. These transformations, which may be implemented byunitary optical interferometers, map coherent states to coherent states: if ˆ U is a passive lineartransformation and U is the unitary matrix describing its action on the creation and annihilationoperators of the modes, an input coherent state | α 〉 is mapped to an output coherent stateˆ U | α 〉 = | U α 〉 , where U α is obtained by multiplying the vector α by the unitary matrix U . Hence,the POVM elements corresponding to a passive linear transformation ˆ U followed by a productof single-mode balanced heterodyne detection are given by ˆ U Π α ˆ U † = Π U α , for all α ∈ (cid:67) m . Thisimplies that the passive linear transformation ˆ U † followed by a product of single-mode heterodynedetections can be simulated by performing the heterodyne detections first, then multiplying thevector of samples obtained by U .A similar property holds with single-mode displacements: since displacements map coherentstates to coherent states, up to a global phase, by displacing their amplitude, a single-modedisplacement followed by a single-mode heterodyne detection can be simulated by performing theheterodyne detection first, then translating the sample obtained according to the displacementamplitude. In particular we have ˆ D ( β ) Π α ˆ D † ( β ) = Π α + β for all α , β ∈ (cid:67) m , where α + β = ( α + β , . . . , α m + β m ).Combining the properties of heterodyne detection we obtain the following result: Lemma 4.15.
Let β , ξ ∈ (cid:67) m and let ˆ V = ˆ S ( ξ ) ˆ D ( β ) ˆ U, where ˆ U is an m-mode passive linear trans-formation with m × m unitary matrix U. For all α ∈ (cid:67) m , let γ = U α + β . Then, Π ξγ = ˆ V Π α ˆ V † . (4.174) Proof.
We have Π ξα = π m | α , ξ 〉〈 α , ξ | , for all α , ξ ∈ (cid:67) m , where | α , ξ 〉 = ˆ S ( ξ ) ˆ D ( α ) | 〉 is a tensorproduct of squeezed coherent states. We also have ˆ U Π α ˆ U † = Π U α ,ˆ D ( β ) Π α ˆ D † ( β ) = Π α + β ,ˆ S ( ξ ) Π α ˆ S † ( ξ ) = Π ξα , (4.175)for all α , β , ξ ∈ (cid:67) m and all m -mode passive linear transformations ˆ U with m × m unitarymatrix U . Writing ˆ V = ˆ S ( ξ ) ˆ D ( β ) ˆ U , we obtainˆ V Π α ˆ V † = ˆ S ( ξ ) ˆ D ( β ) ˆ U Π α ˆ U † ˆ D † ( β ) ˆ S † ( ξ ) = ˆ S ( ξ ) ˆ D ( β ) Π U α ˆ D † ( β ) ˆ S † ( ξ ) = ˆ S ( ξ ) Π U α + β ˆ S † ( ξ ) = Π ξ U α + β . (4.176) (cid:4) HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES
Lemma 4.15 implies that the POVM { ˆ V Π α ˆ V † } α ∈ (cid:67) m can be simulated with the POVM { Π ξγ } γ ∈ (cid:67) m by computing α = U † ( γ − β ), i.e., translating the vector of samples γ by the vector of complexamplitudes − β and multiplying the vector obtained by the m × m unitary matrix U † . Thismeans that a passive linear transformation followed by single-mode Gaussian unitary operationsbefore balanced heterodyne detection can be simulated by performing unbalanced heterodynedetection directly, then post-processing efficiently the classical outcomes. In particular, for such atransformation ˆ V , if a multimode pure product state (cid:78) mi = | ψ i 〉 can be efficiently verified usingbalanced heterodyne detection, then the state ˆ V ( (cid:78) mi = | ψ i 〉 ) can be efficiently verified usingunbalanced heterodyne detection.Formally, let ρ be an m -mode (mixed) state. Let | ψ 〉 , . . . , | ψ m 〉 be single-mode pure states andlet ˆ V = ˆ S ( ξ ) ˆ D ( β ) ˆ U , with β , ξ ∈ (cid:67) m , where ˆ U is a passive linear transformation over m modes withan associated m × m unitary matrix U . Then, F ( ρ , ˆ V | ψ 〉〈 ψ | ⊗ · · · ⊗ | ψ m 〉〈 ψ m | ˆ V † ) = F ( ˆ V † ρ ˆ V , | ψ 〉〈 ψ | ⊗ · · · ⊗ | ψ m 〉〈 ψ m | ) ≥ − m (cid:88) i = (cid:179) − F ( | ψ i 〉〈 ψ i | , ( ˆ V † ρ ˆ V ) i ) (cid:180) , (4.177)where we have used Lemma 4.14 and where ( ˆ V † ρ ˆ V ) i is the i th single-mode reduced densitymatrix of the state ˆ V † ρ ˆ V .The single-mode fidelities F ( | ψ i 〉〈 ψ i | , ( ˆ V † ρ ˆ V ) i ) can be estimated with analytical confidenceintervals by measuring multiple copies of the m -mode state ˆ V † ρ ˆ V with product balanced het-erodyne detection and post-processing the samples for individual subsystems according to theprotocols from Theorems 4.3 and 4.4. By Lemma 4.15, this is equivalent to measuring the state ρ directly with a product of single-mode unbalanced heterodyne detections with unbalancingparameters ξ , translating the vector of samples γ obtained by the vector of complex amplitudes − β and multiplying the vector obtained by the unitary matrix U † . Then, the obtained samplesmay be post-processed according to the heterodyne certification or verification protocols.If all the single-mode fidelity estimates obtained are precise to m and greater than1 − m with high probability, which can be checked in time O (poly m ), then with the unionbound for the failure probabilities, the fidelity between the m -mode state ρ and the targetstate ˆ V | ψ 〉〈 ψ | ⊗ · · · ⊗ | ψ m 〉〈 ψ m | ˆ V † is greater than 1 − m , with high probability. Hence thesingle-mode fidelity estimation protocols give a verification protocol for obtaining tight multimodefidelity witnesses, under or without the i.i.d. assumption.The single-mode protocols from Theorems 4.3 and 4.4 are efficient as long as the energy of thesingle-mode target pure state is constant, i.e., it does not scale with the number of modes. Notethat additional displacements may be introduced to reduce the energy of the single-mode targetpure states, since by modifying their amplitudes these displacements can be braided throughthe transformation ˆ V and accounted for by translating the heterodyne detection samples. Theefficiently verifiable states thus are the pure states of the form ˆ S ( ξ ) ˆ D ( β ) ˆ U ( (cid:78) mi = | ψ i 〉 ), such thatfor all i ∈ {
1, . . . , m } , the state | ψ i 〉 can be displaced onto a state of constant energy.168 .7. CERTIFYING MULTIMODE CONTINUOUS VARIABLE QUANTUM STATES In particular, multimode Gaussian pure states with constant squeezing parameter can beefficiently verified, since these can be written as a product of pure single-mode squeezed coherentstates followed by a passive linear transformation (see section 1.3). Note however that under thei.i.d. assumption the witnesses from [AGKE15] may provide a more efficient certification methodfor Gaussian states.Remarkably, the class of efficiently verifiable states also includes the output states of CVScircuits and Boson Sampling interferometers. Our verification protocol may thus be used to verifyquantum supremacy, as we detail in the following section.
The experimental demonstration of quantum computational supremacy is regarded as an im-portant milestone in the field of quantum information. It involves a quantum device solvingefficiently a computational task which is provably hard for classical computers, together witha verification of its correct functionning [HM17]. While the former has been recently accom-plished with superconducting circuits [AAB + + HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES with single-mode Gaussian measurements under the i.i.d. assumption [AL18]. However, there isno efficient certification nor verification protocol using single-mode Gaussian measurements forBoson Sampling with input single photons: current methods used for validation of Boson samplingare either not scalable, e.g., computing the total variation distance with the ideal probabilitydistribution, or else provide incomplete certificates, e.g., telling apart the tested distribution fromclassical mock-up distributions such as the uniform distribution [BGC +
19, WQD + n = O ( (cid:112) m ). Inparticular, verifying a Boson Sampling quantum supremacy experiment amounts to verifyingthat the experimental quantum device samples from an ideal probability distribution, up to aconstant error in total variation distance.Our verification protocol derived in the previous section can be applied to check efficientlythe fidelity of the output state of an experimental Boson Sampling interferometer with the idealoutput state, using only balanced heterodyne detection. The fidelity witness gives in turn acertificate of the total variation distance with the ideal probability distribution for any observableby Eq. (4.89), therefore allowing for an experimental demonstration of quantum supremacy withBoson Sampling, with a verifier having minimal continuous variable quantum computationalpower, namely the ability to perform single-mode Gaussian measurements.Performing verified Boson Sampling with our protocol, even under i.i.d. assumption, wouldalready provide a convincing evidence of quantum supremacy with photonic quantum computing,as the verification without i.i.d. assumption only comes at the cost of an increased number ofmeasurements, still polynomial in the number of modes m .To that end, we optimise the bounds for the multimode certification protocol under i.i.d.assumption. In particular, the support estimation step in the protocol is no longer necessary,because the single-mode target pure states are either single-photon Fock states or vacuum statesand thus have finite support over the Fock basis. We show that the number of copies neededfor a constant additive precision in the antibunching regime n = O ( (cid:112) m )—which is required fora demonstration of quantum supremacy—scales as O ( m log m ), making reliable verificationof Boson Sampling using single-mode Gaussian measurements within the reach of currentexperiments. Remarkably, this is only a logarithmic factor harder than verifying multimodeGaussian states [AGKE15].Let 0 < η < z ∈ (cid:67) , f ( z , η ) = η exp (cid:183)(cid:181) − η (cid:182) | z | (cid:184) , (4.178)170 .7. CERTIFYING MULTIMODE CONTINUOUS VARIABLE QUANTUM STATES and f ( z , η ) = η ( | z | η −
1) exp (cid:183)(cid:181) − η (cid:182) | z | (cid:184) . (4.179)The verification protocol for Boson Sampling with n photons fed into a unitary interferometer U of size m under i.i.d. assumption reads:1. The verifier chooses two precision parameters 0 < η , η < N = O ( m log m ) copies of the target state U | 〉 .Let ρ ⊗ N be the ( N × m )-mode (mixed) state sent by the prover, where ρ is an m -mode(mixed) state.3. The verifier measures with balanced heterodyne detection all the m subsystems of all the N copies of ρ , obtaining the N vectors of samples γ (1) , . . . , γ ( N ) ∈ (cid:67) m .4. For all k ∈ {
1, . . . , N } , the verifier computes α ( k ) = U † γ ( k ) . We write α ( k ) = ( α ( k )1 , . . . , α ( k ) m ).5. For all i ∈ { n +
1, . . . , m } the verifier computes the mean ˜ F i of the function z (cid:55)→ f ( z , η ) overthe values α (1) i , . . . , α ( N ) i and for all j ∈ {
1, . . . , n } the mean ˜ F j of the function z (cid:55)→ f ( z , η ).6. The verifier computes ˜ W = − (cid:80) mi = (1 − ˜ F i ). Theorem 4.6 (Certification of Boson Sampling using Gaussian measurements) . ˜ W is an estimatewith constant precision of a tight lower bound on the fidelity with the ideal Boson Sampling outputstate, with probability exponentially close to . The estimate ˜ W thus provides an efficient and reliable certificate of the total variation distancewith the ideal probability distribution for any observable by Eq. (1.14). Proof.
Let 0 < η < ρ = (cid:80) k , l ≥ ρ kl | k 〉〈 l | , (cid:69) α ← Q ρ ( α ) [ f ( α , η )] = Tr ( ρ | 〉〈 | ) + η +∞ (cid:88) n = η n ρ n + n + , (4.180)and (cid:69) α ← Q ρ [ f ( α , η )] = Tr ( ρ | 〉〈 | ) + η +∞ (cid:88) n = η n ( n + ρ n + n + , (4.181)where (cid:69) α ← Q ρ [ f ] denotes the expected value of the function f for samples from single-modebalanced heterodyne detection of ρ .Since η ≤ η n + < η n and η n + ( n + < η n ( n +
2) for all n ∈ (cid:78) , so by a simpleinduction η n ≤ η n ( n + ≤
2, for all n ∈ (cid:78) . In particular, (cid:80) +∞ n = η n ρ n + n + ≤ (cid:80) +∞ n = η n ( n + ρ n + n + ≤
2, since Tr ( ρ ) =
1. With Eqs. (4.180) and (4.181) we obtain (cid:175)(cid:175)(cid:175)(cid:175)
Tr ( ρ | 〉〈 | ) − (cid:69) α ← Q ρ [ f ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175) ≤ η , (4.182)171 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES and (cid:175)(cid:175)(cid:175)(cid:175)
Tr ( ρ | 〉〈 | ) − (cid:69) α ← Q ρ [ f ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175) ≤ η . (4.183)We also have, for all z ∈ (cid:67) , 0 < f ( z , η ) ≤ η , (4.184)and − η ≤ f ( z , η ) ≤ e η − η (1 − η ) . (4.185)For η < e η − (1 − η ) <
1. In particular, the range of the function f is less than η .Let N ∈ (cid:78) ∗ , let α , . . . , α N be i.i.d. samples from single-mode balanced heterodyne detec-tion of a single mode state ρ and let β , . . . , β N be i.i.d. samples from single-mode balancedheterodyne detection of a single mode state ρ . Let (cid:178) , η , (cid:178) , η >
0, by Hoeffding inequality,Pr (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) N N (cid:88) p = f ( α p , η ) − (cid:69) α ← Q ρ [ f ( α , η )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≥ (cid:178) (cid:35) ≤ e − N (cid:178) η , (4.186)and Pr (cid:34)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) N N (cid:88) p = f ( β p , η ) − (cid:69) β ← Q ρ [ f ( β , η )] (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≥ (cid:178) (cid:35) ≤ e − N (cid:178) η . (4.187)Let σ be an m -mode state and let σ i denote its k th single-mode subsystem for all k ∈ {
1, . . . , m } .Let α (1) , . . . , α ( N ) ∈ (cid:67) m be samples from product balanced heterodyne detection of N identicalcopies of the state σ . For all p ∈ {
1, . . . , N } , we write α p = ( α ( p )1 , . . . , α ( p ) m ).Combining Eqs. (4.182) and (4.186) we obtain, for any i ∈ { n +
1, . . . , m } , (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) Tr ( σ i | 〉〈 | ) − N N (cid:88) p = f (cid:179) α ( p ) i , η (cid:180)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ (cid:178) + η , (4.188)with probability greater than 1 − (cid:163) − N (cid:178) η (cid:164) . Similarly, combining Eqs. (4.183) and (4.187)we obtain, for any j ∈ {
1, . . . , n } , (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) Tr ( σ j | 〉〈 | ) − N N (cid:88) p = f (cid:179) α ( p ) j , η (cid:180)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ (cid:178) + η , (4.189)with probability greater than 1 − (cid:163) − N (cid:178) η /2 (cid:164) .We now choose (cid:178) , η , (cid:178) , η in order to minimize the error probabilities for a givenprecision. Let λ >
0. Setting (cid:178) + η = λ , the optimal choice, which maximises (cid:178) η , is (cid:178) = η = λ and with Eq. (4.188) we obtain, for any i ∈ { n +
1, . . . , m } , (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) Tr ( σ i | 〉〈 | ) − N N (cid:88) p = f (cid:181) α ( p ) i , λ (cid:182)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ λ , (4.190)172 .7. CERTIFYING MULTIMODE CONTINUOUS VARIABLE QUANTUM STATES with probability greater than 1 − (cid:104) − N λ (cid:105) .Let λ >
0. Setting (cid:178) + η = λ , the optimal choice, which maximises (cid:178) η , is (cid:178) = η = λ and with Eq. (4.189) we obtain, for any j ∈ {
1, . . . , n } , (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) Tr ( σ j | 〉〈 | ) − N N (cid:88) p = f (cid:181) α ( p ) j , λ m (cid:182)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) ≤ λ , (4.191)with probability greater than 1 − (cid:104) − N λ (cid:105) .Let us define the fidelity witness W : = − (cid:195) m (cid:88) k = n + − F ( σ k , | 〉〈 | ) + n (cid:88) l = − F ( σ k , | 〉〈 | ) (cid:33) , (4.192)as in Lemma 4.14, and the witness estimate˜ W ( λ , λ ) : = − (cid:195) m (cid:88) i = n + (cid:34) − N N (cid:88) p = f (cid:181) α ( p ) i , λ (cid:182)(cid:35) + n (cid:88) j = (cid:34) − N N (cid:88) p = f (cid:181) α ( p ) j , λ m (cid:182)(cid:35)(cid:33) . (4.193)Taking the union bound of the failure probabilities for i ∈ { n +
1, . . . , m } and j ∈ {
