Bosonic data hiding: power of linear vs non-linear optics
BBosonic data hiding: power of linear vs non-linear optics
Krishna Kumar Sabapathy
1, 2, ∗ and Andreas Winter
2, 3, † Xanadu, 777 Bay Street, Toronto ON, M5G 2C8, Canada Departament de F´ısica: Grup d’Informaci´o Qu`antica,Universitat Aut`onoma de Barcelona, 08193 Bellaterra (Barcelona), Spain ICREA—Instituci´o Catalana de Recerca i Estudis Avan¸cats,Pg. Lluis Companys, 23, 08010 Barcelona, Spain (Dated: 2 February 2021)We show that the positivity of the Wigner function of Gaussian states and measurements providesan elegant way to bound the discriminating power of “linear optics”, which we formalise as Gaussianmeasurement operations augmented by classical (feed-forward) communication (GOCC). This allowsus to reproduce and generalise the result of Takeoka and Sasaki [PRA 78:022320, 2008], whichtightly characterises the GOCC norm distance of coherent states, separating it from the optimaldistinguishability according to Helstrom’s theorem.Furthermore, invoking ideas from classical and quantum Shannon theory we show that thereare states, each a probabilistic mixture of multi-mode coherent states, which are exponentiallyreliably discriminated in principle, but appear exponentially close judging from the output of GOCCmeasurements. In analogy to LOCC data hiding, which shows an irreversibility in the preparationand discrimination of states by the restricted class of local operations and classical communication(LOCC), we call the present effect
GOCC data hiding .We also present general bounds in the opposite direction, guaranteeing a minimum of distin-guishability under measurements with positive Wigner function, for any bounded-energy states thatare Helstrom distinguishable. We conjecture that a similar bound holds for GOCC measurements.
I. INTRODUCTION
One of the most basic problems of quantum informa-tion theory is the discrimination of two alternatives (“hy-potheses”), each of which represents the possible state ofa system, ρ or ρ . Under the formalism of quantum me-chanics, this calls for the design of a measurement anda decision rule to choose between the two options basedon the measurement outcome. The measurement is abinary resolution of unity, also called a positive opera-tor valued measure (POVM), ( M = M, M = − M )of two semidefinite operators M M ≥ M + M =
1. The outcome M ˆ i of the measurement isintended to correspond to the estimate ˆ i of the true state ρ i . For simplicity, we will assume that the two hypothe-ses come with equal (uniform) prior probabilities, so theerror probability is P e = 12 Tr ρ ( − M ) + 12 Tr ρ M = 12 (cid:0) − Tr( ρ − ρ ) M (cid:1) . (1)The minimum error over all quantum mechanically al-lowed POVMs gives rise to the trace norm,min ≤ M ≤ P e = 12 (cid:18) − (cid:107) ρ − ρ (cid:107) (cid:19) , (2) ∗ [email protected] † [email protected] which formula is nowadays known as Helstrom bound[19, 20] or Holevo-Helstrom bound [22], since it was ini-tially only proved for projective measurements and sub-sequently for generalised measurements.However, from the beginning of quantum detectiontheory, it was understood that – depending on the phys-ical system – the Helstrom optimal measurement maynot be easily implemented. Indeed, the very example ofdiscrimination of to coherent states of an optical modewas already considered by Helstrom [18], who contrastedthe absolutely minimum error probability with the per-formance of reasonable practical measurements. Mathe-matically, this means that the minimisation on the l.h.s.of Eq. (2) is performed over a smaller set of POVMs, therestriction an expression of what is deemed physically fea-sible. Consequently, the error probability becomes larger,in some interesting cases close to even for orthogo-nal, i.e. ideally perfectly distinguishable, states. Thisphenomenon was first observed in bipartite systems un-der the restriction of local operations and classical com-munication (LOCC), and dubbed data hiding [10, 37],which has been generalised to multi-party settings [11],and analysed extensively [29, 31, 42].In the present paper we will look at a different kind ofrestriction, in Bosonic quantum systems, motivated bythe distinction between phase-space linear (aka Gaus-sian ) and non-linear (i.e. non-Gaussian) operations, seealso [32]. It is well-known that a process that starts froma Gaussian state and proceeds only via Gaussian oper-ations, including Gaussian measurements and classicalfeed-forward (GOCC, see below), is in a certain sensevery far away from the full complexity of quantum me-chanics: indeed, such a process can be simulated effi- a r X i v : . [ qu a n t - ph ] F e b ciently on a classical computer [5, 30] and hence, unlessBQP=BPP, is not quantum computationally universal.In other words, non-Gaussianity is a resource for compu-tation, which it becomes quite explicitly in proposals ofoptical quantum computing such as the Knill-Laflamme-Milburn scheme [25] that relies on photon detection andotherwise passive linear optics.Here we show that non-Gaussianity is a resource forthe basic task of binary hypothesis testing. In particularwe show how to leverage simple properties of the Wignerfunction to prove not only a limitation of the power ofGaussian operations, but construct data hiding with re-spect to GOCC.To conclude the introduction, a word on terminology:we refer to Gaussian states and channels as “linear”, be-cause the latter are described by linear transformationsin the phase space of the canonical variables x and p .Conversely, “non-linear” is anything outside the Gaus-sian set. Note however that in parts of quantum optics anarrower concept is used, whereby only channels are con-sidered linear that are built with passive Gaussian uni-taries, and perhaps admitting displacement operators.The rest of paper is structured as follows: In the nextsection (II) we recall the necessary formalism and no-tation of quantum harmonic oscillators and GaussianBosonic states and operations; for our purposes in par-ticular useful will be the phase space methods based onWigner functions. Then, in Section III we specialise thegeneral framework of restricted measurements to Gaus-sian quantum operations and arbitrary classical compu-tations (GOCC), and the important relaxation of thisclass to measurements with non-negative Wigner func-tions (W+). We use these in Section IV to analyze theoptimal GOCC measurement to distinguish two coherentstates, reproducing (with a conceptually much simplerproof) a result of Takeoka and Sasaki [36]. The GOCCdistinguishability of any two distinct coherent states is al-ways a little, but always strictly worse than the optimaldistingishability according to Helstrom [19, 22]. Motivateby this, in Section V we exhibit examples of multimodestates, each a mixture of coherent states (hence “classi-cal” in the quantum-optical sense [14, 34] and in particu-lar preparable by GOCC), whose GOCC distinguishabil-ity is exponentially small while they are almost perfectlydistinguishable under the optimal Holevo-Helstrom mea-surement. From the other side, there are lower limits tohow indistinguishable two orthogonal states on n quan-tum harmonic modes and with bounded energy can be,which we show for W+ measurements and conjecture forGOCC measurements (Section VI). We conclude in Sec-tion VII. II. BOSONIC GAUSSIAN FORMALISM