1, . . . , n } , weobtain with Eqs. (4.190) and (4.191), (cid:175)(cid:175) W − ˜ W ( λ , λ ) (cid:175)(cid:175) ≤ ( m − n ) λ + n λ , (4.194)with probability greater than1 − (cid:195) ( m − n ) exp (cid:34) − N λ (cid:35) + n exp (cid:34) − N λ (cid:35)(cid:33) . (4.195)By Lemma 4.14 we have F ( σ , | 〉〈 | ) ≥ W , (4.196)hence with Eq. (4.194) we obtain F ( σ , | 〉〈 | ) ≥ ˜ W ( λ , λ ) − ( m − n ) λ − n λ , (4.197)with probability greater than1 − (cid:195) ( m − n ) exp (cid:34) − N λ (cid:35) + n exp (cid:34) − N λ (cid:35)(cid:33) . (4.198)Let ρ = ˆ U † σ ˆ U , where ˆ U is an m -mode passive linear transformation with m × m unitarymatrix U . By Lemma 4.15 from the main text, the estimate ˜ W can be computed using173 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES samples of product balanced heterodyne detection of ρ multiplied by the unitary matrix U † ,rather than samples from the balanced heterodyne detection of σ .With Eqs. (4.197) and (4.198) we obtain F ( ρ , ˆ U | 〉〈 | ˆ U † ) ≥ ˜ W ( λ , λ ) − ( m − n ) λ − n λ , (4.199)with probability greater than1 − (cid:195) ( m − n ) exp (cid:34) − N λ (cid:35) + n exp (cid:34) − N λ (cid:35)(cid:33) , (4.200)where ˜ W ( λ , λ ) is computed using samples of product balanced heterodyne detection of N copies of the m -mode state ρ , each of the N vectors of samples being multiplied by theunitary matrix U † .The values of λ and λ must be chosen by the verifier to maximise the above probabilityfor a given precision of the witness. Equivalently, one needs to minimise:( m − n ) exp (cid:34) − N λ (cid:35) + n exp (cid:34) − N λ (cid:35) , (4.201)with the constraint ( m − n ) λ + n λ = (cid:178) , for (cid:178) >
0. For a given experimental setup, Eq. (4.201)should be minimised depending on the values of m and n .For example, setting λ = (cid:178) m − n ) and λ = (cid:178) n gives F ( ρ , ˆ U | 〉〈 | ˆ U † ) ≥ ˜ W (cid:181) (cid:178) m − n ) , (cid:178) n (cid:182) − (cid:178) , (4.202)with probability greater than1 − (cid:181) ( m − n ) exp (cid:183) − N (cid:178) [4( m − n )] (cid:184) + n exp (cid:183) − N (cid:178) n ) (cid:184)(cid:182) , (4.203)and the estimate ˜ W is (cid:178) -close to the actual fidelity witness, which by Lemma 4.14 is a tightwitness of the fidelity. In particular, for a constant precision fidelity witness W , the estimate˜ W yields a constant precision fidelity witness with probability exponentially close (in m ) to1 for N = O (max { ( m − n ) log( m − n ), n log n } ). In the antibunching regime n = O ( (cid:112) m ), thismeans that the estimate has constant precision with exponentially small failure probabilityalready for N = O ( m log m ). (cid:4) A similar Boson Sampling verification protocol without i.i.d. assumption is obtained by usingthe version of the protocol which does not assume i.i.d. state preparation for estimating thesingle-mode fidelities in Theorem 4.4. This comes at the cost of an increased number of samplesnecessary for the same witness precision and confidence interval, which corresponds to a poly-174 .7. CERTIFYING MULTIMODE CONTINUOUS VARIABLE QUANTUM STATES nomial overhead in m and a slightly different classical post-processing: a small fraction of themeasured subsystems have to be discarded at random and an additional support estimation stepis necessary, for which the samples must be randomly chosen and cannot be used to computethe fidelity estimates. More precisely, changing the parameters of the protocol above to thoseof Theorem 4.4, we get that Theorem 4.6 holds without the i.i.d. assumption with polynomialconfidence. . . .. . . . . .. . . | i
0, with | ζ | = Ω (2 − poly m ) allows us to perform efficiently a sampling task whichis hard for classical computers, unless the polynomial hierarchy collapses.An interesting point is that by changing the unbalancing of heterodyne detection of the outputmodes of a Boson Sampling interferometer, one can switch between verification of Boson Samplingoutput states and demonstration of quantum sampling supremacy with continuous variablemeasurements. Indeed, CVS SP circuits introduced in the previous chapter, which correspond toBoson Sampling with unbalanced heterodyne detection, are hard to sample classically when theunbalancing of the heterodyne detection is not too small (see section 3.3.3), but their output canbe efficiently certified simply by switching to balanced heterodyne detection and computing afidelity witness with the above method. This can be done within the same experimental setupusing a reconfigurable beam splitter (Fig. 4.2) and showing the hardness of approximate CVScircuits sampling is an important step before an experimental demonstration.Alternatively, by switching between balanced heterodyne detection and single-photon thresh-old detection, one can switch between verification of Boson Sampling output states and demon-175 HAPTER 4. CERTIFICATION OF CONTINUOUS VARIABLE QUANTUM STATES stration of quantum sampling supremacy with discrete variable measurements, for which ap-proximate sampling hardness is demonstrated, assuming two conjectures on the permanent ofrandom Gaussian matrices and the fact that the polynomial hierarchy of complexity classes doesnot collapse [AA13].
Existing methods for building trust for continuous variable quantum states like homodynequantum state tomography require many different measurement settings, and heavy classicalpost-processing. For that purpose, we have introduced a reliable method for heterodyne quantumstate tomography, which uses heterodyne detection as a single Gaussian measurement settingand allows for the retrieval of the density matrix of an unknown quantum state with analyticalconfidence intervals, without the need for data reconstruction nor binning of the sample space.For data reconstruction methods such as Maximum Likelihood, errors from the reconstructionprocedure are usually indistinguishable from errors coming from the tested quantum device. Forthat reason, such methods do not extend well to the task of verification, unlike our method.Building on these tomography techniques and with the addition of cryptographic techniquessuch as the de Finetti theorem, we have derived a protocol for verifying various copies of acontinuous variable quantum state, without i.i.d. assumption, with Gaussian measurements.This protocol is robust, as it directly gives a confidence interval on an estimate of the fidelitybetween the tested state and the target pure state. We emphasize that, while the target stateis pure, the tested state is not required to be pure. The general protocol may be tailored todifferent uses and assumptions, from tomography to verification, simply by changing the classicalpost-processing.Our verification protocol is complementary to the approach of [TMM + .8. DISCUSSION AND OPEN PROBLEMS crucially relies on being able to revert efficiently, at the stage of classical post-processing, specificquantum operations (passive linear transformations in this case) after a specific measurement(heterodyne detection, i.e., sampling from the Husimi Q function in this case).177 H A PT ER Q UANTUM - PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS D istinguishing two unknown quantum states is central to many quantum applications [MdW13],notably for entanglement testing [MKB05, WRD +
06, HM13], quantum communica-tion [BCWDW01, dB04, KDK17] and quantum machine learning [EAO +
02, LMR13].This task is referred to as unknown quantum state discrimination .The ability to program a fixed computer to perform a variety of computations is especiallyimportant: we do not want to build a new physical device for every different computation. Inparticular, quantum-programmable devices are quantum machines that take additional quantumstates in input as a program, which dictates the rest of the computation. It is not possible to build afixed quantum computer which can be programmed to perform any quantum computation [NC97],but we can design quantum-programmable devices for a restricted set of computations, such asprojective measurements.In this chapter, we show a correspondence between unknown quantum state discriminationand quantum-programmable measurements, by generalising the celebrated swap test [BCWDW01]for quantum state discrimination to an unbalanced setting where multiple copies of only one ofthe two tested states are available.Next, we also generalise a known link between the Hong–Ou–Mandel effect for partiallydistinguishable photons and the swap test [GECP13]: we present the Hadamard interferometerand show that it provides a scheme for performing unknown quantum state discrimination andquantum-programmable measurements with linear optics and single photons.In order to reduce the experimental requirements for implementation, we consider the case ofprojective measurements onto coherent states and simplify the previously derived scheme. Inthis case, we perform a simple analysis of the consequences of experimental imperfections.This chapter is based on [CDM +
18, KCK + HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS
In the previous chapter, we discussed the efficient characterization of a continuous variablequantum state, either by full tomographic reconstruction, by fidelity estimation with a targetstate, or by obtaining a fidelity witness with a target state. In some cases, however, one merelywants to test simple properties of quantum systems. Given two unknown quantum states, oneof the simplest questions one may ask is whether these states are equal or not. In this section,we make a connection between unknown quantum state discrimination schemes and quantum-programmable projective measurement devices.
The swap test [BCWDW01] provides a simple probabilistic tool to compare two unknown quan-tum states. It takes as input two quantum states | φ 〉 and | ψ 〉 that are not entangled and outputs0 with probability + | 〈 φ | ψ 〉 | and 1 with probability − | 〈 φ | ψ 〉 | , where 〈 φ | ψ 〉 is the innerproduct of the states | φ 〉 and | ψ 〉 . When the measurement outcome is 0 (resp. 1), we conclude thatthe states were identical (resp. different), up to a global phase. H H
SWAP | i
2. We introduce the following generalisation of the swap test, in the context where onehas access to various copies of a reference state | ψ 〉 but to only a single copy of the other testedstate | φ 〉 : Definition 5.1 (Swap test of order m ) . The swap test of order m is a binary test that takes asinput a state | φ 〉 and m − | ψ 〉 , and outputs 0 with probability m + m − m | 〈 φ | ψ 〉 | and 1 with probability ( m − m )(1 −| 〈 φ | ψ 〉 | ). If the outcome 0 (resp. 1) is obtained, the test concludesthat the states | φ 〉 and | ψ 〉 were identical (resp. different).Such a test clearly satisfies the one-sided error requirement. HHH HHH S S S n-1 | i
0, . . . , n − } S k = (cid:79) i ∈ [ k − ] , j ∈ [ n − k − − ]SWAP (cid:104) j k + + i , j k + + i + k (cid:105) , (5.2)with SWAP[ i , j ] being the unitary operation that swaps the i th and j th qubits for i , j ∈ {
0, . . . , m − } .These controlled gates are applied to the input states | φ 〉 , | ψ 〉 , . . . , | ψ 〉 (one copy of a state | φ 〉 and m − | ψ 〉 ). Finally, a Hadamard gate is applied to each ancilla, which is thenmeasured in the computational basis. By a simple induction, we obtain that the probability ofobtaining the outcome 0 for all ancilla qubits is the squared norm of the following state:1 m ( | φψ . . . ψ 〉 + | ψφ . . . ψ 〉 + · · · + | ψ . . . ψφ 〉 ), (5.3)which only depends on the overlap between the states | φ 〉 and | ψ 〉 . More precisely,Pr [0, . . . , 0] = m + m − m | 〈 φ | ψ 〉 | . (5.4)The swap circuit of order m thus implements the swap test of order m . Indeed, if the outcome(0, . . . , 0) is obtained, the test outputs 0 and we conclude that the states were identical, while forany other outcome the test outputs 1 and we conclude that the states were different. Note that inthe case where m =
2, the scheme reduces to the original swap test.Because the m − m by replacing the n = log m layers of swap gates inEq. (5.2) by the following n layers S (cid:48) , . . . , S (cid:48) n − , which have to be applied in this order: S (cid:48) k = k − (cid:79) l = SWAP (cid:104) l , l + k (cid:105) . (5.5)This reduces the total number of swap gates from m log m to m − n = log m consecutive swap tests (Fig. 5.3).For k ∈ {
0, . . . , n − } , conditioned on all the previous outputs being 0, the k th swap test comparesthe output state of the previous test and the state | ψ 〉 ⊗ k . Here, the swap test of two multipartitequantum states consists in applying a swap test to each of their corresponding subsystems.However, this multipartite swap test uses only a single ancilla qubit controlling the product ofswap gates, as in Eq. (5.5), instead of an ancilla qubit for each pair of subsystems.We now prove the optimality of the swap test of order m under the one-sided error requirement,i.e., we show that it achieves the lowest error probability in comparing states | φ 〉 and | ψ 〉 given m − | φ 〉 and one copy of | ψ 〉 such that the one-sided error requirement is satisfied.182 .1. TESTING QUANTUM STATES H H
SWAPSWAP
H H
SWAP
H H (a) (b) (c) | i
Under the one-sided error requirement, any identity test of m unknown quantumstates | ψ 〉 , . . . , | ψ m 〉 has an error probability at least m ! (cid:88) σ ∈ S m m (cid:89) k = 〈 ψ k | ψ σ ( k ) 〉 , (5.6) where S m is the symmetric group over {
1, . . . , m } . Proof.
An identity test satisfying the one-sided error requirement can only be wrong whendeclaring identical (outputting 0) states that are not identical. Hence, to prove Theorem 5.1,it suffices to lower bound the probability of outputting 0 for any identity test. This is done byshowing that the optimal identity test consists in a projection onto the symmetric subspaceof the input states Hilbert space.An identity test on a Hilbert space H is a binary test which can be written as a positive-operator valued measure { Π , Π } , with Π + Π = (cid:49) . Such a test takes as input a pure tensorproduct state | ψ . . . ψ m 〉 ∈ H ⊗ m and outputs 0 with probabilityPr[0] = Tr [ Π | ψ . . . ψ m 〉 〈 ψ . . . ψ m | ], (5.7)183 HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS and 1 with probabilityPr[1] = − Pr[0] = Tr [ Π | ψ . . . ψ m 〉 〈 ψ . . . ψ m | ]. (5.8)If the output 0 is obtained we conclude that we had | ψ 〉 = · · · = | ψ m 〉 , whereas if the output 1 isobtained we conclude that the states were not all identical. The one-sided error requirementcan thus be written as ∀ | ψ 〉 , Tr [ Π | ψ 〉 〈 ψ | ⊗ m ] =
0. (5.9)Following [Har13], the symmetric subspace of H ⊗ m is characterised as S = span { | ψ 〉 ⊗ m : | ψ 〉 ∈ H } , (5.10)and the orthogonal projector onto this space can be written as P S = m ! (cid:88) σ ∈ S m P σ , (5.11)where for all σ ∈ S m and all | ψ . . . ψ m 〉 ∈ H ⊗ m we have P σ | ψ . . . ψ m 〉 = | ψ σ (1) . . . ψ σ ( m ) 〉 . Giventhe characterisation of the symmetric subspace, the one-sided error requirement in Eq. (5.9)implies that the supports of P S and Π are disjoint. The support of P S is thus included inthe support of Π , given that Π + Π = (cid:49) and this implies in turn that Π ≥ P S by positivityof Π .The error probability of the identity test under the one-sided error requirement is givenby the probability of outputting the result 0 while the states were not all identical:Pr[0] = Tr [ Π | ψ . . . ψ m 〉 〈 ψ . . . ψ m | ] ≥ Tr [ P S | ψ . . . ψ m 〉 〈 ψ . . . ψ m | ] ≥ m ! (cid:88) σ ∈ S m Tr [ P σ | ψ . . . ψ m 〉 〈 ψ . . . ψ m | ] (5.12) ≥ m ! (cid:88) σ ∈ S m Tr [ | ψ σ (1) . . . ψ σ ( m ) 〉 〈 ψ . . . ψ m | ] ≥ m ! (cid:88) σ ∈ S m m (cid:89) k = 〈 ψ k | ψ σ ( k ) 〉 ,where in the third line we used the expression of the orthogonal projector P S onto thesymmetric subspace. (cid:4) Applying Theorem 5.1 with | ψ . . . ψ k + l 〉 = | φ 〉 ⊗ k ⊗ | ψ 〉 ⊗ l , we obtain the following lower bound for184 .1. TESTING QUANTUM STATES the error probability of any identity test of k + l states | φ 〉 ⊗ k ⊗ | ψ 〉 ⊗ l :1( k + l )! min( k , l ) (cid:88) p = (cid:195) kp (cid:33)(cid:195) lp (cid:33) k ! l ! | 〈 φ | ψ 〉 | p = min( k , l ) (cid:88) p = (cid:161) kp (cid:162)(cid:161) lp (cid:162)(cid:161) k + lk (cid:162) | 〈 φ | ψ 〉 | p , (5.13)where (cid:161) kp (cid:162)(cid:161) lp (cid:162) k ! l ! is the number of partitions of {
1, . . . , k + l } which map exactly k − p elements of {
1, . . . , k } to elements of {
1, . . . , k } . Testing quantum state identity with the input state | φ 〉 ⊗ k ⊗| ψ 〉 ⊗ l amounts to comparing the states | φ 〉 and | ψ 〉 using k copies of | φ 〉 and l copies of | ψ 〉 .In the case where k = l = m −
1, we have | ψ . . . ψ m 〉 = | φψ . . . ψ 〉 and Theorem 5.1 showsthat the value m + m − m | 〈 φ | ψ 〉 | is a lower bound for the error probability of any identity testof m states | φ 〉 , | ψ 〉 , . . . , | ψ 〉 , i.e., one copy of a state | φ 〉 and m − | ψ 〉 . WithDefinition 5.1 we directly obtain the following result: Corollary 5.1.
The swap test of order m has optimal error probability m + m − m | 〈 φ | ψ 〉 | underthe one-sided error requirement. The swap circuit of order m is thus optimal for quantum state identity testing with an input | φ 〉 , | ψ 〉 , . . . , | ψ 〉 , under the one-sided error requirement, since it implements the swap test of order m . In the next section, we show that the swap circuit of order m can be used to implement aprogrammable projective measurement. In a typical experiment performing a quantum measurement, the choice of measurement isencoded in macroscopic, classical, information in the experimental setup. For example it can beencoded into the reflectance of a beam splitter, the phase in the branch of an interferometer or thespacial direction of a Stern Gerlach device. Often these choices are made beforehand and fixed.In some cases they can be programmed in a single set up (for example using thermo-optic phaseshifters [CHS + HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS
A related and, in a sense, more general problem is that of a programmable quantum computer,where a quantum program state is used to encode a unitary to be run on a generic quantumcomputing device (gate array), first proposed by Nielsen and Chuang [NC97]. There it was shownthat to do so deterministically requires orthogonal program states for every different unitary. Touse the continuous parameters available in quantum states to encode more computations, thebest one can do is probabilistic. In principle these techniques can be used to program quantummeasurements. Indeed since the original proposal there have been several alternative schemes,extensions and applications, including programmable quantum state discriminators and measure-ments [VC00, DB02, RBCH03, ZB05, BBF + C
1. Itwill thus suffice to consider, e.g., the first condition. In this context, under the one-sided errorrequirement, a projective measurement with any error (cid:178) always outputs 0 if the input state isequal to the reference state.
Theorem 5.2.
A swap circuit of order m can be used to perform a projective measurement witherror m under the one-sided error requirement. Moreover, it is optimal in the sense that it uses theminimum number of copies of the reference state for achieving such an error. HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS P o s t - p r o ce ss i n g | i
Figure 5.5: The swap circuit of order m used as a programmable projective measurement device.It takes as input a state | φ 〉 and the internal measurement outcomes are post-processed such thatthe device outputs 0 with probability m + m − m | 〈 φ | ψ 〉 | and 1 with probability m − m (1 − | 〈 φ | ψ 〉 | ).The programmable resource is the state | ψ 〉 and the process uses m − n = log m ancillas. Proof.
For the swap circuit of order m , we have Pr [0, . . . , 0] = m + m − m | 〈 φ | ψ 〉 | by Eq. (5.4),so we can consider the whole circuit except the state | φ 〉 as a black box in Fig. 5.2, andpost-process the measurement outcomes d as follows: if d = (0, . . . , 0), output 0, and output 1otherwise. The setup now takes a single state | φ 〉 in input and outputs 0 with probabilityPr[0] = m + m − m | 〈 φ | ψ 〉 | , and 1 with probability Pr[1] = − Pr[0]. We have | Pr[0] − ( | 〈 φ | ψ 〉 | ) | ≤ m and when | φ 〉 = | ψ 〉 , we have Pr[0] = = | 〈 φ | ψ 〉 | , hence this device performs a projectivemeasurement with error m and meets the one-sided error requirement.We now prove the optimality of this device in terms of resources, i.e., we show that anydevice implementing a projective measurement with error m and meeting the one-sidederror requirement cannot use less than m − (cid:178) , with respectto a reference state | ψ 〉 , using p copies of this reference state. This device takes as inputa quantum state | φ 〉 and outputs 0 with probability Pr φ [0] and 1 with probability Pr φ [1] = − Pr φ [0]. By Definition 5.2, the probability of outputting 0 satisfies | Pr φ [0] − ( | 〈 φ | ψ 〉 | ) | ≤ (cid:178) .When the input state | φ 〉 is orthogonal to the reference state | ψ 〉 , the probability Pr φ , ⊥ [0] of188 .2. UNIVERSAL PROGRAMMABLE PROJECTIVE MEASUREMENTS WITH LINEAR OPTICS outputting 0 thus satisfies Pr φ , ⊥ [0] ≤ (cid:178) . (5.14)On the other hand, we can use this device to perform an identity test of p + | φ 〉 , | ψ 〉 , . . . , | ψ 〉 (one copy of the state | φ 〉 and p copies of the state | ψ 〉 ): if the output 0(resp. 1) is obtained we conclude that the states were identical (resp. different). This devicemeets the one-sided error requirement, so by Theorem 5.1 it has error probability at least p + + pp + | 〈 φ | ψ 〉 | . This error probability corresponds to the probability of outputting 0 whenthe input states are different. In particular, when the input state | φ 〉 is orthogonal to thereference state | ψ 〉 , the probability Pr φ , ⊥ [0] of outputting 0 thus satisfiesPr φ , ⊥ [0] ≥ p + p + ≤ (cid:178) or equivalently p ≥ (cid:178) −
1. For (cid:178) = m , this amounts to p ≥ m −
1, which completes the proof. (cid:4)
Theorem 5.2 implies that given a large enough swap circuit and the ability to produce manycopies of a state | ψ 〉 , one can projectively measure any state with respect to the state | ψ 〉 up toarbitrary small error. This error scales as the inverse of the number of copies. The circuit can thusbe used as a programmable projective measurement device, where the programmable resourceis the reference state | ψ 〉 whose number of copies can be adjusted to control the precision of themeasurement (Fig. 5.5).The implementation of the swap circuit of order m is however challenging, due to the presenceof many controlled-swap gates. In order to lower the implementation requirements, we study inthe next section the linear optical Hadamard interferometer [Cre15, COR +
16] and show that itsstatistics can be efficiently post-processed to reproduce those of a swap circuit of order m , withoutthe need for ancillas. This comes at the cost that the device no longer has a quantum output,which however does not matter for most applications. In particular, we show that the Hadamardinterferometer provides a simple linear optical platform for implementing the programmableprojective measurement that we have described. The swap test has been shown equivalent to the linear optical Hong-Ou-Mandel effect [GECP13](see section 1.4.4), in the sense that one can use Hong-Ou-Mandel effect to perform a statediscrimination test between two partially distinguishable photons, whose statistics reproduce189
HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS those of a swap test. Generalising this equivalence, we present a practical solution to our problemwith linear optics, using the Hadamard interferometer [Cre15, COR + In what follows, we consider optical unitary interferometers of size m which take as input onesingle photon in a quantum state | φ 〉 in the first mode and m − | ψ 〉 , one in each other spatial mode (the spatial modes of the interferometers areindexed from 1 to m ). These states should be thought of as encoded in additional degrees offreedom of the photons (e.g., polarisation, time bins). The output modes are measured usingphoton number-resolving detection.There exist complex amplitudes α and β and a state | ψ ⊥ 〉 with 〈 ψ | ψ ⊥ 〉 = | φ 〉 = α | ψ 〉 + β | ψ ⊥ 〉 , (5.16)where α = 〈 ψ | φ 〉 and | α | + | β | =
1. We have the following homomorphism property for singlephoton states: | φ 〉 = | αψ + βψ ⊥ 〉 = α | ψ 〉 + β | ψ ⊥ 〉 , (5.17)where for any state | χ 〉 , | χ 〉 is the state of a single photon encoding the state | χ 〉 . The singlephoton encoding maps identity of quantum states to distinguishability of single photons. Inorder to test the distinguishability of the photons, we look for detection events that do not occurwhen the photons are indistiguishable. In that case, it suffices to compute the output statisticsseparately when | φ 〉 = | ψ 〉 ( indistinguishable case ) and when | φ 〉 = | ψ ⊥ 〉 ( distinguishable case ) toobtain the output statistics in the general case by linearity. The probability of detecting a photonnumber pattern d = ( d , . . . , d m ) which does not occur in the indistiguishable case, or equivalentlythat the k th detector detects d k photons for all k ∈ {
1, . . . , m } , is thenPr ( d ) = | α | Pr i ( d ) + | β | Pr d ( d ) = (cid:161) − | 〈 φ | ψ 〉 | (cid:162) Pr d ( d ), (5.18)where Pr i ( d ) = d ( d ) is the probability inthe distinguishable case. We thus have (cid:88) d Pr i ( d ) = Pr ( d ) = (cid:161) − | 〈 φ | ψ 〉 | (cid:162) (cid:88) d Pr i ( d ) = Pr d ( d ). (5.19)Note that for any measurement outcome d = ( d , . . . , d m ), we have d + · · · + d m = m since aninterferometer is a passive device that does not change the total number of photons. For anyinterferometer of size m , we also obtain the following result: Lemma 5.1.
For any detection pattern d ,Pr d ( d ) ≥ Pr i ( d ) m , (5.20)190 .2. UNIVERSAL PROGRAMMABLE PROJECTIVE MEASUREMENTS WITH LINEAR OPTICS Proof.
We consider optical unitary interferometers of size m which take as input one singlephoton in a quantum state | φ 〉 and m − | ψ 〉 ,one in each spatial mode, indexed from 1 to m . The output modes are measured usingphoton number detection. A measurement outcome thus has the form d = ( d , . . . , d m ), with d + · · · + d m = m .Recall that the permanent of an m × m matrix A = ( a i j ) ≤ i , j ≤ m is defined byPer ( A ) = (cid:88) σ ∈ S m m (cid:89) k = a k σ ( k ) , (5.21)where S m is the symmetric group over {
1, . . . , m } . We now compute Pr i ( d ) and Pr d ( d ) for alldetection patterns d .In the indistinguishable case, m indistinguishable photons, one in each mode, are sentthrough a linear optical network described by an m × m unitary matrix U = ( u i j ) ≤ i , j ≤ m . Theprobability of a detection event d can be computed as (see, section 1.4.5 and [AA13])Pr i ( d ) = | Per ( U d ) | d ! , (5.22)where d ! = d ! . . . d m ! and where U d is the matrix obtained from U by repeating d k timesthe k th column for k ∈ {
1, . . . , m } .In the distinguishable case, m − m through a linear optical network described by an m × m unitary matrix U = ( u i j ) ≤ i , j ≤ m ,along with one additional photon in the first mode in an orthogonal state. Since it is fullydistinguishable from the others, the additional photon behaves independently, hence theprobability of detecting the photon number pattern d for one distinguishable photon and m − d ( d ) = m (cid:88) k = d k (cid:54)= Pr i ( d − k ) · Pr i ( k ), . (5.23)This last expression formalises the fact that the m − d − k which, completed by the additional distinguishable photon in the k th output mode, forms the pattern d . Developing this expression with Eq. (5.22) yieldsPr d ( d ) = d ! m (cid:88) k = d k (cid:54)= d k | u k Per ( U d − k ) | (5.24)where U d − k is the matrix obtained from U by removing the first row, then by repeating d l times the l th column for l (cid:54)= k and by repeating d k − k th column.191 HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS
In order to obtain more readable expressions, we define for all k ∈ {
1, . . . , m } and for anydetection pattern d , p k ( d ) = u k Per( U d − k ) (cid:112) d ! if d k (cid:54)= i ( d ) = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) m (cid:88) k = d k p k ( d ) (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) , (5.26)and Pr d ( d ) = m (cid:88) k = d k | p k ( d ) | . (5.27)Since (cid:80) mk = d k = m , we obtain, using Cauchy-Schwarz inequality with the complex vectors (cid:110)(cid:112) d k (cid:111) ≤ k ≤ m and (cid:110)(cid:112) d k p k ( d ) (cid:111) ≤ k ≤ m ,Pr d ( d ) ≥ Pr i ( d ) m , (5.28)for any detection pattern d . (cid:4) For all d we have (cid:88) d Pr i ( d ) = Pr d ( d ) + (cid:88) d Pr i ( d ) (cid:54)= Pr d ( d ) =
1. (5.29)Combining Lemma 5.1 with Eqs. (5.19) and (5.29) yields (cid:88) d Pr i ( d ) (cid:54)= Pr ( d ) ≥ (cid:181) m + m − m | 〈 φ | ψ 〉 | (cid:182) . (5.30)This last expression is valid for any interferometer and can be used it to retrieve, in the contextof linear optics, the error probability bound for state identity testing under the one-sided errorrequirement obtained in Corollary 5.1. Indeed, assume that d is a detection event such thatPr i ( d ) (cid:54)=
0, which could be a disjoint union of multiple detection events, used for an identity test:if d is obtained we conclude that the states were identical (or equivalently that the photonswere indistinguishable), otherwise we assume that the states were different (or equivalentlythat the first photon was distinguishable from the others). The one-sided error requirementcan thus be written as (cid:80) d ,Pr i ( d ) (cid:54)= Pr i ( d ) =
1: indistinguishable photons always pass the test. Fordifferent input states | φ 〉 and | ψ 〉 , the error probability of the corresponding test is then given by (cid:80) d ,Pr i ( d ) (cid:54)= Pr ( d ), which by Eq. (5.30) is lower bounded by m + m − m | 〈 φ | ψ 〉 | .We now study a particular unitary interferometer, when the size m is a power of 2: theHadamard interferometer [Cre15, COR + .2. UNIVERSAL PROGRAMMABLE PROJECTIVE MEASUREMENTS WITH LINEAR OPTICS of the swap test of order m . For m = (cid:112) (cid:195) H HH − H (cid:33) (5.31)where H is a Hadamard matrix, see Eq. (5.1). | i
The output statistics of the Hadamard interferometer of order m can be classicallypost-processed in time O ( m log m ) to reproduce those of the swap test of order m. Proof.
Let us define S = ( s i j ) ≤ i , j ≤ m = (cid:112) m H n , (5.33)193 HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS thus omitting the normalisation factor. We have S = (cid:112) H ⊗ · · · ⊗ (cid:112) H (cid:124) (cid:123)(cid:122) (cid:125) n times , (5.34)where H is a Hadamard matrix. The rows of (cid:112) H , together with the element-wise multipli-cation, form a group isomorphic to (cid:90) /2 (cid:90) , thus the rows of S together with the element-wisemultiplication form a group isomorphic to ( (cid:90) /2 (cid:90) ) n . As a consequence, multiplying element-wise all the rows of S by its i th row for a given i amounts to permuting the rows of S .Let d = ( d , . . . , d m ) and k ∈ {
1, . . . , m } such that d k (cid:54)=
0. Let also S d − k be the matrixobtained from S by repeating d l times the l th column for l (cid:54)= k and d k − k th column.For all i ∈ {
1, . . . , m } , one can obtain the matrix S d − k (with the first row removed) fromthe matrix S i , d − k (with the i th row removed) by multiplying element-wise all rows by the i th row and permuting the rows. Since the permanent is invariant by row permutation weobtain, for all i ∈ {
1, . . . , m } and all k ∈ {
1, . . . , m } such that d k (cid:54)= S i , d − k ) = (cid:178) ik ( d ) Per ( S d − k ), (5.35)where (cid:178) ik ( d ) = s ik (cid:81) mj = (cid:161) s i j (cid:162) d j . Let us define for all d = ( d , . . . , d m ) π ( d ) = m (cid:88) i = m (cid:89) j = (cid:161) s i j (cid:162) d j . (5.36)We use the Laplace row expansion formula for the permanent of S d to obtain, for all d = ( d , . . . , d m ) and all k ∈ {
1, . . . , m } such that d k (cid:54)= S d ) = m (cid:88) i = s ik Per ( S i , d − d k ) = (cid:195) m (cid:88) i = s ik (cid:178) ik ( d ) (cid:33) Per ( S d − k ) = (cid:195) m (cid:88) i = m (cid:89) j = (cid:161) s i j (cid:162) d j (cid:33) Per ( S d − k ) = π ( d ) Per ( S d − k ), (5.37)where we used Eq. (5.35) in the second line. With the general expressions of Pr i ( d ) (5.22)and Pr d ( d ) (5.24), this equation implies m Pr i ( d ) = π ( d ) Pr d ( d ). (5.38)With the Laplace column expansion formula for the permanent of S d and the last line ofEq. (5.37), we also obtain m Pr i ( d ) = π ( d ) Pr i ( d ). (5.39)194 .2. UNIVERSAL PROGRAMMABLE PROJECTIVE MEASUREMENTS WITH LINEAR OPTICS In particular, combining Eqs. (5.38) and (5.39), m π ( d ) Pr d ( d ) = π ( d ) Pr d ( d ). (5.40)Now Pr d ( d ) is non-zero for all d , since by Eq. (5.24) it is a sum of moduli squared ofpermanents of (2 n − × (2 n −
1) matrices, which in turn cannot vanish by a result of [SS83].Hence the previous equation rewrites m π ( d ) = π ( d ) . (5.41)As a consequence, π ( d ) = m or π ( d ) = d . Combining Eqs. (5.38) and (5.41) we obtain π ( d ) (cid:54)= ⇔ π ( d ) = m ⇔ Pr i ( d ) (cid:54)= ⇔ Pr d ( d ) = Pr i ( d ) m , (5.42)and thus Pr i [ π ( d ) = m ] = (cid:88) π ( d ) = m Pr i ( d ) = (cid:88) Pr i ( d ) (cid:54)= Pr i ( d ) =
1. (5.43)We also obtain Pr d [ π ( d ) = m ] = (cid:88) π ( d ) = m Pr d ( d ) = m (cid:88) π ( d ) = Pr i ( d ) = m . (5.44)We finally conclude by combining Eqs. (5.43), (5.44) and (5.19):Pr [ π ( d ) = m ] = (cid:88) π ( d ) = m Pr ( d ) = m + m − m (cid:175)(cid:175) 〈 φ | ψ 〉 (cid:175)(cid:175) . (5.45)The post-processing (i.e., computing π ( d )) can be done efficiently in time O ( m log m ) for anydetection pattern d = ( d , . . . , d m ). Indeed, let S d be the m × m matrix obtained from S byrepeating d k times the k th column for k ∈ {
1, . . . , m } . The expression π ( d ) in Eq. (5.36) is thesum of the product of the elements of each row of S d . Since the entries of the matrix S areonly + − π ( d ) = m if and only if the number of − S d is even for allrows. The condition π ( d ) = m can thus be written as a system of m linear equations modulo2. Since ( (cid:90) /2 (cid:90) ) n is finitely generated by n elements, the m rows of S d can be generated with195 HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS at most n rows using element-wise multiplication, for any measurement outcome d . Hence,computing the parity of the number of − S d , which is equivalent to testing π ( d ) = m , can be done by computing at most n = log m parity equations, with a number ofterms in each equation which is at most m .A simple induction shows that a possible choice for the rows whose parity has to be testedis the rows with index 2 k + k ∈ {
0, . . . , n − } (the rows of the matrix being indexed from 1to m ). (cid:4) Note that the group structure invoked in the proof is preserved under permutations, so Theo-rem 5.3 also applies to the unitary interferometers described by permutations of the Hadamard-Walsh transform.The conclusion to be drawn from Theorem 5.3 is that as long as a state | ψ 〉 can be encodedusing single photons, then one can perform a swap test of order m with respect to the state | ψ 〉 using the Hadamard interferometer of order m and an efficient classical post-processing ofthe measurement outcomes. The post-processing consists in the following parity test: given themeasurement outcome d = ( d , . . . , d m ), where d +· · ·+ d m = m , construct the matrix S d from thematrix S = (cid:112) m H n by keeping the k th column only if d k is odd. If the rows (2, 3, 5, . . . , 2 n − +
1) of S d all have an even number of −
1, output 0. Output 1 otherwise. This means that the post-processingonly requires the parity of the photon number in each output mode.In particular, the photon number-resolving detectors can be replaced by detecting the parityof the number of photons in each output mode. Detecting this parity can for example be achievedwith microwave technology [HBR07, VKL +
13, SPL + m − m . If only the parity is measured,the discrimination test is non-destructive and the remaining single-mode state is a mixture ofeither even or odd photon-number states, depending on the measured parity and the total numberof photons.Using the argument developed in the proof of Theorem 5.2, by considering the m − Corollary 5.2.