We briefly review the formalism of Bosonic systemsand Gaussian states, which has been laid out in manyreview articles and textbooks, such as [38] and [4, 26], which two emphasise the quantum information aspect.For our particular choice of normalisations, see [7].Each elementary system, called a (harmonic) mode,is characterised by a pair of canonical variables x and p , satisfying the canonical commutation relation (CCR)[ x, p ] = i (customarily choosing units where (cid:126) = 1) andgenerating the CCR algebra of Heisenberg and Weyl. Bythe Stone-von-Neumann theorem, each irreducible rep-resentation of this algebra on a separable Hilbert space H is isomorphic to the usual position and momentumoperators x and p , respectively.It is convenient to introduce the annihilation and cre-ation operators a = x + ip √ , a † = x − ip √ , (3)respectively. They can be used to define the numberoperator, N = a † a = 12 ( x + p ) −
12 = ∞ (cid:88) n =0 n | n (cid:105)(cid:104) n | , (4)which up to the energy shift of − to bring the groundstate energy to zero, is equivalent to the quantum har-monic Hamiltonian (at fixed frequency), and has pre-cisely the non-negative integers as eigenvalues; the eigen-states are known as Fock states or number states, | n (cid:105) . Inthe number basis, a = ∞ (cid:88) n =0 √ n | n − (cid:105)(cid:104) n | , a † = ∞ (cid:88) n =0 √ n | n (cid:105)(cid:104) n − | . (5)All these operators are unbounded, and one might havejustified hesitations against the algebraic operations per-formed above. The established solution to all of the po-tential problems associated to the unboundedness andassociated restricted domains is to pass to the displace-ment operators, D ( α ) = e αa † − αa , for α ∈ C , (6)which are bona fide unitaries, hence bounded operators.So far, we have discussed our quantum system at handas if it were a single mode, but we can of course con-sider multi-mode systems, which again by the Stone-von-Neumann theorem are characterised uniquely as ir-reducible representations of the CCR algebra generatedby x , . . . , x m and p , . . . , p m such that [ x j , p k ] = iδ jk .This means that its Hilbert space can be identified with H ⊗ · · · ⊗ H m , where H j is the Hilbert space of the j -th mode, carrying the representation of x j and p j . Inparticular, each mode has its own annihilation opera-tor a j and displacement operator D ( α ); for an m -tuple α = ( α , . . . , α m ) of displacements, we write D ( α ) = D ( α ) ⊗ · · · ⊗ D ( α m ) for the m -mode displacement oper-ator. The subspace spanned by these operators is densein the bounded operators B ( H ) on the Hilbert space.For a general density operator ρ ∈ S ( H ) = { ρ ≥ , Tr ρ = 1 } , or more generally for a trace class oper-ator, the characteristic function is defined as χ ρ ( α ) := Tr ρD ( α ) . (7)This is a bounded complex function, uniquely specifying ρ . A state ρ is called Gaussian if its characteristic func-tion χ ρ is of Gaussian form. For our purposes, we needanother, particularly useful representation of the state asa quasi-probability function, the so-called Wigner func-tion W ρ [21, 39], see also [4, 7] for many fundamental anduseful relations such as the following two. It is definedas the (multidimensional complex) Fourier transform ofthe characteristic function χ ρ , W ρ ( x, p ) = (cid:18) π (cid:19) m (cid:90) d m ξ e α · ξ † − ξ · α † χ ρ ( ξ ) , (8)where we reparametrise the argument in phase space co-ordinates, α j = √ ( x j + ip j ), and α · ξ † = (cid:80) j α j ξ j is theHermitian inner product of the complex coordinate tu-ples. This is a real-valued function, and its normalisationis chosen in such a way that (cid:90) d m x d m p W ρ ( x, p ) = Tr ρ, (9)hence for a state we can address it as a quasi-probabilityfunction as it integrates to 1, and if the Wigner functionis positive it is a genuine probability density. In general,can be expressed as an expectation value, cf. [4, 7], W ρ ( x, p ) = π − m Tr ρD ( α )( − N + ... + N m D ( α ) † . (10)where as before α j = √ ( x j + ip j ). It shows that W ρ is well-defined and indeed a continuous bounded func-tion for all trace class operators: indeed, | W ρ ( x, p ) | ≤ π − m (cid:107) ρ (cid:107) . The above formula can be used to give mean-ing to more general operators (such as POVM elements);for instance the Wigner function of the identity operatoris a constant, W π ) − m . The Wigner transformationpreserves the Frobenius (Hilbert-Schmidt) inner product,Tr ρσ = (2 π ) m (cid:90) d m x d m p W ρ ( x, p ) W σ ( x, p ) . (11)Unitary transformations of the Hilbert space preservethe canonical commutation relations; but the subset ofunitaries that map the Lie algebra of the canonical vari-ables, which is span { , x j , p k } , to itself, are called Gaus-sian unitaries. We address them also as “linear” trans-formations, since they are correspond to an affine linearmap of phase space, and are described by a displace-ment vector and a symplectic matrix. Gaussian chan-nels are precisely the completely positive and trace pre-serving (cptp) maps taking Gaussian states to Gaussianstates. It is a fundamental fact that a quantum channel N : S ( H ) → S ( H (cid:48) ) is Gaussian if and only if it has a Gaussian unitary dilation U , with the environment ini-tialised in the vacuum state: N ( ρ ) = Tr E U (cid:0) ρ ⊗ | (cid:105)(cid:104) | ⊗ (cid:96) (cid:1) U † , (12)where the environment has (cid:96) modes.For the following, we need the (Glauber-Sudarshan)coherent states, also known as minimal dispersion states | α (cid:105) , which are eigenstates of the annihilation operator: a | α (cid:105) = α | α (cid:105) , for α ∈ C . This defines the states uniquely,and one can show that they are related by displacements: | α (cid:105) = D ( α ) | (cid:105) , where | (cid:105) is both the coherent state cor-responding to α = 0 and the vacuum, i.e. the groundstate of the Hamiltonian, in other words the zeroth Fockstate. In the Fock basis, | α (cid:105) = e − | α | ∞ (cid:88) n =0 α n √ n ! | n (cid:105) , (13)a relation that reassuringly shows that the coherentstates are well-defined unit vectors, written in a genuineorthonormal basis. However, what is more relevant arethe following expressions for the first and second mo-ments. For α = α R + iα I written in terms of real andimaginary parts, (cid:104) α | x | α (cid:105) = α R √ , (cid:104) α | p | α (cid:105) = α I √ , (14) (cid:104) | x | (cid:105) = (cid:104) | p | (cid:105) = 12 , (15)the latter “vacuum fluctuations” consistent with theHeisenberg-Robertson uncertainty relation. Further-more, the inner product, easily confirmed from the ex-pansion in the Fock basis, |(cid:104) α | β (cid:105)| = e −| α − β | . (16)And finally, we record1 π (cid:90) d α | α (cid:105)(cid:104) α | = , (17)showing that the family of operators d απ | α (cid:105)(cid:104) α | forms aPOVM, known as heterodyne measurement. III. STATE DISCRIMINATION BYGAUSSIAN MEASUREMENTS