The Hadamard interferometer of order m can be used to perform a projectivemeasurement with error m , using a classical post-processing of its measurement outcomes thattakes time O ( m log m ) . Interestingly, the unitary interferometers described by the Hadamard-Walsh transform and itspermutations are not the only unitary interferometers which can reproduce the statistics ofa swap test with efficient post-processing, and indeed we present a generalisation in the nextsection. However, it is the simplicity of the Hadamard interferometer in terms of experimental196 .2. UNIVERSAL PROGRAMMABLE PROJECTIVE MEASUREMENTS WITH LINEAR OPTICS P o s t - p r o ce ss i n g H n
The m × m Hadamard interferometer can be implemented using m log m balancedbeam splitters. Proof.
The size m is a power of 2, with n = log m . We prove by induction over n that thereexist P ( n ), . . . , P n − ( n ) permutation matrices of order m /2, such that H n = n − (cid:89) k = P k ( n ) ( (cid:49) m /2 ⊗ H ) P k ( n ) T . (5.46)Since multiplying matrices is equivalent to setting up experimental devices in sequence, andgiven that H is the matrix describing a balanced beam splitter, Eq. (5.46) implies the resultwe want to prove.For n =
1, we have m = P (1) = (cid:49) . For brevity, we define for all kH ( k ) = (cid:49) k ⊗ H . (5.47)Assuming that Eq. (5.46) is true for n , we use the recursive definition of the Hadamard-Walsh197 HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS transform H n + = H ⊗ H n , (5.48)along with properties of the tensor product of matrices in order to obtain H n + = ( H n ⊗ (cid:49) ) H ( m ) = Q ( (cid:49) ⊗ H n ) Q T H ( m ) = Q (cid:34) (cid:49) ⊗ n − (cid:89) k = P k ( n ) H ( m /2) P k ( n ) T (cid:35) Q T H ( m ) (5.49) = Q (cid:34) n − (cid:89) k = ( (cid:49) ⊗ P k ( n )) H ( m ) (cid:179) (cid:49) ⊗ P k ( n ) T (cid:180)(cid:35) Q T H ( m ) = n − (cid:89) k = [ Q ( (cid:49) ⊗ P k ( n ))] H ( m ) [ Q ( (cid:49) ⊗ P k ( n ))] T H ( m ) ,where Q is a permutation matrix of order m and where in the third line we have usedEq. (5.46). Setting P k ( n + = Q ( (cid:49) ⊗ P k ( n )) for k ∈ {
0, . . . , n − } and P n ( n + = (cid:49) m provesEq. (5.46) for n +
1, since these matrices are permutation matrices of order m . This completesthe induction and the proof of the result. (cid:4) Using the Hadamard interferometer requires the size parameter m to be a power of 2. Thisrequirement can be relaxed, possibly raising the experimental requirements at the same time.Indeed, for any value of m , one can associate to any abelian group of order m an interferometerof size m which gives the desired statistics. This is the object of the following result that uses theinvariant factor decomposition of an abelian group: Theorem 5.4.
Let G be an abelian group of order m. Then, there exists n ∈ (cid:78) ∗ and a , . . . , a n ∈ (cid:78) ∗ ,where a i | a i + for i ∈ {
1, . . . , n − } and a · · · a n = m, such that the interferometer described by them × m unitary matrix U G : = (cid:112) m F a ⊗ · · · ⊗ F a n , (5.50) where F a = ( e i π a ( k − l − ) ≤ k , l ≤ a is the quantum Fourier transform of order a for all a ∈ (cid:78) ∗ , canperform a m -approximate projective measurement with a post-processing of its measurementoutcomes that takes time at most mn. The rows of F G : = (cid:112) m U G together with the element-wisemultiplication form a group isomorphic to G . Proof.
We use the notations of the theorem. The invariant factor decomposition of G gives G (cid:39) ( (cid:90) / a (cid:90) ) ⊗ · · · ⊗ ( (cid:90) / a n (cid:90) ) , (5.51)198 .3. PROGRAMMABLE PROJECTIVE MEASUREMENTS WITH COHERENT STATES where n ∈ (cid:78) ∗ and a , . . . , a n ∈ (cid:78) ∗ are unique, satisfying a i | a i + for i ∈ {
1, . . . , n − } and a · · · a n = m . Given that the rows of F a together with the element-wise multiplication forma group isomorphic to ( (cid:90) / a (cid:90) ) for all a ∈ (cid:78) ∗ , the rows of F G = ( f i j ) ≤ i , j ≤ m = (cid:112) m U G togetherwith the element-wise multiplication form a group isomorphic to G .Since the group structure was the only argument invoked in the proof of Theorem 5.3,the same conclusion can be drawn here, by following the same argument: for any detectionevent d = ( d , . . . , d m ), Pr [ π ( d ) = m ] = m + m − m (cid:175)(cid:175) 〈 φ | ψ 〉 (cid:175)(cid:175) , (5.52)where π ( d ) = m (cid:88) i = m (cid:89) j = (cid:161) f i j (cid:162) d j . (5.53)The group G is finitely generated by n elements, so n rows of F G are sufficient to generateall its rows by element-wise multiplication. The condition π ( d ) = m can thus be checked intime O ( mn ). (cid:4) In particular, for G (cid:39) ( (cid:90) / m (cid:90) ), the corresponding interferometer is described by the (normalised)quantum Fourier transform of order m , while for G (cid:39) ( (cid:90) /2 (cid:90) ) n , we retrieve Theorem 5.3 and theHadamard interferometer. The previous scheme for performing programmable measurements with linear optics requirescreation and manipulation of high-dimensional superposition states. In order to simplify theexperimental requirements, we adapt this scheme to an encoding of quantum states in coherentstates of light. Since coherent states are natural realisations of states produced by lasers, theycan be efficiently produced and manipulated experimentally. The coherent state scheme takesas input a generic single-mode continuous variable quantum state, the test state, and m − { | β 〉〈 β | , (cid:49) − | β 〉〈 β | } on the input state in a single run, using only threshold detectors. In particular,we obtain a more faithful projective measurement using coherent states than using a single-photon encoding.In what follows, we introduce three different schemes for performing state discriminationand programmable projective measurement with coherent states: the Hadamard scheme , the merger scheme , and the looped merger scheme . Further, we give the proof for the optimality ofthe coherent state projective measurement performed by all three schemes, under the one-sidederror requirement. 199
HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS
The swap test discriminates between two unknown states. If the unknown states are coherentstates instead, then an analogous test can be performed by mixing the states on a balanced beamsplitter and measuring the lower output branch with a single-photon threshold detector (Fig. 5.8). | ↵ i
1. Thus the last m − (cid:59) ] that none of the m − α , β , m [ (cid:59) ] = m − (cid:89) k = (1 − Pr [click in k th mode]) = m − (cid:89) k = [1 − (1 − exp( −| δ k | ))] = exp (cid:181) − m − m | α − β | (cid:182) = ( | 〈 α | β 〉 | ) − m . (5.61)In particular, for all α , β ∈ (cid:67) , Pr α , β , +∞ [ (cid:59) ] = | 〈 α | β 〉 | , which corresponds to a perfect projectivemeasurement of the states | α 〉 and | β 〉 . Writing x = | 〈 α | β 〉 | the overlap of the test and referencestates, we obtain Pr x , m [ (cid:59) ] = x − m . (5.62)201 HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS
Assigning to the event ‘none of the detectors clicks’ the value 0 and to other detection events (‘atleast one of the m − x , m [0] = − Pr x , m [1] = x − m . (5.63)With Theorem 5.3 and Eq. (5.4), for an m -mode input state | φψ . . . ψ 〉 the corresponding statisticswith single-photon encoding arePr x , m [0] = − Pr x , m [1] = m + (cid:181) − m (cid:182) x . (5.64)The single-photon encoding implies having m − m − x : ∀ x ∈ [0, 1], x ≤ x − m ≤ m + (cid:181) − m (cid:182) x . (5.65)In particular, for a given size m , the maximal statistical gap with a perfect projective measure-ment is, e SP ( m ) = max x ∈ [0,1] (cid:175)(cid:175)(cid:175)(cid:175)(cid:183) m + (cid:181) − m (cid:182) x (cid:184) − x (cid:175)(cid:175)(cid:175)(cid:175) = m , (5.66)for the single-photon encoding, and e CS ( m ) = max x ∈ [0,1] (cid:175)(cid:175)(cid:175)(cid:179) x − m (cid:180) − x (cid:175)(cid:175)(cid:175) = ( m − m − m m ∼ e · m , (5.67)for the coherent state encoding, which is lower than the single-photon encoding gap. This happensbecause for the single-photon encoding no assumption is made about the states | φ 〉 and | ψ 〉 , whilethe states | α 〉 and | β 〉 are assumed to be coherent states. This additional information about thestates allows us to better approximate a perfect projective measurement with the same numberof input states. We show in the next section that there exists a simpler measurement settingthan the Hadamard interferometer, achieving the same performance in the test, due to coherentstate encoding. 202 .3. PROGRAMMABLE PROJECTIVE MEASUREMENTS WITH COHERENT STATES | ↵ i
1, 2, 3, 4 } is described by thefollowing unitary matrix: U =
12 12 12 121 (cid:112) − (cid:112)
12 12 − − (cid:112) − (cid:112) = (cid:112) (cid:112)
00 1 0 0 (cid:112) − (cid:112)
00 0 0 1 × (cid:112) (cid:112) (cid:112) − (cid:112) (cid:112) (cid:112) (cid:112) − (cid:112) = H × ( H ⊕ H ), (5.68)where H i , j corresponds to the balanced beam splitter operation acting on modes i and j (wherethe modes are indexed from 1 to m ) and identity on the other modes (Fig. 5.9).203 HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS
The generalised merger interferometer is defined by induction: U m = H m /2 + × ( U m /2 ⊕ U m /2 ), (5.69)where U = H = H is a Hadamard matrix. This induction relation is illustrated in Fig. 5.10. | ↵ i
1. A simple induction shows that the output state inthe 2 k + (cid:175)(cid:175)(cid:175) α − β (cid:112) k + (cid:69) . Hence, the probability that none of the n = log m detectorsclicks is given by Pr α , β , m [ (cid:59) ] = n − (cid:89) k = (1 − Pr [click in the 2 k th mode]) = n − (cid:89) k = (cid:183) − (cid:181) − exp (cid:181) − (cid:175)(cid:175)(cid:175)(cid:175) α − β k + (cid:175)(cid:175)(cid:175)(cid:175) (cid:182)(cid:182)(cid:184) = exp (cid:195) − n − (cid:88) k = (cid:181) (cid:182) k + | α − β | (cid:33) = exp (cid:181) − m − m | α − β | (cid:182) = ( | 〈 α | β 〉 | ) − m , (5.70)204 .3. PROGRAMMABLE PROJECTIVE MEASUREMENTS WITH COHERENT STATES thus retrieving the statistics obtained with the Hadamard scheme, using only n = log m detectors.Moreover, a simple induction shows that the merger interferometer can be implemented withonly m − | ↵ i
The Hadamard interferometer and the merger interferometer are optimal forcoherent states discrimination, under the one-sided error requirement.
Proof.