Now that we have the Bosonic formalism in place, wecan discuss the problem of binary hypothesis testing un-der Gaussian restrictions on the measurement. Indeed,going back to Eqs. (1) and (2) in the introduction, almostany restriction M on the set of possible measurements,be they physically motivated or purely mathematical, re-sults in a larger error probability than the Helstrom ex-pression, which is most conveniently expressed in termsof a distinguishability norm on states:min ( M, − M ) ∈ M P e =: 12 (cid:18) − (cid:107) ρ − ρ (cid:107) M (cid:19) . (18)How to define the set M appropriately and what exactlyis necessary for it to define a norm is explained in detail in[31]. An example exceedingly well-studied in quantum in-formation theory is the set of measurements implementedby local operations and classical communication (LOCC)in a bi- or multi-partite system, as well as its relaxationsseparable POVM elements (SEP) and POVM elementswith positive partial transpose (PPT), cf. [9, 31]Here, we consider restrictions motivated from the factthat Gaussian operations are distinguished among theones allowed by quantum mechanics generally, followingTakeoka and Sasaki [36]. Concretely, we are interested inthe measurements implemented by any sequence of par-tial Gaussian POVMs and classical feed-forward (Gaus-sian operations and classical computation, GOCC). Verymuch like LOCC, there is no concise way of writing downa general GOCC transformation, but for a binary mea-surement the prescription is as follows. Definition 1 A GOCC measurement protocol on m modes consists of the repetition of the following steps,for r = 1 , . . . , R (“rounds”), after initially setting ξ = ∅ and m ∅ = m . Here, ξ r − is the collection of all measure-ment outcomes prior to round r .(r.1) create a number k ξ r − of Bosonic modes in the vac-uum state;(r.2) perform a Gaussian unitary U ξ r − on the m ξ r − + k ξ r − modes;(r.3) perform homodyne detection on the last (cid:96) ξ r − modes, keeping the first m ξ r := m ξ r − + k ξ r − − (cid:96) ξ r − ; call the outcome x ( r ) = x ( r )1 . . . x ( r ) (cid:96) ξr − andset ξ r := { ξ r − , x ( r ) } .Each r is called a “round”, and in the R -th round allremaining modes are measured, i.e. (cid:96) ξ r − = m ξ r − + k ξ r − . The final measurement outcome is a measurablefunction f ( ξ R ) ∈ Ω , taking values in a prescribed set Ω ,which for simplicity we assume to be discrete.This defines a POVM ( M ω : ω ∈ Ω) , and every POVMthat arises in the above way, or as a limit of such POVMsin the strong topology is called a GOCC POVM . Now, returning to equiprobable hypotheses ρ and ρ ,and enforcing the POVMs to be implemented by GOCCprotocols, we arrive at the GOCC norm:inf ( M, − M )GOCC POVM P e =: 12 (cid:18) − (cid:107) ρ − ρ (cid:107) GOCC (cid:19) . (19)Note that (cid:107) · (cid:107) GOCC is indeed a norm, since the set ofGOCC measurements is tomographically complete. In-deed, heterodyne detection on every available mode isa tomographically complete measurement, meaning thatfor every pair of distinct quantum states, there exists abinary coarse graining of the heterodyne detection out-comes that discriminates the states with some non-zerobias.
Remark 2
In the definition of a GOCC protocol, wecould have allowed the k ξ r − ancillary modes to be pre-pared in any Gaussian state in step (r.1), but that doesnot add any more generality, since every Gaussian statecan be prepared from the vacuum by a suitable Gaussianunitary. We could also have allowed an arbitrary Gaus-sian quantum channel in step (r.2), but again that doesnot add any more generality since every Gaussian chan-nel has a dilation to a Gaussian unitary with an environ-ment prepared in the vacuum state. Finally, in step (r.3)we could have allowed any Gaussian measurement, butevery Gaussian measurement can be implemented by ad-joining suitable ancilla modes in the vacuum, performinga Gaussian unitary and a homodyne measurement.From the point of view of the discussion of classes ofoperations, of which measurements are a special case, itis interesting to distinguish certain subclasses of GOCC:what we actually have defined are the measurements im-plemented by a GOCC protocol with finitely many rounds,as well as the closure of this set. One could also definethe POVMs implemented by a GOCC protocol with un-bounded rounds (but probability 1 to stop), which wouldsit between the former two, cf. [9] for the case of LOCC.While it is interesting to study these three classes, in par-ticular whether they coincide or are separated (as theyare in the analogous case of LOCC [9]) this is beyond thescope of the present work. Indeed, for the case of hypothe-sis testing, thanks to the infimum in the error probability,all three classes will give rise to the same GOCC norm. An elementary observation about GOCC is that thefine-grained measurement (i.e. before coarse-graining toa discrete POVM) consists of operators each of which isa positive scalar multiple of a Gaussian pure state. Inparticular, they have non-negative Wigner function, andbecause the coarse-graining amounts to summing POVMelements, also the final POVM has non-negative Wignerfunctions. We thus call a binary POVM ( M, − M )with non-negative Wigner functions W M and W − M a W+ POVM , and denote their set W + .Just as the restriction to GOCC leads to the distin-guishability norm (cid:107) · (cid:107) GOCC [Eq. (19)], the restrictionto W+ POVMs gives rise to the distinguishability norm (cid:107) · (cid:107) W+ : inf ( M, − M )W+ POVM P e =: 12 (cid:18) − (cid:107) ρ − ρ (cid:107) W+ (cid:19) . (20)Since every GOCC measurement is automatically W+,we have by definition (cid:107) ρ − ρ (cid:107) GOCC ≤ (cid:107) ρ − ρ (cid:107) W+ ≤ (cid:107) ρ − ρ (cid:107) . (21)The rest of the paper is concerned with the comparisonof these norms. The questions guiding us are: are theydifferent, and how large are the gaps? IV. SEPARATION BETWEEN GOCC ANDUNRESTRICTED MEASUREMENTS
Our first result shows a simple upper bound on theGOCC and W+ distinguishability norms in terms of theWigner functions of the two states.