The proof extends results from [SZP + { Π , Π } be a POVM for discriminating coherent states | α 〉 and | β 〉 under the one-sidederror requirement, when provided a single copy of | α 〉 and m − | β 〉 (the proofof [SZP +
07] assumes m = Π corresponds to saying that the states | α 〉 and205 HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS | β 〉 are the same, while the operator Π corresponds to saying that they are different. Theseoperators thus verify the following conditions: Π , Π ≥ (cid:48) , Π + Π = (cid:49) , (5.71)and ∀ α ∈ (cid:67) , Tr (cid:163) Π | α 〉 〈 α | ⊗ m (cid:164) =
0, (5.72)where the last condition is the one-sided error requirement. Integrating this condition over (cid:67) yields 0 = (cid:90) d α Tr (cid:163) Π | α 〉 〈 α | ⊗ m (cid:164) = Tr [ Π ∆ m ] , (5.73)where we have defined ∆ m = (cid:90) d α | α 〉 〈 α | ⊗ m ≥
0. (5.74)Note that the condition in (5.73) is equivalent to the one-sided requirement in (5.72) becausethe operators Π and | α 〉 〈 α | ⊗ m are positive.The operator m π ∆ m is actually a projector. This result can be obtained by writing thestate | α 〉 in the Fock basis and an integration in polar coordinates, where α = re i θ , as follows:writing | α 〉 = e − | α | +∞ (cid:88) k = α k (cid:112) k ! | k 〉 , (5.75)we obtain ∆ m = (cid:90) d α exp[ − m | α | ] ∞ (cid:88) k j , l j = ∀ j ∈ [ m ] α (cid:80) j k j ( α ∗ ) (cid:80) j l j (cid:112) k ! . . . k m ! l ! . . . l m ! | k . . . k m 〉 〈 l . . . l m |= ∞ (cid:88) k j , l j = | k . . . k m 〉 〈 l . . . l m | (cid:112) k ! . . . k m ! l ! . . . l m ! (cid:90) ∞ r = dr exp[ − mr ] r + (cid:80) j k j + l j (cid:90) πθ = d θ exp[ i θ (cid:88) j ( k j − l j )] = π m ∞ (cid:88) k j , l j = δ (cid:80) j k j , (cid:80) j l j m (cid:80) j kj m (cid:80) j l j (cid:115) ( (cid:80) j k j )!( (cid:80) j l j )! k ! . . . k m ! l ! . . . l m ! | k . . . k m 〉 〈 l . . . l m |= π m ∞ (cid:88) p = (cid:88) (cid:80) j k j = p (cid:80) j l j = p m − p (cid:115) p ! k ! . . . k m ! (cid:115) p ! l ! . . . l m ! | k . . . k m 〉 〈 l . . . l m |= π m ∞ (cid:88) p = | χ mp 〉 〈 χ mp | , (5.76)where we have defined for all p ≥ | χ mp 〉 = m − p /2 (cid:88) (cid:80) j k j = p (cid:115) p ! k ! . . . k m ! | k . . . k m 〉 . (5.77)206 .3. PROGRAMMABLE PROJECTIVE MEASUREMENTS WITH COHERENT STATES With the multinomial formula, we obtain 〈 χ mp | χ mp 〉 = p ≥
0, and since the states | χ mp 〉 have exactly p photons, we have 〈 χ mp | χ mq 〉 = δ pq for all p , q ≥
0. The states | χ mp 〉 thus areorthonormal and with Eq. (5.76), the operator m π ∆ m is a projector.By Eq. (5.73), the supports of Π and m π ∆ m are disjoint, and by Eq. (5.72) we have Π + Π = (cid:49) , so the support of m π ∆ m is included in the support of Π . The optimal POVM { Π opt , Π opt } for state discrimination minimises the error probability, hence with the one-sidederror requirement Π opt must have minimal support, meaning that Π opt = m π ∆ m = +∞ (cid:88) p = | χ mp 〉 〈 χ mp | and Π opt = (cid:49) − Π opt . (5.78)Note that, with the same proof, this choice of POVM is also optimal in the generalised settingwhere one is given one unknown generic state and m − { Π h , Π h } corresponding to the Hadamard interferometer witha threshold detection of the last m − Π h = Π opt , (5.79)where Π opt is defined in Eq. (5.78). We have Π h = ˆ H † n Π d ˆ H n , (5.80)where ˆ H n is the unitary evolution corresponding to the action of the interferometer oforder m defined in Eq. (5.59), with n = log m , and Π d = (cid:49) ⊗ | 〉 〈 | ⊗ m − is the POVM operatorcorresponding to the event where none of the m − Π h = ˆ H † n (cid:161) (cid:49) ⊗ | 〉 〈 | ⊗ m − (cid:162) ˆ H n = +∞ (cid:88) p = ˜ H † n (cid:161) | p 〉 〈 p | ⊗ | 〉 〈 | ⊗ m − (cid:162) ˆ H n . (5.81)For k = m , we write a † k the creation operator for the k th mode. For all p ≥ H † n (cid:161) | p 〉 ⊗ | 〉 ⊗ m − (cid:162) = (cid:112) p ! ˆ H † n ( ˆ a † ) p | 〉 ⊗ m = (cid:112) p ! ( ˆ H † n ˆ a † ˆ H n ) p | 〉 ⊗ m = m − p /2 (cid:112) p ! ( ˆ a † + · · · + ˆ a † m ) p | 〉 ⊗ m HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS = m − p /2 (cid:112) p ! (cid:88) k +···+ k m = p p ! k ! . . . k m ! ( ˆ a † ) k . . . ( ˆ a † m ) k m | 〉 ⊗ m (5.82) = m − p /2 (cid:88) k +···+ k m = p (cid:115) p ! k ! . . . k m ! | k . . . k m 〉= | χ mp 〉 ,where we have used ˆ H n | 〉 ⊗ m = | 〉 ⊗ m , ˆ H † n ˆ H n = (cid:49) , ˆ H † n ˆ a † ˆ H n = ˆ a † +···+ ˆ a † m (cid:112) m , the multinomialformula, and Eq. (5.77). With Eqs. (5.78) and (5.81), this concludes the proof.Given that the statistics obtained with the merger scheme and the looped merger schememimic those of the Hadamard scheme, these schemes are also optimal for the same discrimi-nation task. (cid:4) While these devices are relatively easy to implement, any implementation will suffer fromexperimental imperfections. In the next section, we investigate how such imperfections affect theperformance of the merger scheme, for m = In this section, we analyse the performance of the merger scheme in presence of experimentalimperfections. Our error model is the following, with three major sources of error: ( i ) the limiteddetector efficiency and channel transmission loss, characterized by a parameter 0 ≤ η ≤
1, whichchanges the coherent state | α 〉 to |(cid:112) η α 〉 thus reducing the probability of obtaining a click using asingle-photon threshold detector by a factor η ; ( ii ) the limited beam-splitter visibility 0 ≤ ν ≤ iii ) the dark count in the detectors charac-terized by a probability p dark . For our analysis, the click probability due to the coherent statesis of O (1) and thus significantly larger than the dark count probability p dark ∼ − . The darkcounts can thus be safely ignored. η ν p dark Exp. 0.9 (98.8 ± ± ∗ − Table 5.1: Table illustrating the experimental parameters used in simulation of our results.The dark-count rate, p dark , achievable with super-conducting detectors [SJZ + η , and beam splitter visibility ν are from [KKD19].For m =
2, when the input | α 〉 , | β 〉 is fed in an imperfect beam splitter, the transformation frominput modes { ˆ a † , ˆ b † } into the output modes { ˆ c † , ˆ d † } , is the following: | α 〉 a ⊗ | β 〉 b (cid:55)→ (cid:175)(cid:175)(cid:175)(cid:175) (cid:112) ν α + β (cid:112) + (cid:112) − ν α − β (cid:112) (cid:192) c ⊗ (cid:175)(cid:175)(cid:175)(cid:175) (cid:112) ν α − β (cid:112) + (cid:112) − ν α + β (cid:112) (cid:192) d . (5.83)208 .3. PROGRAMMABLE PROJECTIVE MEASUREMENTS WITH COHERENT STATES The corresponding unitary transformation is H (cid:48) = (cid:112) (cid:195) A BA − B (cid:33) , (5.84)where A = (cid:112) ν + (cid:112) − ν , and B = (cid:112) ν − (cid:112) − ν .We consider the case of m = | αβββ 〉 . This results in | αβββ 〉 (cid:55)→ U (cid:48) | αβββ 〉 = | δ δ δ δ 〉 , (5.85)where from Eq. (5.68) we derive U (cid:48) = H (cid:48) × ( H (cid:48) ⊕ H (cid:48) ) = A AB AB B (cid:112) A − (cid:112) B A AB − AB − B (cid:112) A − (cid:112) B , (5.86)with A = (cid:112) ν + (cid:112) − ν and B = (cid:112) ν − (cid:112) − ν . We obtain δ = A α − B β (cid:112) , and δ = A α − B β . Adding thechannel and detector losses η , the output is mapped as δ k (cid:55)→ (cid:112) η δ k , for all k .Similar to the analysis without experimental imperfection, we detect the output modes 1and 2 of the imperfect merger interferometer, with the coherent state input being | αβββ 〉 . Theprobability that none of the two detectors clicks is given byexp (cid:161) − η ( | δ | + | δ | ) (cid:162) . (5.87)Assigning to the detection event no detector clicks the value 0, and to other detection events, i.e., at least one of the detectors clicks , the value 1, we obtain a device whose statistics approximatethose of a projective measurement.When the states are the same, the completeness, which is the probability of not obtaining thedetection event 1 is c exp = exp( − η (1 − ν )(1 + ν ) | α | ). (5.88)We observe that if ν = c exp =
1, thus we obtain perfect completeness.For the imperfection values of Table 5.1, the value of c exp is close to 1 for small | α | values.The analogous completeness for m = c exp = exp( − η (1 − ν ) | α | ). (5.89)From Eq. (5.89) and Eq. (5.88), we observe that c exp ≤ c exp , which implies that the complete-ness for the m = m = HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS in completeness probability for the m = Lemma 5.3. s exp = − exp (cid:104) (4 ν − | α − β | + (cid:104) (1 + ν )(1 − ν ) + (cid:112) ν (1 − ν ) (cid:105) | α | + (cid:104) (1 + ν )(1 − ν ) − (cid:112) ν (1 − ν ) (cid:105) | β | (cid:180)(cid:105) . (5.90) Proof.
When the states are different, the probability of obtaining the detection event 0(failure probability) is 1 − s = exp (cid:179) − η A (cid:180) , (5.91)where A = (cid:175)(cid:175)(cid:175) (cid:112) ν ( α − β ) + (cid:112) − ν ( α + β ) (cid:175)(cid:175)(cid:175) + (cid:175)(cid:175)(cid:175) α − β + (cid:112) ν (1 − ν ) ( α + β ) (cid:175)(cid:175)(cid:175) = (1 + ν ) | α − β | + + ν )(1 − ν ) | α + β | + (cid:112) ν (1 − ν ) ( | α | − | β | ), (5.92)where we used ( α − β )( α + β ) ∗ + ( α − β ) ∗ ( α + β ) = | α | − | β | . Using | α + β | = | α | + | β | −| α − β | we obtain A = (4 ν − | α − β | + (cid:104) (1 + ν )(1 − ν ) + (cid:112) ν (1 − ν ) (cid:105) | α | + (cid:104) (1 + ν )(1 − ν ) − (cid:112) ν (1 − ν ) (cid:105) | β | .(5.93) (cid:4) The analogous soundness in m = s exp = − exp (cid:104) − η (cid:179) ν − (cid:180) | α − β | − η (cid:179) − ν + (cid:112) ν (1 − ν ) (cid:180) | α | − η (cid:179) − ν − (cid:112) ν (1 − ν ) (cid:180) | β | (cid:105) . (5.94)We then obtain: Lemma 5.4.
For all experimental parameters,s exp ≤ s exp . (5.95) Proof.
We have s exp = − exp (cid:104) − η (cid:179) ν − (cid:180) | α − β | − η (cid:179) − ν + (cid:112) ν (1 − ν ) (cid:180) | α | − η (cid:179) − ν − (cid:112) ν (1 − ν ) (cid:180) | β | (cid:105) : = − exp[ − η A ], (5.96)210 .4. DISCUSSION AND OPEN PROBLEMS and s exp = − exp (cid:104) − η (cid:179) ν − (cid:180) | α − β | − η (cid:179) (1 + ν )(1 − ν ) + (cid:112) ν (1 − ν ) (cid:180) | α | − η (cid:179) (1 + ν )(1 − ν ) − (cid:112) ν (1 − ν ) (cid:180) | β | (cid:105) : = − exp[ − η A ]. (5.97)Since the function x (cid:55)→ − e − x is increasing, it is sufficient to show that A ≤ A for all α , β .Writing α = re i φ and β = te i ψ , where r , t ≥ φ , ψ ∈ [0, 2 π ], we obtain A − A = (cid:179) + ν (1 − ν ) + (cid:112) ν (1 − ν ) (cid:180) r + (cid:179) + ν (1 − ν ) − (cid:112) ν (1 − ν ) (cid:180) t − rt (cid:179) − ν (1 − ν ) (cid:180) cos( φ − ψ ). (5.98)This last expression is a polynomial of degree 2 in r , with a positive leading coefficient. Thusif its discriminant is negative, then the expression is always positive. The discriminant is ∆ = t (cid:104)(cid:179) − ν (1 − ν ) (cid:180) cos( φ − ψ ) − (cid:179) + ν (1 − ν ) + (cid:112) ν (1 − ν ) (cid:180)(cid:179) + ν (1 − ν ) − (cid:112) ν (1 − ν ) (cid:180)(cid:105) ≤ − t ν (1 − ν ) ≤
0, (5.99)where the second line is obtained by using cos( φ − ψ ) ≤
1. Hence for all experimental parame-ters within the error model we consider, we have s exp ≤ s exp . (cid:4) Hence, the experimental m = m = We have identified a connection between unknown quantum state discrimination and quantum-programmable measurements. We have presented an optimal scheme for a programmable projec-tive measurement device, and a linear optical implementation, with the Hadamard interferometerand single-photon encoding, which is straightforward and efficient. This could for example beused to design a photonic circuit which would act as a universal projective measurement devicefor a broad range of potential applications from quantum information and cryptography to testsof contextuality. 211
HAPTER 5. QUANTUM-PROGRAMMABLE MEASUREMENTS WITH LINEAR OPTICS
Our scheme can also be interpreted as an optimal swap test when one has a single copy ofone state, and m − m − | ψ 〉 . In principle we could have chosen any other encoding of the quantum input into m − m − | ψ 〉 in m − m − | ψ 〉 —anything more can be done afterwards. This result alsoprovides a natural interpretation of the notion of projective measurement in quantum mechanics,as a comparison between one state and several copies of another state using an interferometer:in the macroscopic limit, when many copies of a reference eigenstate are available, we retrieve amacroscopic classically programmable quantum measurement set up.In order to reduce the experimental requirements, we have also presented an optimal pro-grammable measurement scheme that projects the incoming single mode state in the test registerinto a local coherent state basis of the program registers. Our scheme is implemented usingbalanced beam splitters and single-photon threshold detectors. Threshold detectors with highefficiency and ultra low dark counts are commercially available [SJZ + +
08, MKB05, WRD +
06, HM13, EAO +
02, LMR13].For completeness, it would be interesting to characterise the full class of interferometers thatare optimal for state identity testing under the one-sided error requirement, as we only gave abroad class of such interferometers using a group construction. It would be also interesting toconsider the influence of real experimental conditions in a more general setting.212 H A PT ER Q UANTUM WEAK COIN FLIPPING WITH LINEAR OPTICS W eak coin flipping is among the fundamental cryptographic primitives which ensurethe security of modern communication networks. It allows two mistrustful parties toremotely agree on a random bit when they favor opposite outcomes. Unlike other two-party computations, one can achieve information-theoretic security using quantum mechanicsonly: both parties are prevented from biasing the flip with probability higher than 1/2 + (cid:178) , where (cid:178) is arbitrarily low. Classically, the dishonest party can always cheat with probability 1 unlesscomputational assumptions are used. Despite its importance, no physical implementation hasbeen proposed so far for quantum weak coin flipping.In this chapter, we present a practical protocol for quantum weak coin flipping that requires asingle photon and linear optics only. We show that it is secure even when threshold single-photondetectors are used, and reaches a bias as low as (cid:178) = (cid:112) − ≈ Compared to weak coin flipping, where two mistrustful parties wish to remotely agree on theoutcome of a coin flip when they favor different outcomes, the cryptographic task of strong coinflipping corresponds to the case where they want to agree on an unbiaised random bit whenthey do not necessarily favor a particular outcome. Despite its name, strong coin flipping is lessgeneral than weak coin flipping in the sense that optimal strong coin flipping protocols may bedesigned which use weak coin flipping protocols as a subroutine [CK09].213
HAPTER 6. QUANTUM WEAK COIN FLIPPING WITH LINEAR OPTICS
While quantum strong coin flipping protocols have been experimentally demonstrated [MTVUZ05,BBB +
11, PJL + +
16, ARW19, ARV19] translate triviallyinto a simple experiment: they all involve performing single-shot generalized measurements orgenerating beyond-qubit states.We introduce a family of quantum weak coin flipping protocols, inspired by [SR02], whichachieve biases as low as (cid:178) = (cid:112) − ≈ + ≈ x ∈ [0, ] as a free protocol parameter, these read:• Alice mixes a single photon with the vacuum on a beam splitter of reflectance x .• Alice keeps the first half of the state obtained, and sends the second half to Bob.• Bob mixes the half he receives with the vacuum on a beam splitter of reflectance y = − − x ) .• Bob measures the second register of his state with a single-photon detector, and broadcaststhe outcome c ∈ {
0, 1 } .• The last step is a verification step, which splits into two cases. If c =
0, Alice sends her halfof the state to Bob, who mixes it with his half on a beam splitter of reflectance z = x . Hethen measures the two output modes with single-photon detectors. He declares Alice thewinner if the outcome (1, 0) is obtained. If c =
1: Bob discards his half, and Alice measuresher half with a single-photon detector. If the outcome is (0), Bob is declared winner.214 .1. WEAK COIN FLIPPING PROTOCOL WITH LINEAR OPTICS ( x )
Representation of the honest protocol.
The dashed boxes indicate Alice andBob’s laboratories, respectively. Dashed red lines represent beam splitters, with the reflectanceindicated in red. | 〉 and | 〉 are the vacuum and single photon Fock states, respectively. Curlylines represent fiber used for quantum communication from Alice to Bob, or delay lines withinAlice’s or Bob’s laboratory, when waiting for the other party’s communication. Bob broadcaststhe classical outcome c , which controls an optical switch on Alice’s side. The protocol when Bobdeclares c = In what follows, we let the parameters x , y , z vary freely, and derive the relations these parametersneed to satisfy to enforce a honest protocol without abort cases. We show that for the specificrelations indicated above the protocol is fair, i.e., the probability of winning for each party is when they are both honest.Single photons are quantized excitations of the electromagnetic field, which are describedby the action of the creation operator onto the vacuum. Beam splitters act linearly on creationoperators and leave invariant the vacuum. More precisely, a beam splitter of reflectance r actingon modes k , l maps the creation operators ˆ a † k and ˆ a † l of the input modes onto ˆ b † k and ˆ b † l , where (cid:195) ˆ b † k ˆ b † l (cid:33) = H ( r ) kl (cid:195) ˆ a † k ˆ a † l (cid:33) , (6.1)with H ( r ) kl = (cid:195) (cid:112) r (cid:112) − r (cid:112) − r −(cid:112) r (cid:33) . (6.2)215 HAPTER 6. QUANTUM WEAK COIN FLIPPING WITH LINEAR OPTICS
Hence, the evolution of the quantum state over the three modes up to Bob’s first measurementreads: | 〉 → ( x ),12 (cid:112) x | 〉 + (cid:112) − x | 〉→ ( y ),23 (cid:112) x | 〉 + (cid:112) (1 − x ) y | 〉+ (cid:112) (1 − x )(1 − y ) | 〉 , (6.3)where the notation ( r ), kl indicates the reflectance of the beam splitter and the correspondingspatial modes. The probability that Bob obtains outcome c = = (1 − x )(1 − y ), while the probability of outcome c = = − Pr [1]. Setting y = − − x ) ensures Pr [0] = Pr [1] = .When c =
1, the state on modes 1 and 2 is projected onto | 〉 , while c = (cid:112) x | 〉 + (cid:112) − x | 〉 . In the first case, the measurement performed by Alice outputs (0)with probability 1. In the second case, the measurement performed by Bob outputs (1, 0) withprobability 1 when z = x − (1 − x )(1 − y ) = x . (6.4)In that case, the probability that Alice (resp. Bob) wins is directly given by P ( A ) h = Pr [0] (resp. P ( B ) h = Pr [1]). This shows that the protocol is fair, since Pr [0] = Pr [1] = .In the following, we make use of a simple reduction which allows us to simplify calculations inthe proofs: Lemma 6.1.
Let U = ( H ( z ) ⊗ (cid:49) )( (cid:49) ⊗ H ( y ) ) , with z > . For all density matrices τ , Tr [( τ ⊗ | 〉 〈 | ) U † ( (cid:49) ⊗ | 〉 〈 | ) U ] = Tr [( τ ⊗ | 〉 〈 | ) V † ( | 〉 〈 | ⊗ (cid:49) ⊗ | 〉 〈 | ) V ], (6.5) where V = ( (cid:49) ⊗ H ( b ) )( H ( a ) ⊗ (cid:49) )( (cid:49) ⊗ R ( π ) ⊗ (cid:49) ) , with a = y (1 − z )1 − (1 − y )(1 − z ) and b = − (1 − y )(1 − z ) , and R ( π ) a phase shift of π acting on mode . Proof.
The action of U on the creation operators is given by U = (cid:112) z (cid:112) − z (cid:112) − z −(cid:112) z
00 0 1 (cid:112) y (cid:112) − y (cid:112) − y −(cid:112) y = (cid:112) z (cid:112) y (1 − z ) (cid:112) (1 − y )(1 − z ) (cid:112) − z −(cid:112) yz − (cid:112) (1 − y ) z (cid:112) − y −(cid:112) y . (6.6)Linear interferometers map product coherent states onto product coherent states, and, for216 .1. WEAK COIN FLIPPING PROTOCOL WITH LINEAR OPTICS all α ∈ (cid:67) , we have that U † | α 〉 = | β β β 〉 , where β β β = α (cid:112) z α (cid:112) y (1 − z ) α (cid:112) (1 − y )(1 − z ) . (6.7)We have V = ( (cid:49) ⊗ H ( b ) )( H ( a ) ⊗ (cid:49) )( (cid:49) ⊗ R ( π ) ⊗ (cid:49) ), with a , b ∈ [0, 1], and R ( π ) a phase shift of π acting on mode 2. The action of V on the creation operators is given by V = (cid:112) b (cid:112) − b (cid:112) − b −(cid:112) b (cid:112) a (cid:112) − a (cid:112) − a −(cid:112) a
00 0 1 − = (cid:112) a −(cid:112) − a (cid:112) b (1 − a ) (cid:112) ab (cid:112) − b (cid:112) (1 − a )(1 − b ) (cid:112) a (1 − b ) −(cid:112) b . (6.8)For all α ∈ (cid:67) , V † | α 〉 = | γ γ γ 〉 , where γ γ γ = α (cid:112) b (1 − a ) α (cid:112) ab α (cid:112) − b . (6.9)Since a = y (1 − z )1 − (1 − y )(1 − z ) and b = − (1 − y )(1 − z ), we have b (1 − a ) = z , ab = y (1 − z ), and 1 − b = (1 − y )(1 − z ), so ( β , β , β ) = ( γ , γ , γ ).Then, Tr [( τ ⊗ | 〉 〈 | ) U † ( (cid:49) ⊗ | 〉 〈 | ) U ] = π (cid:90) (cid:67) d α Tr [( τ ⊗ | 〉 〈 | ) U † | α 〉 〈 α | U ] = π (cid:90) (cid:67) d α Tr [( τ ⊗ | 〉 〈 | ) V † | α 〉 〈 α | V ] = Tr [( τ ⊗ | 〉 〈 | ) V † ( | 〉 〈 | ⊗ (cid:49) ⊗ | 〉 〈 | ) V ], (6.10)where we used the completeness relation of coherent states (cid:49) = π (cid:82) (cid:67) | α 〉 〈 α | d α . (cid:4) We now derive the soundness of the protocol. Namely, we obtain the maximal winning probabilitieswhen Bob is dishonest and Alice is honest, and vice versa.
Lemma 6.2.
Bob’s optimal cheating probability is given byP ( B ) d = − x . (6.11)217 HAPTER 6. QUANTUM WEAK COIN FLIPPING WITH LINEAR OPTICS
Proof.
Dishonest Bob should always declare the outcome c = x , so that Bob’s winningprobability is upper bounded by (1 − x ). This upper bound is reached if Bob discards his halfof the state and broadcasts c =
1. Bob’s optimal cheating probability thus is P ( B ) d = − x . (cid:4) Alice wins when Bob declares c = σ , while Bob performs the restof the protocol honestly. Assuming honest Bob has number-resolving detectors, we obtain thefollowing result: Lemma 6.3.
Alice’s optimal cheating probability when Bob has number resolving detectors isgiven by P ( A ) d = − (1 − y )(1 − z ). (6.12) ( y )
Dishonest Alice.
Alice aims to maximize the outcome (1, 0, 0): an outcome 0 on thethird mode means that Bob declared Alice the winner, while an outcome (1, 0) for modes 1 and2 means that Alice passed Bob’s verification. The reflectances of the beam splitter are given by y = − − x ) and z = x . Proof.
When using number-resolving single-photon detectors, any projection onto the n > | 〉 〈 | only (Fig. 6.2).Let σ be the state sent by Alice. Let U = ( H ( z ) ⊗ (cid:49) )( (cid:49) ⊗ H ( y ) ), with z = x − (1 − x )(1 − y ) . Aliceneeds to maximize the probability of the overall outcome (1, 0, 0), which is given by P ( A ) d = Tr [ U ( σ ⊗ | 〉 〈 | ) U † | 〉 〈 | ], (6.13)since Bob uses number-resolving detectors. By convexity of the probabilities, we may assume218 .1. WEAK COIN FLIPPING PROTOCOL WITH LINEAR OPTICS without loss of generality that Alice sends a pure state σ = | ψ 〉 〈 ψ | , which allows us to write: P ( A ) d = Tr [ U ( | ψ 〉 〈 ψ | ⊗ | 〉 〈 | ) U † | 〉 〈 | ] = Tr [( | ψ 〉 〈 ψ | ⊗ | 〉 〈 | ) U † | 〉 〈 | U ] = Tr [ 〈 ψ | ⊗ 〈 | U † | 〉 〈 | U | ψ 〉 ⊗ | 〉 ]. (6.14)We have: U † | 〉 = ( (cid:49) ⊗ H ( y ) )( H ( z ) ⊗ (cid:49) ) | 〉= ( (cid:49) ⊗ H ( y ) )( (cid:112) z | 〉 + (cid:112) − z | 〉 ) = (cid:112) z | 〉 + (cid:112) y (1 − z ) | 〉 + (cid:112) (1 − y )(1 − z ) | 〉 , (6.15)and therefore: U † | 〉 〈 | U = z | 〉 〈 | + y (1 − z ) | 〉 〈 | + (1 − y )(1 − z ) | 〉 〈 |+ (cid:112) yz (1 − z ) ( | 〉 〈 | + | 〉 〈 | ) + (cid:112) z (1 − y )(1 − z ) ( | 〉 〈 | + | 〉 〈 | ) + (1 − z ) (cid:112) y (1 − y ) ( | 〉 〈 | + | 〉 〈 | ) . (6.16)Substituting back into Eq. (6.14) then reduces to: P ( A ) d = 〈 ψ | (cid:179) z | 〉 〈 | + y (1 − z ) | 〉 〈 | + (cid:112) yz (1 − z ) ( | 〉 〈 | + | 〉 〈 | ) (cid:180) | ψ 〉= 〈 ψ | (cid:179) (cid:112) z | 〉 + (cid:112) y (1 − z ) | 〉 (cid:180) (cid:179) (cid:112) z 〈 | + (cid:112) y (1 − z ) 〈 | (cid:180) | ψ 〉= (cid:175)(cid:175)(cid:175) 〈 ψ | (cid:179) (cid:112) z | 〉 + (cid:112) y (1 − z ) | 〉 (cid:180)(cid:175)(cid:175)(cid:175) . (6.17)Using Cauchy-Schwarz inequality then allows us to upper bound P ( A ) d as: P ( A ) d ≤ (cid:107) ψ (cid:107) (cid:176)(cid:176)(cid:176)(cid:179) (cid:112) z | 〉 + (cid:112) y (1 − z ) | 〉 (cid:180)(cid:176)(cid:176)(cid:176) ≤ (1 − (1 − y )(1 − z )) (cid:107) ψ (cid:107) , (6.18)which is maximized for (cid:107) ψ (cid:107) =
1. Hence we finally get: P ( A ) d ≤ − (1 − y )(1 − z ). (6.19)In order to find Alice’s optimal cheating strategy (i.e., the optimal pure state | φ 〉 thatshe must send to achieve this bound), we remark that the unnormalized state (cid:112) z | 〉 + (cid:112) y (1 − z ) | 〉 maximizes the expression in Eq. (6.18). Normalizing this state then providesAlice’s optimal strategy, which is to prepare the state | φ 〉 : = (cid:114) z − (1 − y )(1 − z ) | 〉 + (cid:115) y (1 − z )1 − (1 − y )(1 − z ) | 〉 . (6.20)219 HAPTER 6. QUANTUM WEAK COIN FLIPPING WITH LINEAR OPTICS
Hence, P ( A ) d = − (1 − y )(1 − z ). (6.21)In the case of a fair protocol, y = − − x ) and z = x , so P ( A ) d = − x ) , (6.22)and Alice’s optimal strategy is to prepare the state | φ x 〉 : = (cid:112) x (1 − x ) | 〉 + (1 − x ) | 〉 . (6.23) (cid:4) Remarkably, the protocol is still secure even when Bob only uses single photon threshold detectors,which is essential to the practicality of the protocol. Moreover, Alice’s optimal cheating probabilityremains the same:
Lemma 6.4.
Alice’s optimal cheating probability when Bob has threshold detectors is given byP ( A ) d = − (1 − y )(1 − z ). (6.24) | i
Equivalent picture for dishonest Alice.
In the original dishonest setup of Fig. 6.2,Alice aims to maximize the outcome (1, 0, 0). This is equivalent to Alice maximizing outcome 0on spatial modes 1 and 3, independently of what is detected on mode 2. The outcomes indicatedcorrespond to Alice winning. The reflectance is b = − (1 − y )(1 − z ). Proof.
Unlike the previous case, incorrect outcomes with higher photon number could stillpass the test: for n ≥
1, the threshold detectors cannot discriminate between a | 〉 and | n 〉 projection. We show in the following that this doesn’t help a dishonest Alice, and thatthe strategy described previously for the case of number resolving detectors is still optimalin the case of threshold detectors.With the same notations as in the previous proof, Alice needs to maximize the probabil-ity of the overall outcome (1, 0, 0), hence the overlap with the projector (cid:80) ∞ n = | n 〉 〈 n | = .1. WEAK COIN FLIPPING PROTOCOL WITH LINEAR OPTICS ( (cid:49) − | 〉 〈 | ) ⊗ | 〉 〈 | . This allows us to write: P ( A ) d = Tr [ U ( | ψ 〉 〈 ψ | ⊗ | 〉 〈 | ) U † (( (cid:49) − | 〉 〈 | ) ⊗ | 〉 〈 | )], (6.25)since Bob uses threshold detectors, where U = ( H ( z ) ⊗ (cid:49) )( (cid:49) ⊗ H ( y ) ), with z = x − (1 − x )(1 − y ) .Linear optical evolution conserves photon number. Hence if Alice sends the vacuumstate, the detectors will never click. Removing the two-mode vacuum component of the stateprepared by Alice and renormalizing therefore always increases her winning probability.Since we are looking for the maximum winning probability, we can assume without loss ofgenerality that 〈 ψ | 〉 =
0, i.e.,Tr [ U ( | ψ 〉 〈 ψ | ⊗ | 〉 〈 | ) U † | 〉 〈 | ] = | 〈 ψ | 〉 | , (6.26)So maximizing the winning probability in Eq. (6.25) is equivalent to maximizing˜ P ( A ) d = Tr [ U ( | ψ 〉 〈 ψ | ⊗ | 〉 〈 | ) U † ( (cid:49) ⊗ | 〉 〈 | )], (6.27)given the constraint 〈 ψ | 〉 =
0. We have˜ P ( A ) d = Tr [ U ( | ψ 〉 〈 ψ | ⊗ | 〉 〈 | ) U † ( (cid:49) ⊗ | 〉 〈 | )] = Tr [( | ψ 〉 〈 ψ | ⊗ | 〉 〈 | ) U † ( (cid:49) ⊗ | 〉 〈 | ) U ]. (6.28)With Lemma 6.1 and Eq. (6.28), we may thus write:˜ P ( A ) d = Tr [( | ψ 〉 〈 ψ | ⊗ | 〉 〈 | ) V † ( | 〉 〈 | ⊗ (cid:49) ⊗ | 〉 〈 | ) V ], (6.29)where V = ( (cid:49) ⊗ H ( b ) )( H ( a ) ⊗ (cid:49) )( (cid:49) ⊗ R ( π ) ⊗ (cid:49) ), with a = y (1 − z )1 − (1 − y )(1 − z ) and b = − (1 − y )(1 − z ). Letus now define: | ψ a 〉 : = H ( a ) ( (cid:49) ⊗ R ( π )) | ψ 〉 . (6.30)The constraints 〈 ψ | 〉 = 〈 ψ a | 〉 = P ( A ) d = Tr [( | ψ a 〉 〈 ψ a | ⊗ | 〉 〈 | )( (cid:49) ⊗ H ( b ) )( | 〉 〈 | ⊗ (cid:49) ⊗ | 〉 〈 | )( (cid:49) ⊗ H ( b ) )], (6.31)with the constraint 〈 ψ a | 〉 =
0. Maximizing this expression thus corresponds to maximizingthe probability of the outcome (0, 0) when measuring modes 1 and 3 of the state obtainby mixing the second half of | ψ a 〉 with the vacuum on a beam splitter of reflectance b = − (1 − y )(1 − z ) (Fig. 6.3).We now show that an optimal strategy for Alice is to ensure that | ψ a 〉 = | 〉 . Let us write | ψ a 〉 = (cid:88) p + q > ψ pq | pq 〉 , (6.32)221 HAPTER 6. QUANTUM WEAK COIN FLIPPING WITH LINEAR OPTICS where we take into account the constraint 〈 ψ x | 〉 =
0. Then, with Eq. (6.31) we obtain˜ P ( A ) d = (cid:88) p + q > p (cid:48) + q (cid:48) > ψ pq ψ ∗ p (cid:48) q (cid:48) Tr [ | pq 〉 〈 p (cid:48) q (cid:48) | ( | 〉 〈 | ⊗ H ( b ) ( (cid:49) ⊗ | 〉 〈 | ) H ( b ) )] = (cid:88) q > q (cid:48) > ψ q ψ ∗ q (cid:48) Tr [ | q 〉 〈 q (cid:48) | H ( b ) ( (cid:49) ⊗ | 〉 〈 | ) H ( b ) ] = (cid:88) n ≥ q > q (cid:48) > ψ q ψ ∗ q (cid:48) Tr [ | q 〉 〈 q (cid:48) | H ( b ) | n 〉 〈 n | H ( b ) ] (6.33) = (cid:88) n > | ψ n | | 〈 n | H ( b ) | n 〉 | = (cid:88) n > | ψ n | b n ,where we used in the fourth line the fact that H ( b ) doesn’t change the number of photons.Since b ∈ [0, 1], this shows that ˜ P ( A ) d ≤ b (cid:88) n > | ψ n | = b , (6.34)since | ψ a 〉 is normalized, and this bound is reached for | ψ | =
1, i.e., | ψ a 〉 = | 〉 . WithEq. (6.30), this implies that an optimal strategy for Alice is to prepare the state | ψ 〉 = ( (cid:49) ⊗ R ( π )) H ( a ) | 〉= (cid:112) − a | 〉 + (cid:112) a | 〉= (cid:114) z − (1 − y )(1 − z ) | 〉 + (cid:115) y (1 − z )1 − (1 − y )(1 − z ) | 〉= | φ 〉 , (6.35)where | φ 〉 is the state that dishonest Alice needs to send to maximize her winning probabilitywhen Bob uses number-resolving detectors (Eq. (6.20)). Her winning probability is then P ( A ) d = − (1 − y )(1 − z ). (6.36)We therefore recover the same result as for number-resolving detectors. Once again, if theprotocol is fair then y = − − x ) and z = x , so P ( A ) d = − x ) , (6.37)and an optimal strategy for Alice is to prepare the state | φ x 〉 : = (cid:112) x (1 − x ) | 〉 + (1 − x ) | 〉 . (6.38) (cid:4) Alice’s cheating probability equals − x ) for y = − − x ) and z = x . In particular, for all values222 .1. WEAK COIN FLIPPING PROTOCOL WITH LINEAR OPTICS of x , we retrieve the property shared by the protocols of [SR02]: P ( A ) d P ( B ) d = . Setting x = − (cid:112) (cid:112) (cid:178) = (cid:112) − ≈ Following [CK09], we show that our family of quantum weak coin flipping protocols allows us toconstruct a quantum strong coin flipping protocol:
Lemma 6.5.