Lemma 3
For any two states ρ and ρ of an m -modesystem, with associated Wigner functions W and W ,respectively, (cid:107) ρ − ρ (cid:107) GOCC ≤ (cid:107) ρ − ρ (cid:107) W+ ≤ (cid:107) W − W (cid:107) L = (cid:90) d m x d m p | W ( x, p ) − W ( x, p ) | . Note that, unlike the inequalities (21), the third term inthe chain is not a trace norm of density matrices, butan L norm of real functions, which we may interpret asgeneralised densities. Proof.
Only the second inequality remains to be proved.Consider any W+ POVM ( M, − M ), meaning that thestochastic response functions F = (2 π ) m W M and 1 − F = (2 π ) m W − M are bounded between 0 and 1. By theFrobenius inner product formula for the Wigner function,Eq. (11), we haveTr( ρ − ρ ) M = (cid:90) d m x d m p (cid:0) W ( x, p ) − W ( x, p ) (cid:1) F ( x, p ) , (22)where the left hand side appears in Eq. (1), its supre-mum over W+ POVMs being (cid:107) ρ − ρ (cid:107) W+ ; while theright hand side is upper bounded by (cid:107) W − W (cid:107) L ,where we made use of the fact that (cid:82) d m x d m p ( W ( x, p ) − W ( x, p )) = Tr( ρ − ρ ) = 0. (cid:50) Remark 4
The lemma assumes measurements with W+POVMs, but it gives interesting information also in caseswhere the POVM has some limited Wigner negativity.Namely, looking at Eq. (22), we subsequently use that ≤ F ( x, p ) ≤ , which is the property W‘+ of the mea-surement.If we do not have “too much” Wigner negativity inthe measurement operators, this could be expressed by abound | F ( x, p ) − | ≤ B , and then we would get (cid:12)(cid:12) Tr( ρ − ρ ) M (cid:12)(cid:12) ≤ B (cid:107) W − W (cid:107) L . (23) The right hand side can still be small when the L -distance is really small, and at the same time B not toolarge. In the next section we shall see an example of this. As one might expect, the inequality in Lemma 3 is of-ten crude, or even trivial since one can find states wherethe right had side exceeds 2. However, if ρ and ρ areboth states with non-negative Wigner function, for in-stance probabilistic mixtures of Gaussian states, then W and W are bona fide probability densities, and the righthand side is ≤
2. In that case, we have the followingcorollary for the quantum Chernoff coefficient when mea-surements are restricted to GOCC or W+ POVMs. Re-call that the Chernoff coefficient is the exponential rateof the minimum error probability in distinguishing twoi.i.d. hypotheses. I.e., in the case of two quantum states ξ ( ρ , ρ ) := lim n →∞ − n ln (cid:18) − (cid:13)(cid:13) ρ ⊗ n − ρ ⊗ n (cid:13)(cid:13) (cid:19) , (24)which generalises the analogous question for probabilitydistributions [8]. Amazingly, there is a formula for thisexponent [3], generalising in its turn the classical answer[8]: ξ ( ρ , ρ ) = − ln inf
For two states ρ and ρ with non-negativeWigner functions W , W ≥ (meaning that they areprobability density functions), ξ GOCC ( ρ , ρ ) ≤ ξ W+ ( ρ , ρ ) ≤ ξ ( W , W ) , where according to Chernoff ’s theorem [8], ξ ( W , W ) = − ln inf
Example 6
Consider ρ and ρ as two coherent statesof a single mode, say ρ = | + α (cid:105)(cid:104) + α | , ρ = | − α (cid:105)(cid:104)− α | for α > . Then, (cid:107) ρ − ρ (cid:107) = (cid:112) − e − α . (27) while by Lemma 3, (cid:107) ρ − ρ (cid:107) GOCC = 12 (cid:107) ρ − ρ (cid:107) W+ = erf( α √ , (28) with the error function erf( x ) = √ π (cid:82) x d x e − x . Theequality follows from homodyning the x -coordinate anddeciding depending on the sign of the measurement out-come. The norms are compared in Fig. 1.Furthermore, in the asymptotic i.i.d. setting of theChernoff bound, ξ ( ρ , ρ ) = − ln F ( ρ , ρ ) = 4 | α | , (29) while by Corollary 5, ξ GOCC ( ρ , ρ ) = ξ W+ ( ρ , ρ ) = 2 | α | . (30) The equality follows from homodyning each mode sepa-rately in the x direction, and classical post-processing. FIG. 1. Plot of the trace distance (red) versus the GOCC dis-tance (green) against α on the horizontal axis. While thereis a nonzero gap for all α >
0, it vanishes for asymptoticallylarge and small displacements, as expected. The largest dif-ference between (cid:107) ρ − ρ (cid:107) and (cid:107) ρ − ρ (cid:107) GOCC is ≈ . α ≈ . Example 7
More generally, for any one-mode Gaussianstate and its displacement along one of the principal axesof the covariance matrix, (cid:107) ρ − ρ (cid:107) GOCC = (cid:107) ρ − ρ (cid:107) W+ = (cid:107) W − W (cid:107) L , (31) and the latter can be expressed in terms of the error func-tion and the shared variance of the two states in the di-rection of the displacement connecting them.The equality follows from homodyning in the directionof the line connecting the two first moment vectors inphase space, and deciding depending on which of the twopoints is closer to the outcome. V. DATA HIDING SECURE AGAINSTGAUSSIAN ATTACKER
As soon as we realize that it is possible to get largegaps between (cid:107) · (cid:107) and (cid:107) · (cid:107) GOCC , we have to ask our-selves, just how large the gap can be. In particular, is itpossible to find state pairs which are almost maximallydistant in the trace norm, yet almost indistinguishablein the GOCC norm? In other words, can we protect theinformation against an adversary who attempts the hy-pothesis testing on the two states but with only access toGaussian operations and classical communication? Thisis the definition of data hiding, first explored in the con-text of the LOCC restriction, and then later abstractlyfor an arbitrary restriction on the possible measurements.Next we shall show that data hiding is possible alsounder GOCC, at least when going to multiple modes.
Theorem 8
Let
E > . Then, there is a constant c > such that for all sufficiently large integers m there exist m -mode states ρ and ρ , each a mixture of a finite set ofcoherent states and with average energy (photon number)per mode bounded by E + o (1) , such that (cid:107) ρ − ρ (cid:107) ≥ − e − cm , (32) e − cm ≥ (cid:107) ρ − ρ (cid:107) W+ ≥ (cid:107) ρ − ρ (cid:107) GOCC . (33) Proof .
Consider the m -mode coherent states | α ( λ ) (cid:105) = | α ( λ )1 (cid:105)| α ( λ )2 (cid:105) · · · | α ( λ ) m (cid:105) ( λ = 1 , . . . , L ), where the parame-ters α ( λ ) j ∈ C are chosen i.i.d according to a normal dis-tribution with mean 0 and variance E | α ( λ ) j | = E . Thendefine ρ = 1 L L (cid:88) λ =1 | α (2 λ ) (cid:105)(cid:104) α (2 λ ) | ,ρ = 1 L L (cid:88) λ =1 | α (2 λ − (cid:105)(cid:104) α (2 λ − | , (34)so these are random states. Note that with high proba-bility, indeed asymptotically converging to 1 as m (cid:29) E + o (1). Also, E ρ = E ρ = γ ( E ) ⊗ m , (35)where γ ( E ) = (1 − e − β ) e − βN is the thermal state of asingle Bosonic mode of mean photon number E , i.e. with β = ln (cid:16) E (cid:17) .The rest of the proof will consist in showing that wecan fix L in such a way that with probability close to 1, ρ and ρ are distinguishable except with exponentiallysmall error probability, and that with probability closeto 1, the Wigner functions W and W are exponentiallyclose to W ⊗ mγ ( E ) , the Wigner function of γ ( E ) ⊗ m , in thetotal variational distance. Eq. (32):
The ensemble of coherent states | α ( λ ) (cid:105) isthe well-studied random coherent state modulation of thenoiseless Bosonic channel with input power (photon num-ber) E , whose classical capacity is well-known [13], withthe strong converse proved in [40]. C (id , E ) = g ( E )= ( E + 1) ln( E + 1) − E ln E = ln(1 + E ) + E ln (cid:18) E (cid:19) . (36)Thus, when 2 L ≤ e m (cid:0) C (id ,E ) − δ (cid:1) , it follows fromthe Holevo-Schumacher-Westmoreland theorem [23, 24,33] that with probability close to 1, there exists aPOVM ( D λ ) Lλ =1 that decodes λ reliably from the state | α ( λ ) (cid:105)(cid:104) α ( λ ) | :12 L L (cid:88) λ =1 Tr | α ( λ ) (cid:105)(cid:104) α ( λ ) | D λ ≥ − e − c (cid:48) m , (37)with a suitable constant c (cid:48) > m . Thus, with ρ i ( i = 0 ,
1) as defined above and M i = L (cid:88) λ =1 D λ − i ( i = 0 , , (38)it follows 12 Tr ρ M + 12 Tr ρ M ≥ − e − c (cid:48) m , (39)which implies Eq. (32). Eq. (33):
The Wigner functions W α ( λ ) of the coher-ent states | α ( λ ) (cid:105)(cid:104) α ( λ ) | are 2 m -dimensional real Gaussianprobability densities centered at z ( λ ) , where z ( λ )2 j − = (cid:60) α ( λ ) j √ z ( λ )2 j = (cid:61) α ( λ ) j √ α ( λ ) j , respectively; they have variance in each direction. We read them as output distribu-tions of an i.i.d. additive white Gaussian noise (AWGN)channel on 2 m inputs z ( λ ) , and with noise power . Notethat all z ( λ ) j are themselves Gaussian distributed randomvariables with E z ( λ ) j = 0 and E | z ( λ ) j | = E . This channel,which we denote (cid:102) W since its output distributions comefrom the Wigner functions of the coherent states α ( λ ) ,thanks to Shannon’s famous formula with the signal-to-noise ratio has the capacity C ( (cid:102) W , E ) = 12 ln(1 + 2 E ) . (40)Thus, by the theory of approximation of output statistics[15], adapted to the AWGN channel [16], it follows that when 2 L ≥ e m (cid:0) C ( (cid:102) W ,E )+ δ (cid:1) , then with probability closeto 1 (cid:13)(cid:13)(cid:13) W i − W ⊗ mγ ( E ) (cid:13)(cid:13)(cid:13) L ≤ e − c (cid:48)(cid:48) m , (41)for i = 0 ,
1, with a suitable constant c (cid:48)(cid:48) > m . See [17, Thm. 6.7.3] for the concretestatement. Hence, by the triangle inequality and Lemma3, we get Eq. (33).It remains to put the two parts together: We observethat 2 C ( (cid:102) W , E ) < C (id , E ) for all E >
0. Indeed, a well-known elementary inequality statesln(1 + t ) ≥ t t , (42)which we apply to t = E , yielding E ln (cid:18) E (cid:19) ≥ E E E = E E > ln (cid:18) E E (cid:19) , (43)which is equivalent to the claim. This means that we canchoose δ > C ( W, E ) + 2 δ < C (id , E ) − δ, meaning we can satisfy e m (cid:0) C ( (cid:102) W ,E )+ δ (cid:1) ≤ L ≤ e m (cid:0) C ( (cid:102) W ,E )+ δ (cid:1) (44)simultaneously for all sufficiently large m . Finally, set-ting c = min { c (cid:48) , c (cid:48)(cid:48) } concludes the proof. (cid:50) Remark 9
While we didn’t make any attempt to give anumerical value for c (which is a function of E ), in prin-ciple it can be extracted from the HSW coding theorem forthe noiseless Bosonic channel and the resolvability codingtheorem for the AWGN channel.Likewise, we presented the theorem as an asymptoticresult, but the proofs of the two coding theorems will yieldfinite values of m for which the constructions work withprobability > , and so we get the existence of the datahiding states for that number of modes. Corollary 10
For the two m -mode states ρ and ρ fromTheorem 8, ξ ( ρ , ρ ) ≥ c m − ln √ , whereas ξ GOCC ( ρ , ρ ) ≤ ξ W+ ( ρ , ρ ) ≤ ξ ( W , W ) ≤ − (cid:0) − e − cm (cid:1) ∼ e − cm . Proof .