There exists a quantum strong coin flipping protocol achieving bias (cid:178) ≈ whichuses an unbalanced linear optical weak coin flipping protocol as a subroutine. Proof.
An unbalanced quantum weak coin flipping protocol can be turned into a quantumstrong coin flipping protocol using an additional classical protocol, as described in [CK09]. Inparticular, let us consider a weak coin flipping protocol such that: P ( A ) h = pP ( B ) h = − pP ( A ) d = p + (cid:178) P ( B ) d = − p + (cid:178) , (6.39)for p ∈ [0, 1] and (cid:178) >
0. Then, the corresponding strong coin flipping protocol has bias [CK09]max (cid:181) −
12 ( p − (cid:178) ), 12 − ( p + (cid:178) ) − (cid:182) . (6.40)For our weak coin flipping protocol, we have: P ( A ) h = − (1 − x )(1 − y ) P ( B ) h = (1 − x )(1 − y ) P ( A ) d = − (1 − y )(1 − z ) P ( B ) d = − x , (6.41)with the constraint z = x − (1 − x )(1 − y ) (so that the protocol does not abort in the honest case,Eq. (6.4)). Enforcing the conditions in Eq. (6.39), and optimizing over the correspondingstrong coin flipping bias implies x = y (1 − y )(1 − y ) z = y (1 − y ) − x = − y − z + yz , (6.42)223 HAPTER 6. QUANTUM WEAK COIN FLIPPING WITH LINEAR OPTICS which in turn give the values x ≈ y ≈ z ≈ x , y , z ∈ [0, 1], and a bias of ≈ + (cid:4) We investigate how imperfect state generation, non-ideal beam splitters and single-photondetector dark counts affect the correctness and security of the protocol. While we fixed theparameter values to y = − − x ) and z = x in the ideal setting, we now allow the threeparameters x , y , z to vary freely.The vacuum/single-photon encoding is very robust to noise, in comparison to polarization orphase encoding for instance: the only property that must be preserved through propagation isphoton number. This implies that photon indistinguishability and purity are not required in anydegree of freedom other than photon number. In this case, Alice may simply produce a heraldedsingle photon via spontaneous parametric down-conversion (SPDC) [Cou18], which generates aphoton pair: one may be used for the flip, while the other may herald the presence of the first one.Given the photon-pair emission probability p , accidentally emitting two pairs at the same timeusing SPDC occurs with probability p . Since p may be arbitrarily tuned by changing the pumppower, p —and therefore the probability of two photons being accidentally generated by Alice atonce—may then be decreased to negligible values.Note that, in the case where Alice’s single photon source is probabilistic but heralded (as inSPDC), she may always inform Bob of a successful state generation prior to his announcement of c without compromising security. In what follows, we may therefore assume that both partieshave agreed on the presence of an initial state, and hence know when the protocol occurs.Noise will therefore stem from the non-ideal reflectances of the beam splitters, and the non-zero detector dark count probability p dc . For each party, these may affect the protocol correctnessin two ways: an undesired bias of the flip, and an added abort probability during the verificationprocess.Deviations on the beam splitter reflectances x , y , and z will first change the honest winningprobabilities: these may be re-calculated by replacing the ideal reflectance r ∈ { x , y } with animperfect r (cid:48) . As regards to honest aborts, a beam splitter with reflectance z (cid:48) instead of z maybe applied on the resulting state when c =
0. Noisy detectors may cause an unwanted abort224 .2. EXPERIMENTAL IMPERFECTIONS corresponding to a click because of dark counts. However, with superconducting nanowire single-photon detectors, this probability is typically very low, of the order of p dc < − [Had09].We can therefore conclude that any source of noise may be incorporated in the securityanalysis by simply replacing parameters x , y , and z with x (cid:48) , y (cid:48) , and z (cid:48) . Furthermore, this sourceof error will most likely be negligible with current technology. We therefore solely focus on themore consequential effects of losses. Losses can be due to the channel transmission and to non-unit delay line transmission anddetection efficiencies. We label η t the transmission efficiency of the quantum channel from Aliceto Bob. We also define as η ( i ) f the transmission of party i ’s fiber delay, while η ( i ) d denotes thedetection efficiency of party i ’s single-photon detectors. Here, we assume the efficiencies of Bob’sdetectors to be the same, and that each party introduces a fiber delay whenever they are waitingfor the other party’s communication. The delay time therefore depends on the distance betweenthe two parties. We give a representation of the honest protocol with losses, in Fig. 6.4.We recall a useful simple property, which we will use extensively in the following: Lemma 6.6.
Equal losses on various modes can be commuted through passive linear opticalelements acting on these modes.
This result was proven, e.g., in [BL10], and we give hereafter a quick proof.
Proof.
One way to prove this statement is to use the fact that any interferometer may bedecomposed as beam splitters and phase shifters [RZBB94]. Then, losses trivially commutewith phase shifters, and are easily shown to commute with beam splitters. Indeed, considera beam splitter of reflectance t acting on modes 1 and 2. Its action on the creation operatorsof the modes is given by ˆ a † , ˆ a † → (cid:112) t ˆ a † + (cid:112) − t ˆ a † , (cid:112) − t ˆ a † − (cid:112) t ˆ a † , (6.44)while equal losses η on both modes act asˆ a † , ˆ a † → (cid:112) η ˆ a † , (cid:112) η ˆ a † . (6.45)Hence, the action of the beam splitter followed by losses is given byˆ a † , ˆ a † → (cid:112) η ( (cid:112) t ˆ a † + (cid:112) − t ˆ a † ), (cid:112) η ( (cid:112) − t ˆ a † − (cid:112) t ˆ a † ), (6.46)while losses followed by the beam splitter act asˆ a † , ˆ a † → (cid:112) t ( (cid:112) η ˆ a † ) + (cid:112) − t ( (cid:112) η ˆ a † ), (cid:112) − t ( (cid:112) η ˆ a † ) − (cid:112) t ( (cid:112) η ˆ a † ), (6.47)225 HAPTER 6. QUANTUM WEAK COIN FLIPPING WITH LINEAR OPTICS which is equal to the previous evolution. (cid:4) ( x )
Representation of the honest protocol with losses.
The dashed boxes indicateAlice and Bob’s laboratories, respectively. Dashed red lines represent beam splitters, with thereflectance indicated in red. The efficiencies of the detectors, are indicated in white. Curly linesrepresent fiber used for quantum communication from Alice to Bob, or delay lines within Alice’sor Bob’s laboratory. | 〉 and | 〉 are the vacuum and single photon Fock states, respectively. Bobbroadcasts the classical outcome c , which controls an optical switch on Alice’s side. The protocolwhen Bob declares c = P ( A ) h and P ( B ) h , andhence the probability P ab of abort, in the presence of losses: Lemma 6.7. P ( A ) h = η t η ( B ) d (cid:181)(cid:114) xz η ( A ) f + (cid:114) (1 − x ) y (1 − z ) η ( B ) f (cid:182) P ( B ) h = η t η ( B ) d (1 − x )(1 − y ) P ab = − P ( A ) h − P ( B ) h . (6.48)226 .2. EXPERIMENTAL IMPERFECTIONS Proof.
The honest winning probability for Bob is directly given by his chance of detectingthe photon (the photon gets to his detector and doesn’t get lost): P ( B ) h = η t η ( B ) d (1 − x )(1 − y ). (6.49)On the other hand, Alice wins if the photon, starting from her first input mode, is detectedby Bob in the last step.The evolution of the creation operator of the first mode during the lossy honest protocolis given by:ˆ a † → (cid:112) x ˆ a † + (cid:112) − x ˆ a † → (cid:114) x η ( A ) f ˆ a † + (cid:112) (1 − x ) η t ˆ a † → (cid:114) x η ( A ) f ˆ a † + (cid:112) (1 − x ) η t y ˆ a † + (cid:112) (1 − x )(1 − y ) η t ˆ a † → (cid:114) x η ( A ) f ˆ a † + (cid:112) (1 − x ) η t y ˆ a † + (cid:113) (1 − x )(1 − y ) η t η ( B ) d ˆ a † → (cid:114) x η ( A ) f η t ˆ a † + (cid:114) (1 − x ) η t y η ( B ) f ˆ a † + (cid:113) (1 − x )(1 − y ) η t η ( B ) d ˆ a † (6.50) → (cid:181)(cid:114) x η ( A ) f η t z + (cid:114) (1 − x ) η t y η ( B ) f (1 − z ) (cid:182) ˆ a † + (cid:181)(cid:114) x η ( A ) f η t (1 − z ) − (cid:114) (1 − x ) η t y η ( B ) f z (cid:182) ˆ a † + (cid:113) (1 − x )(1 − y ) η t η ( B ) d ˆ a † → (cid:181)(cid:114) x η ( A ) f η t z η ( B ) d + (cid:114) (1 − x ) η t y η ( B ) f (1 − z ) η ( B ) d (cid:182) ˆ a † + (cid:181)(cid:114) x η ( A ) f η t (1 − z ) η ( B ) d − (cid:114) (1 − x ) η t y η ( B ) f z η ( B ) d (cid:182) ˆ a † + (cid:113) (1 − x )(1 − y ) η t η ( B ) d ˆ a † .In particular, the photon reaches Bob’s uppermost detector with probability P ( A ) h = (cid:181)(cid:114) x η ( A ) f η t z η ( B ) d + (cid:114) (1 − x ) η t y η ( B ) f (1 − z ) η ( B ) d (cid:182) = η t η ( B ) d (cid:181)(cid:114) xz η ( A ) f + (cid:114) (1 − x ) y (1 − z ) η ( B ) f (cid:182) . (6.51)Finally, the protocol aborts for all other detection events: P ab = − P ( A ) h − P ( B ) h . (6.52) (cid:4) Note that the overall correctness does not depend on Alice’s detection efficiency η ( A ) d , since thedeclaration of outcome c depends solely on Bob’s detector and the verification step on Alice’s sideinvolves detecting vacuum. 227 HAPTER 6. QUANTUM WEAK COIN FLIPPING WITH LINEAR OPTICS
The soundness of the protocol is also affected by the presence of losses.Dishonest Bob’s best strategy is to perform the same attack as in the lossless case, becausehe has no control over Alice’s half of the subsystem. His winning probability is then given by thefollowing result:
Lemma 6.8.
Dishonest Bos’s maximum winning probability is given by:P ( B ) d = − x η ( A ) f η ( A ) d . (6.53)In a more general game-theoretic scenario, Bob’s best strategy will in fact depend on the rewardsand sanctions associated with honest aborts and ‘getting caught cheating’ aborts. In other words,Bob has to minimize his risk-to-reward ratio. Maximizing his winning probability makes him runthe risk of getting caught cheating with probability x η ( A ) f η ( A ) d .Dishonest Alice must still generate the state which maximizes the (1, 0, 0) outcome on Bob’sdetectors after his honest transformations have been applied. However, the expression for Bob’scorresponding projector now changes, as there is a finite probability (1 − η ( B ) d ) n that the n -photoncomponent is projected onto the vacuum. The 0 outcome on one spatial mode is therefore triggeredby the projection Π = (cid:80) ∞ n = (1 − η ( B ) d ) n | n 〉 〈 n | . The total projector responsible for the (1, 0, 0) outcomethen reads Π = ( (cid:49) − Π ) ⊗ Π ⊗ Π . Lemma 6.9.
Dishonest Alice’s maximum winning probability is given by:P ( A ) d = max l > (cid:183)(cid:179) − (1 − y η ( B ) f )(1 − z ) η ( B ) d (cid:180) l − (cid:179) − η ( B ) d (cid:180) l (cid:184) ≤ − (1 − y )(1 − z ). (6.54)The value of the upper bound in the second line is Alice’s cheating probability in the lossless case.This shows that Alice cannot take advantage of Bob’s imperfect detectors or his lossy delay linein order to increase her cheating probability.