With (cid:15) = e − cm as in Theorem 8, we use theFuchs-van de Graaf relation between trace distance andfidelity [12]:1 − F ( ρ , ρ ) ≤ (cid:107) ρ − ρ (cid:107) ≤ (cid:112) − F ( ρ , ρ ) , (45)where the mixed-state fidelity is given F ( ρ , ρ ) := (cid:107)√ ρ √ ρ (cid:107) . (46)Now, we get first 1 − F ( ρ , ρ ) ≥ (1 − (cid:15) ) . And thenwe can estimate: e − ξ ( ρ ,ρ ) = inf
So far, we have seen examples of separations, includinglarge ones, between the trace norm and GOCC and W+norms. Especially about the construction in the previoussection we can ask, whether and in which sense it uses theavailable resources optimally: these would be the numberof modes and the energy. Here we show lower bounds onthe distinguishability of general states when restricted toW+, compared to the trace norm. They are motivated bysimilar studies under the LOCC, SEP or PPT constraint,or an abstract constraint on the allowed measurements[29, 31], see also [27].
Proposition 11
For any two m -mode states ρ and ρ , (cid:107) ρ − ρ (cid:107) W+ ≥ − m − (cid:107) ρ − ρ (cid:107) = 2 − m − Tr( ρ − ρ ) . Proof.
We will write down a specific W+ POVM thatachieves the r.h.s. as its statistical distance. In fact, with∆ = ρ − ρ , for our POVM ( M, − M ) we make theansatz M = 12 ( η ∆) , − M = 12 ( − η ∆) , (49) with a suitable constant η > η ≤ W π ) − m .Recall furthermore that the Wigner functions of statesare bounded, | W ρ ( x, p ) | ≤ π − m , see Eq. (10).This means that | W ∆ ( x, p ) | ≤ π − m , and so W M/ − M = W ± ηW ∆ ≥ η = 2 − m − .Thus we have 2 M − η ∆, and can calculate (cid:107) ρ − ρ (cid:107) W+ ≥ Tr ∆(2 M −
1) = 2 − m − (cid:107) ρ − ρ (cid:107) , (50)concluding the proof. (cid:50) Corollary 12
Consider two m -mode states ρ and ρ ,with average energy (photon number) per mode boundedby E and (cid:107) ρ − ρ (cid:107) ≥ t > . Then, with t = 4 c + r , (cid:107) ρ − ρ (cid:107) W+ ≥ r − m − (cid:18) Ec (cid:19) − m . Thus, to achieve the kind of separation as in Theo-rem 8, between a “large” trace norm and “small” W+norm, with bounded energy per mode, their number nec-essarily has to grow; or else, the energy per mode has togrow very strongly.
Proof.
Construct the projector P onto the space of all m -mode number states with photon number ≤ m Ec . Denote D = rank P . By the assumption of the energy bound,and Markov’s inequality,Tr ρ P, Tr ρ P ≥ − c . (51)Hence, by the gentle measurement lemma [41], (cid:107) ρ − P ρ P (cid:107) , (cid:107) ρ − P ρ P (cid:107) ≤ c, (52)and so by the triangle inequality (cid:107) P ρ P − P ρ P (cid:107) ≥ t − c = r .Now, (cid:107) ρ − ρ (cid:107) ≥ (cid:107) P ρ P − P ρ P (cid:107) ≥ √ D (cid:107) P ρ P − P ρ P (cid:107) ≥ r √ D , (53)where the first inequality follows from the fact that theFrobenius norm squared is the sum of the modulus-squared of the all the matrix entries, and the projector P simply gets rid of some of those; the second is the well-known comparison between (Schatten) 1- and 2-normson a D -dimensional space. Thus, by Proposition 11 wehave (cid:107) ρ − ρ (cid:107) W+ ≥ − m − (cid:107) ρ − ρ (cid:107) ≥ r − m − D , (54)and it remains to control D . Note that by its definition,it has an exact expression as a binomial coefficient, D = (cid:18)(cid:106) Ec (cid:107) + mm (cid:19) ≤ (cid:18) Ec + mm (cid:19) ≤ (cid:18) Ec (cid:19) m , (55)concluding the proof. (cid:50) We believe that a lower bound like the one of Corol-lary 12 should hold for the GOCC norm, too. To getsuch a bound, we need to find a “pretty good” Gaussianmeasurement to distinguish two given states.A possible strategy might be provided by [31, Thms. 13and 14], where it is shown that in dimension
D < ∞ ,a fixed rank-one POVM M whose elements form a(weighted) 2-design, provides a bound (cid:107) · (cid:107) M ≥ D + 2 (cid:107) · (cid:107) . (56)This should hold with corrections for approximate de-signs, too, cf. [2].Obviously, as with Bosonic systems we are in infinitedimension, the dimension bound is a priori not going tobe useful. However, we can take inspiration from Corol-lary 12 and its proof, where we assume energy-boundedstates, which we cut off at a finite photon number, re-stricting them thus to a finite-dimensional subspace.The more serious obstacle is that we would have to con-struct a Gaussian measurement, or a probabilistic mix-ture of Gaussian measurements, that approximates a 2-design. But while the set of all Gaussian states has a lo-cally compact symmetry group (symplectic and displace-ment transformations in phase space) that is consistentwith a 2-design, notorious normalisation and convergenceissues prevent us from treating it as such [6].A different approach would be to analyse an even sim-pler measurement, which however must be tomographi-cally complete. A nice candidate would be heterodynedetection on each mode, Eq. (17). VII. DISCUSSION
By analysing the Wigner functions of Bosonic quan-tum states, we showed that there can be arbitrarily largegaps between the GOCC norm distance and the tracedistance. In terms of the norm based on POVMs withpositive Wigner functions, we could show that the sepa-ration necessarily requires many modes, if we are in theregime of states with bounded energy per mode.Our results beg several questions, among them the fol-lowing: first, is it possible to derandomise the construc-tion of Theorem 8, in the sense that we would like to haveconcrete (not random) states with guaranteed separationof GOCC vs trace norm? Secondly, while our construc-tion requires multiple modes, is it possible to have GOCCdata hiding in a fixed number of modes, or even a singlemode, at the expense of larger energy (cf. Corollary 12)?Fortuitously, the recent work by Lami [28] goes someway towards addressing these questions: Indeed, [28,Ex. 5] shows two orthogonal Fock-diagonal states, calledthe even and odd thermal states , which while being atmaximum possible trace distance, have arbitrarily smallGOCC (and indeed W+) distance for sufficiently large energy (temperature). The resulting upper bound [28,Eq. (23)] even compares well with our lower bound fromCorollary 12, when m = 1.The main difference to our scheme in Theorem 8 is thatthose even and odd thermal states are not Gaussian, oreven mixtures of Gaussian states, in fact they have nega-tive Wigner function, indicating the difficulty in creatingthem. Instead our states, while undoubtedly complex(being multi-mode and requiring subtle arrangements ofpoints in phase space, are simply uniform mixtures ofcoherent states, so in a certain sense they are easy toprepare (an experimental implementation would be how-ever still be challenging).As a matter of fact, this is best expressed in resourcetheoretic terms, noticing that GOCC actually can bedefined as a class of quantum maps (to be precise: in-struments), beyond our Definition 1 of only GOCC mea-surements. This point of view is clearly evident in ear-lier references [36], even if it is not formalised. Butrecently, several attempts have been made to createfully-fledged resource theories of non-Gaussianity and ofWigner-negativity [1, 35, 43]. While these works specif-ically focus on state transformations, and in particularthe distillation of some form of “pure” non-Gaussian re-source, our problem of the creation and discriminationof data hiding states are naturally phrased in the gen-eral resource theory. Indeed, in the framework of [1, 35],our GOCC measurements are free operations, and soare the state preparation of ρ and ρ from Theorem 8.Thus, our results can be interpreted as contributions to-wards assessing the non-Gaussianity (Wigner negativity)of a measurement that distinguishes two states optimally.Here is the largest difference to the cited recent papers,which formalise the resource character of states, whereasour focus is on quantum operations. In that sense, The-orem 8 (and equally [28, Ex. 5]) provides a benchmarkfor the realisation of non-Gaussian quantum informationprocessing, simply because optimal, or even decent dis-crimination of the states requires considerable abilitiesbeyond the Gaussian (“linear”) realm. ACKNOWLEDGMENTS
The authors are grateful to Toni Ac´ın and GaelSent´ıs for prompting the first formulation of the ques-tion treated in the present paper, during and after thedoctoral defence of Gael, and in particular for sharingRef. [36]. Thanks to John Calsamiglia and LudovicoLami for asking many further questions which directedthe present research, in particular about the GOCCChernoff coefficient. After the present work having beensuspended for many years, we especially thank LudovicoLami, whose keen interest in data hiding in general,and recent work [28] in particular, have eventually pro-vided the motivation to finish and publish the presentmanuscript. Finally, we thank Prof. Luitpold Blumen-duft for elucidating an optical phenomenon that bears0a certain analogy to the phenomenon of Gaussian datahiding.The authors’ work was supported by the EuropeanCommission (STREP “RAQUEL”), the ERC (AdvancedGrant “IRQUAT”), the Spanish MINECO (grants FIS2008-01236, FIS2013-40627-P, FIS2016-86681-P andPID2019-107609GB-I00), with the support of FEDERfunds, and by the Generalitat de Catalunya, CIRITprojects 2014-SGR-966 and 2017-SGR-1127. [1] F. Albarelli, M. G. Genoni, M. G. A. Paris and A. Fer-raro, “Resource theory of quantum non-Gaussianity andWigner negativity”,
Phys. Rev. A :052350 (2018).[2] A. Ambainis and J. Emerson, “Quantum t-designs: t-wise independence in the quantum world”, in: Proc.22nd Annual IEEE Conference on Computational Com-plexity (CCC07) , pp. 129-140 (2007); arXiv:quant-ph/0701126v2.[3] K. M. R. Audenaert, J. Calsamiglia, Ll. Masanes, R.Mu˜noz-Tapia, A. Ac´ın, E. Bagan and F. Verstraete, “TheQuantum Chernoff Bound”,
Phys. Rev. Lett. :160501(2007).[4] S. Barnett and P. M. Radmore, Methods in TheoreticalQuantum Optics , Oxford Series in Optical and ImagingSciences, Clarendon Press, 2002.[5] S. D. Bartlett, B. C. Sanders, S. L. Braunstein and K.Nemoto, “Efficient Classical Simulation of ContinuousVariable Quantum Information Processes”,
Phys. Rev.Lett. :097904 (2002).[6] R. Blume-Kohout and P. S. Turner, “The Curious Nonex-istence of Gaussian 2-Designs”, Commun. Math. Phys. (3):755-771 (2014).[7] K. E. Cahill and R. J. Glauber, “Density Opera-tors and Quasiprobability Distributions”,