The losses η correspond to a probability 1 − η of losing a photon. These can bemodelled as a mixing with the vacuum on a beam splitter of reflectance η . We first show thatwe can obtain Alice’s cheating probability by solving the case with perfect delay line, andreplacing the parameter y by y η f , independently of the efficiency η d of his detectors.The lossy delay line of efficiency η f may be modelled as a mixing with the vacuum on a beamsplitter of transmission η f .Alice prepares a state σ , which goes through the interferometer depicted in Fig. 6.5, andwins if the measurement outcome obtained by Bob is (1, 0, 0).In particular, note that the outcome 0 must be obtained for the third mode. Hence Alice’swinning probability is always lower than if the third mode was mixed with the vacuum on229 HAPTER 6. QUANTUM WEAK COIN FLIPPING WITH LINEAR OPTICS a beam splitter of transmission amplitude η f just before the detection (Fig. 6.6), since thisincreases the probability of the outcome 0 for this mode.Let us assume that this is the case. Then, by Lemma 6.6, the losses η f on output modes2 and 3 may be commuted back through the beam splitter of reflectance y , acting on modes 2and 3.Since the input state on mode 3 is the vacuum, the losses on this mode may then beremoved (Fig. 6.7). In that case, the probability of winning is clearly lower than when thedelay line is perfect (Fig. 6.8), because Alice is now restricted to lossy state preparationinstead of ideal state preparation.This reduction shows that Alice’s maximum winning probability when Bob is using alossy delay line is always lower than when Bob’s delay line is perfect, independently of theefficiency η d of his detectors.Moreover, Alice’s maximum cheating probability and optimal cheating strategy may beinferred from the case where Bob has a perfect delay line, as we show in what follows.By convexity of the probabilities, Alice’s best strategy is to send a pure state | ψ 〉 = (cid:80) k , l ≥ ψ kl | kl 〉 . Let us denote by W the interferometer depicted in Fig. 6.5, including thedetection losses. Let us consider the evolution of Alice’s state and the vacuum on the thirdinput mode through the interferometer W . The creation operator for the first mode evolvesas ˆ a † → (cid:112) z ˆ a † + (cid:112) − z ˆ a † → (cid:112) z η d ˆ a † + (cid:112) (1 − z ) η d ˆ a † = W ˆ a † W † , (6.55)while the creation operator for the second mode evolves asˆ a † → (cid:112) y ˆ a † + (cid:112) − y ˆ a † → (cid:112) y η f ˆ a † + (cid:112) − y ˆ a † → (cid:113) y (1 − z ) η f ˆ a † − (cid:112) yz η f ˆ a † + (cid:112) − y ˆ a † (6.56) → (cid:113) y (1 − z ) η f η d ˆ a † − (cid:112) yz η f η d ˆ a † + (cid:112) (1 − y ) η d ˆ a † = W ˆ a † W † .Hence, the output state (before the ideal threshold detection) is given by W | ψ 〉 = W (cid:88) k , l ≥ ψ kl | kl 〉= W (cid:34) (cid:88) k , l ≥ ψ kl (cid:112) k ! l ! ( ˆ a † ) k ( ˆ a † ) l (cid:35) | 〉 .2. EXPERIMENTAL IMPERFECTIONS = (cid:34) (cid:88) k , l ≥ ψ kl (cid:112) k ! l ! ( W ˆ a † W † ) k ( W ˆ a † W † ) l (cid:35) | 〉 (6.57) = (cid:183) (cid:88) k , l ≥ ψ kl (cid:112) k ! l ! (cid:179) (cid:112) z η d ˆ a † + (cid:112) (1 − z ) η d ˆ a † (cid:180) k × (cid:179)(cid:113) y (1 − z ) η f η d ˆ a † − (cid:112) yz η f η d ˆ a † + (cid:112) (1 − y ) η d ˆ a † (cid:180) l (cid:184) | 〉 .Now Alice’s maximum cheating probability is given by P ( A ) d = Tr [ W | ψ 〉 〈 ψ | W † ( (cid:49) − | 〉 〈 | ) | 〉 〈 | ]. (6.58)Hence, the state after a successful projection ( (cid:49) − | 〉 〈 | ) | 〉 〈 | , which has norm P ( A ) d , reads (cid:34) (cid:88) k + l > ψ kl (cid:112) k ! l ! ( z η d ) k /2 [ y (1 − z ) η f η d ] l /2 ( ˆ a † ) k + l (cid:35) | 〉 . (6.59)When Bob has a perfect delay line ( η f =
1) this state reads (cid:34) (cid:88) k + l > ψ kl (cid:112) k ! l ! ( z η d ) k /2 [ y (1 − z ) η d ] l /2 ( ˆ a † ) k + l (cid:35) | 〉 , (6.60)and its norm is the winning probability of Alice in that case. Hence, P ( A ) d [ η f , η d , y , z ] = P ( A ) d [1, η d , y η f , z ], (6.61)i.e., we can obtain Alice’s cheating probability by solving the case with perfect delay line, andreplacing the parameter y by y η f . In the following, we thus derive Alice’s optimal strategyin that case. ⌘ d
Let us write | ψ x 〉 = (cid:80) +∞ k , l ≥ ψ kl | kl 〉 . With the expression of the POVM in Eq. (6.62) the lastequation reads P ( A ) d = (cid:88) k , l ≥ | ψ kl | (1 − η d ) k [(1 − η d (1 − b )) l − (1 − η d ) l ] ≤ max k , l ≥ (1 − η d ) k [(1 − η d (1 − b )) l − (1 − η d ) l ] (cid:88) k , l ≥ | ψ kl | = max k , l ≥ (1 − η d ) k [(1 − η d (1 − b )) l − (1 − η d ) l ] (6.70) = max l ≥ [(1 − η d (1 − b )) l − (1 − η d ) l ] = max l ≥ [(1 − η d (1 − y )(1 − z )) l − (1 − η d ) l ],where we used b = y + z − yz . Let l ∈ (cid:78) ∗ such that max l ≥ [(1 − η d (1 − b )) l − (1 − η d ) l ] = (1 − η d (1 − b )) l − (1 − η d ) l . This last expression is an upperbound for P ( A ) d , which is attainedfor ψ kl = δ k ,0 δ l , l , i.e., | ψ x 〉 = | l 〉 . Thus, the best strategy for Alice is to send the state | ψ 〉 = ( (cid:49) ⊗ R ( π )) H ( a ) | ψ x 〉= ( (cid:49) ⊗ R ( π )) H ( a ) | l 〉 , (6.71)where a = y (1 − z ) y + z − yz , and her winning probability is then P ( A ) d = (1 − η d (1 − y )(1 − z )) l − (1 − η d ) l , (6.72)when Bob has a perfect delay line. Recalling Eq. (6.61), the best strategy for Alice when Bobhas a lossy delay line of efficiency η f is to send the state | ψ 〉 = ( (cid:49) ⊗ R ( π )) H ( a ) | ψ x 〉= ( (cid:49) ⊗ R ( π )) H ( a ) | l 〉 , (6.73)where a = y (1 − z ) η f y η f + z − yz η f , and l ∈ (cid:78) ∗ maximizes (1 − η d (1 − y η f )(1 − z )) l − (1 − η d ) l . Her winningprobability is then P ( A ) d = max l > (cid:104)(cid:161) − (1 − y η f )(1 − z ) η d (cid:162) l − (cid:161) − η d (cid:162) l (cid:105) = (1 − η d (1 − y η f )(1 − z )) l − (1 − η d ) l = η d [1 − (1 − y η f )(1 − z )] l − (cid:88) j = (1 − η d ) j (1 − η d (1 − y η f )(1 − z )) l − j − ≤ η d [1 − (1 − y η f )(1 − z )] l − (cid:88) j = (1 − η d ) j (6.74) = η d [1 − (1 − y η f )(1 − z )] 1 − (1 − η d ) l − (1 − η d )234 .2. EXPERIMENTAL IMPERFECTIONS = [1 − (1 − y η f )(1 − z )][1 − (1 − η d ) l ] ≤ − (1 − y η f )(1 − z ) ≤ − (1 − y )(1 − z ),and this last expression is her winning probability when there are no losses. (cid:4) Let us derive the value of l for which the maximum is achieved in Eq. (6.54). For this, we define: r = − η d (1 − y η f )(1 − z ) s = − η d . (6.75)We then consider a λ ∈ (cid:82) ∗+ which maximizes ( r λ − s λ ) for λ ∈ (cid:82) ∗+ . We have that: dd λ ( r λ − s λ ) = ⇔ λ = log log s − log log r log r − log s , (6.76)for strictly non-zero r and s . This allows us to deduce: l = (cid:40) (cid:98) λ (cid:99) if r (cid:98) λ (cid:99) − s (cid:98) λ (cid:99) ≥ r (cid:100) λ (cid:101) − s (cid:100) λ (cid:101) (cid:100) λ (cid:101) if r (cid:100) λ (cid:101) − s (cid:100) λ (cid:101) ≥ r (cid:98) λ (cid:99) − s (cid:98) λ (cid:99) . (6.77) We now analyze the performance of our protocol in a practical setting, by enforcing three con-ditions on the free parameters: the protocol must be fair, balanced, and perform strictly betterthan any classical protocol. The latter condition is not required in an ideal implementation,since quantum weak coin flipping always provides a security advantage over classical weakcoin flipping. Allowing for abort cases, however, may enable some classical protocols to performbetter than quantum ones. This is because increasing the abort probability effectively decreasesAlice and Bob’s cheating probabilities. We say that the protocol allows for quantum advantagewhen it provides a strictly lower cheating probability than any classical protocol with the sameabort probability. This is obtained using the bounds from [HW11], which yield the best classicalcheating probability P Cd = − (cid:112) P ab for our protocol. Condition (i): the first condition enforces a fair protocol, i.e., P ( A ) h = P ( B ) h . With Eq. (6.48), weaim to solve for y as a function of x and z :( i ) ⇔ η t η ( B ) d (cid:181)(cid:114) xz η ( A ) f + (cid:114) (1 − x ) y (1 − z ) η ( B ) f (cid:182) = η t η ( B ) d (1 − x )(1 − y ) ⇔ (1 − x ) (cid:104) (1 − z ) η ( B ) f + (cid:105) y + (cid:114) x (1 − x ) z (1 − z ) η ( A ) f η ( B ) f (cid:112) y + xz η ( A ) f − (1 − x ) =
0. (6.78)235
HAPTER 6. QUANTUM WEAK COIN FLIPPING WITH LINEAR OPTICS
We make the substitution Y = (cid:112) y in order to transform Eq. (6.78) into a second-order polynomialequation. We then take only the positive solution (since y must be positive) which reads: Y = (cid:114) xz (1 − z ) η ( A ) f η ( B ) f − (cid:104) (1 − z ) η ( B ) f + (cid:105) (cid:104) xz η ( A ) f − (1 − x ) (cid:105) − (cid:113) xz (1 − z ) η ( A ) f η ( B ) f (cid:112) − x (cid:104) (1 − z ) η ( B ) f + (cid:105) . (6.79)We may finally write: ( i ) ⇔ y = f (cid:179) x , z , η ( i ) f , η d , η t (cid:180) , (6.80)where f (cid:179) x , z , η ( i ) f , η d , η t (cid:180) = (cid:181)(cid:114) (1 − x ) (cid:104) (1 − z ) η ( B ) f + (cid:105) − xz η ( A ) f − (cid:113) xz (1 − z ) η ( A ) f η ( B ) f (cid:182) (1 − x ) (cid:104) (1 − z ) η ( B ) f + (cid:105) . (6.81)Note that y should be a real number, and hence we require that the expression under the firstsquare root of f (cid:179) x , z , η ( i ) f , η d , η t (cid:180) is positive, i.e., z ≤ (1 − x )(1 + η ( B ) f ) x η ( A ) f + (1 − x ) η ( B ) f . (6.82)Furthermore, note that, for η ( A ) f = η ( B ) f = η f , y should be an increasing function of η f , and thereforea decreasing function of d when η f = − d . Mathematically speaking, this is to prevent y (cid:48) ( d ) → ∞ and y ( d ) >
1. Physically speaking, this condition ensures that, as the probability oftransmitting the photon (and of preserving it for verification) gets smaller, Bob should encouragea detection on the third mode, which evens out the honest probabilities of winning.
Condition (ii): the second condition enforces a balanced protocol, i.e., P ( A ) d = P ( B ) d . With Eqs. (6.53)and (6.54), this translates into the following expression for x :( ii ) ⇔ x = g (cid:179) y , z , η ( i ) f , η ( i ) d (cid:180) , (6.83)where g (cid:179) y , z , η ( i ) f , η ( i ) d (cid:180) = η ( A ) f η ( A ) d (cid:183) − max l ≥ [(1 − η ( B ) d (1 − y η ( B ) f )(1 − z )) l − (1 − η ( B ) d ) l ] (cid:184) . (6.84) Condition (iii): we recall the general coin flipping formalism from [HW11], in which any classicalor quantum coin flipping protocol may be expressed as: CF ( p , p , p ∗ , p ∗ , p ∗ , p ∗ ) , (6.85)where p ii is the probability that two honest players output value i ∈ {
0, 1 } , p ∗ i is the probabilitythat Dishonest Alice forces Honest Bob to declare outcome i , and p i ∗ is the probability that236 .2. EXPERIMENTAL IMPERFECTIONS Dishonest Bob forces Honest Alice to declare outcome i . In this formalism, a perfect strong coinflipping protocol can then be expressed as CF (cid:161) , , , , , (cid:162) , while a perfect weak coin flippingmay be expressed as CF (cid:161) , , , 1, 1, (cid:162) . We may now express our quantum weak coin flippingprotocol in the lossless setting as: CF (cid:181)
12 , 12 , (cid:183) − x ) (cid:184) , 1, 1, [1 − x ] (cid:182) . (6.86)In the lossy setting, note that the probabilities that Alice and Bob each choose to lose (i.e., p ∗ and p ∗ , respectively), both remain 1. When Dishonest Bob chooses to lose, he may always declareoutcome 0 regardless of what he detects, which yields p ∗ =
1. When Dishonest Alice chooses tolose, she may send a state | n 〉 to Bob, and so: p ∗ = Tr (cid:104) H ( y ) | n 〉 〈 n | H ( y ) I ⊗ ( I − Π ) (cid:105) = − Tr (cid:104) H ( y ) | n 〉 〈 n | H ( y ) ( I ⊗ Π ) (cid:105) , (6.87)where Π = (cid:80) l ≥ (1 − η ) l | l 〉 〈 l | and H ( y ) = (cid:195) (cid:112) y (cid:112) − y (cid:112) − y −(cid:112) y (cid:33) .Now, H ( y ) | n 〉 = H ( y ) ( ˆ a † ) n (cid:112) n ! | 〉= (cid:112) n ! ( (cid:112) y ˆ a † + (cid:112) − y ˆ a † ) n | 〉= (cid:112) n ! n (cid:88) k = (cid:195) nk (cid:33) y k (1 − y ) n − k ˆ a † k ˆ a † ( n − k )2 | 〉= n (cid:88) k = (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) nk (cid:33) y k (1 − y ) n − k | k ( n − k ) 〉 . (6.88)We thus obtain, by linearity of the trace: p ∗ = − (cid:88) l , l (cid:48) ≥ (1 − η ) l n (cid:88) k , k (cid:48) = (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) nk (cid:33) y k (1 − y ) n − k (cid:118)(cid:117)(cid:117)(cid:116)(cid:195) nk (cid:48) (cid:33) y k (cid:48) (1 − y ) n − k (cid:48) Tr (cid:163) | k ( n − k ) 〉 〈 k (cid:48) ( n − k (cid:48) ) | | l (cid:48) l 〉 〈 l (cid:48) l | (cid:164) = − n (cid:88) k = (1 − η ) n − k (cid:195) nk (cid:33) y k (1 − y ) n − k = − (cid:163) y + (1 − η )(1 − y ) (cid:164) n , (6.89)which goes to 1 when n goes to infinity, for y <
1. Hence, in the lossy setting, the protocol becomesa: CF (cid:179) P ( A ) h , P ( B ) h , P ( A ) d , 1, 1, P ( B ) d (cid:180) , (6.90)where P ( A ) d = max l > (cid:179) − (1 − y η ( A ) f )(1 − z ) η ( B ) d (cid:180) l − (cid:179) − η ( B ) d (cid:180) l and P ( B ) d = − x η ( A ) f η ( A ) d .237 HAPTER 6. QUANTUM WEAK COIN FLIPPING WITH LINEAR OPTICS
Using Theorem 1 from [HW11], there exists a classical protocol that implements an information-theoretically secure coin flip with our parameters if and only if the following conditions hold: P ( A ) h ≤ P ( A ) d P ( B ) h ≤ P ( B ) d P ab = − P ( A ) h − P ( B ) h ≥ (1 − P ( A ) d )(1 − P ( B ) d ). (6.91)Our quantum protocol therefore presents an advantage over classical protocols if at least oneof these conditions cannot be satisfied. Since we are interested in fair and balanced protocols,setting P h = P ( A ) h = P ( B ) h , and P d = P ( A ) d = P ( B ) d (6.92)allows us to rewrite (6.91) as: P h ≤ P d P ab = − P h ≥ (1 − P d ) ⇔ P h ≤ [1 − (1 − P d ) ]. (6.93)Let us finally remark that for all x we have [1 − (1 − x ) ] = x − x ≤ x , so the first inequality aboveis implied by the second. The system is thus equivalent to the second inequality: P ab = − P h ≥ (1 − P d ) , (6.94)provided that P ( A ) h = P ( B ) h = P h and P ( A ) d = P ( B ) d = P d .In order to get a clearer insight into the meaning of quantum advantage, we express this conditionin terms of cheating probability: our protocol displays quantum advantage if and only if thelowest classical cheating probability P Cd = − (cid:112) − P h = − (cid:112) P ab (6.95)exceeds our quantum cheating probability P Qd .The three conditions may then be translated into the following system of equations, where wedefine P Qd = P ( A ) d = P ( B ) d : ( i ) P ( A ) h = P ( B ) h fairness( ii ) P ( A ) d = P ( B ) d balance( iii ) P Qd < P Cd quantum advantage (6.96)Fig. 6.12 shows a choice of parameters obtained numerically for which the system in Eq. (6.96) issatisfied, up to a distance of d km. 238 .3. DISCUSSION AND OPEN PROBLEMS Figure 6.12: Practical quantum advantage for a fair and balanced protocol: numerical values forthe lowest classical and quantum cheating probabilities, P Cd and P Qd , are plotted as a function ofdistance d in blue and red, respectively. Honest abort probability P ab (responsible for P Qd beinglower than our ideal quantum cheating probability 1/ (cid:112) P Qd < P Cd . We set η f = η s η t ,where η s is the fiber delay transmission corresponding to 500ns of optical switching time, and η t = (10 − d ) is the fiber delay transmission associated with travelling distance d twice (once forquantum, once for classical) in single-mode fibers with attenuation 0.2 dB/km. We have η d = z = By noticing a non-trivial connection between the early protocol from [SR02] and linear opticaltransformations, we answer the question of the implementability of quantum weak coin flipping,and show that it is achievable with current technology over a few hundred meters. As thedistance increases, the issue of stability of the interferometric setup should also be taken intoaccount. Both parties require a set of beam splitters and single photon threshold detectors. Stategeneration on Alice’s side can be performed with any heralded probabilistic single-photon source,for which photon indistinguishability and state purity do not matter. Only Alice requires anoptical switch, which is commercially available. Although short-term quantum storage is needed,a spool of optical fiber with twice the length of the quantum channel suffices, and provides therequired storage/retrieval efficiency.On the fundamental level, our results also raise the question of a potentially deeper connectionbetween the large family of protocols from [Moc04, Moc05, Moc07]—which achieves biases aslow as 1/6—and linear optics. Recalling that the protocol from [SR02], and hence our protocol, isconjectured optimal for this family, its extension to many rounds should be necessary in order tolower the bias. The optimality of the one-round protocol is crucial, as a recent result shows thatthe weak coin flipping bias decreases very inefficiently with the number of rounds [Mil20].239
ONCLUSION AND OUTLOOK G uided by three general questions about the use of quantum information in existingand upcoming technologies, this thesis has provided some answers in the context ofcontinuous variable quantum information theory and linear quantum optics.Firstly, what leads to a quantum advantage? We have considered the case of non-Gaussian states as a resource for outperforming classicalcomputing capabilities. Introducing the stellar formalism, we have characterised single-modenon-Gaussian states by the number of elementary non-Gaussian operations needed to engineerthem [CMG20]. Apart from providing insights about the structure of these states, we have seendirect consequences of the use of our formalism for Gaussian convertibility of states, comparingphoton addition and photon subtraction, and cat state engineering [CRW + + how do we check the correct functioning of a quantum device? We have developped a variety of certification and verification protocols for continuous variablequantum states with single-mode Gaussian measurements. As a first step, we showed how toperform efficient reliable tomography and fidelity estimation for any single-mode continuousvariable quantum state, under the assumption of identical copies, and with no assumption what-soever [CDG + + ONCLUSION AND OUTLOOK
Thirdly, what useful advantages can we obtain from the use of quantum information?
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