Phys. Rev. (5):1882-1902 (1969).[8] H. Chernoff, “A measure of asymptotic efficiency for testsof a hypothesis based on the sum of observations”, Ann.Math. Statistics (4):493-507 (1952).[9] E. Chitambar, D. Leung, L. Manˇcinska, M. Ozols andA. Winter, “Everything You Always Wanted to KnowAbout LOCC (But Were Afraid to Ask)”, Commun.Math. Phys. (1):303-326 (2014).[10] D. P. DiVincenzo, D. Leung and B. M. Terhal, “Quan-tum data hiding”,
IEEE Trans. Inf. Theory (3):580-599 (2002).[11] T. Eggeling and R. F. Werner, “Hiding Classical Datain Multipartite Quantum States”, Phys. Rev. Lett. :097905 (2002).[12] C. A. Fuchs and J. van de Graaf, “Cryptographic Distin-guishability Measures for Quantum Mechanical States”, IEEE Trans. Inf. Theory (4):1216-1227 (1999).[13] V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H.Shapiro and H. P. Yuen, “Classical Capacity of the LossyBosonic Channel: The Exact Solution”, Phys. Rev. Lett. :027902 (2004).[14] R. J. Glauber, “Coherent and Incoherent States of theRadiation Field”, Phys. Rev. (6):2766-2788 (1963).[15] T. S. Han and S. Verd´u, “Approximation theory of out-put statistics”,
IEEE Trans. Inf. Theory (3):752-772(1993).[16] T. S. Han and S. Verd´u, “The resolvability and the capac-ity of AWGN channels are equal”, in: Proc. ISIT 1994,p. 463 (1994).[17] T. S. Han, Information-Spectrum Methods in Informa- tion Theory , Ser. Applications of Mathematics: Stochas-tic Modelling and Applied Probability, vol. 50, SpringerVerlag, Berlin Heidelberg New York, 2003.[18] C. W. Helstrom, “ Quantum Limitations on the Detec-tion of Coherent and Incoherent Signals”,
IEEE Trans.Inf. Theory (4):482-490 (1965).[19] C. W. Helstrom, “Detection Theory and Quantum Me-chanics”, Inform. Control (3):254-291 (1967).[20] C. W. Helstrom, Quantum Detection and EstimationTheory , Math. Science Engineering, vol. 123, AcademicPress, New York, 1976.[21] M. Hillery, R. F. O’Connell, M. O. Scully and E. P.Wigner, “Distribution Functions in Physics: Fundamen-tals”,
Phys. Reports (3):121-167 (1984).[22] A. S. Holevo, “Statistical Decision Theory for QuantumSystems”,
J. Multivar. Anal. (4):337-394 (1973).[23] A. S. Holevo, “The Capacity of the Quantum Channelwith General Signal States”, IEEE Trans. Inf. Theory (1):269-273 (1998).[24] A. S. Holevo, “On the constrained classical capacityof infinite-dimensional covariant quantum channels”, J.Math. Phys. :015203 (2016); see also arXiv:quant-ph/9705054 (1997).[25] E. Knill, R. Laflamme and G. J. Milburn, “A schemefor efficient quantum computation with linear optics”, Nature :46-52 (2001).[26] P. Kok and B. W. Lovett,
Introduction to OpticalQuantum Information Processing , Cambridge UniversityPress, 2010.[27] L. Lami, C. Palazuelos and A. Winter, “Ultimate datahiding in quantum mechanics and beyond”,
Commun.Math. Phys. (2):661-708 (2018).[28] L. Lami, “Quantum data hiding with continuous variablesystems”, arXiv[quant-ph]:2021.TODAY (2021).[29] C. Lancien and A. Winter, “Distinguishing multi-partitestates by local measurements”,
Commun. Math. Phys. :555-573 (2013).[30] A. Mari and J. Eisert, “Positive Wigner Functions Ren-der Classical Simulation of Quantum Computation Effi-cient”,
Phys. Rev. Lett. :230503 (2012).[31] W. Matthews, S. Wehner and A. Winter, “Distinguisha-bility of Quantum States Under Restricted Families ofMeasurements with an Application to Quantum DataHiding”,
Commun. Math. Phys. (3):813-843 (2009).[32] K. K. Sabapathy and A. Winter, “Non-Gaussian opera-tions on bosonic modes of light: Photon-added Gaussianchannels”,
Phys. Rev. A :062309 (2017).[33] B. Schumacher and M. D. Westmoreland, “Sending clas-sical information via noisy quantum channels”, Phys.Rev. A (1):131-138 (1997).[34] E. C. G. Sudarshan, “Equivalence of Semiclassical andQuantum Mechanical Descriptions of Statistical LightBeams”, Phys. Rev. Lett. (7):277-279 (1963).[35] R. Takagi and Q. Zhuang, “Convex resource theory of non-Gaussianity”, Phys. Rev. A :062337 (2018).[36] M. Takeoka and M. Sasaki, “Discrimination of the bi-nary coherent signal: Gaussian-operation limit and sim-ple non-Gaussian near-optimal receivers”, Phys. Rev. A :022320 (2008).[37] B. M. Terhal, D. P. DiVincenzo and D. Leung, “HidingBits in Bell States”, Phys. Rev. Lett. (25):5807-5810(2001).[38] C. Weedbrook, S. Pirandola, R. Garc´ıa-Patr´on, N. J.Cerf, T. C. Ralph, J. H. Shapiro and S. Lloyd, “Gaussianquantum information”, Rev. Mod. Phys. (2):621-669(2012). [39] E. Wigner, “On the Quantum Correction For Thermo-dynamic Equilibrium”, Phys. Rev. (5):749-759 (1932).[40] M. M. Wilde and A. Winter, “Strong converse for theclassical capacity of the pure-loss Bosonic channel”, Probl. Inf. Transm. (2):117-132 (2014).[41] A. Winter, “Coding Theorem and Strong Conversefor Quantum Channels”, IEEE Trans. Inf. Theory (7):2481-2485 (1999).[42] A. Winter, “Information efficiency of local data hiding inquantum systems”, in preparation (2014-2021).[43] Q. Zhuang, P. W. Shor and J. H. Shapiro, “Resource the-ory of non-Gaussian operations”. Phys. Rev. A97