On the optimal certification of von Neumann measurements
Paulina Lewandowska, Aleksandra Krawiec, Ryszard Kukulski, Łukasz Pawela, Zbigniew Puchała
CCERTIFICATION OF QUANTUM MEASUREMENTS
PAULINA LEWANDOWSKA , ALEKSANDRA KRAWIEC , RYSZARD KUKULSKI ,(cid:32)LUKASZ PAWELA* , AND ZBIGNIEW PUCHA(cid:32)LA , Abstract.
In this report we study certification of quantum measurements, which canbe viewed as the extension of quantum hypotheses testing. This extensions involves thestudy of the input state and the measurement procedure. Here, we will be interestedin two-point (binary) certification scheme in which the null alternative hypotheses aresingle element sets. Our goal is to minimize the probability of the type II error givensome fixed statistical significance. In this report, we begin with studying the two-pointcertification of pure quantum states and unitary channels to later use them to proveour main result, which is the certification of von Neumann measurements in single-shot and parallel scenarios. From our main result follow the conditions when two purestates, unitary operations and von Neumann measurements cannot be distinguishedperfectly but still can be certified with a given statistical significance. Moreover, weshow the connection between the certification of quantum channels and the notion of q -numerical range. Introduction
The validation of sources producing quantum states and measurement devices, en-abling computation performance, is a necessary step of quantum technology [1–3]. Thesearch for practical and reliable tools for validation of quantum architecture has attracteda lot of attention in recent years [4–8]. Rapid technology development and increasinginterest in quantum computers paved the way towards creating more and more efficientvalidation methods of Noisy Intermediate-Scale Quantum devices (NISQ) [9, 10]. Sucha growth comes along with challenging prescriptions about the precision of the compo-nents of quantum devices. The tasks of ensuring the correctness of quantum devices arereferred to as certification .Our approach to the problem of validating of quantum architectures can be describedas follows. Imagine you are given a black box and are promised two things. First, itcontains a pure quantum state (or a unitary matrix or a von Neumann POVM), andsecond it contains one of two possible choices of these objects. The owner of the box, Eve,tells you which of the two possibilities is contained within the box. Yet, for some reason,you don’t completely trust her and decide to perform some kind of hypothesis testingscheme on the black box. You decide to take Eve’s promise as the null hypothesis, H , forthis scheme, and the second of the possibilities as the alternative hypothesis, H . Since Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, ul.Ba(cid:32)ltycka 5, 44-100 Gliwice, Poland Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian Univer-sity, ul. (cid:32)Lojasiewicza 11, 30-348 Krak´ow, Poland
E-mail addresses : [email protected] . a r X i v : . [ qu a n t - ph ] S e p CERTIFICATION OF QUANTUM MEASUREMENTS now you own the box and are free to proceed as you please, you need to prepare someinput into the box and perform a measurement on the output. A particular input stateand final measurement (or only the measurement, for the case when the box contains aquantum state) will be called a certification strategy.Of course, just like in classical hypothesis testing, in our certification scheme we havetwo possible types of errors. The type I error happens if we reject the null hypothesiswhen it was in reality true. The type II error happens if we accept the null hypothesiswhen we should have rejected it. The main aim of certification is finding the optimalstrategy which minimizes one type of error when the other one is fixed. In this work weare interested in the minimization of the type II error given a fixed type I error. Wewill begin with considering one-shot scenario and later extend it to the multiple-shotscenario. For this purpose, we will often make use of the notions of numerical range and q -numerical range as essential tools in the proofs [11–14].Certification of quantum objects is closely related with the problem of discriminationof those objects. Intuitively, in the discrimination problem we are given one of twoquantum objects sampled according to a given probability distribution. Hence, theprobability of making an error in the discrimination task is equal to the average ofthe type I and type II errors over the assumed probability distribution. Therefore,the discrimination problem can be seen as symmetric distinguishability , as opposed tocertification, that is asymmetric distinguishability . In other words, the main task ofdiscrimination is the minimization over the average of both types of errors while thecertification concerns the minimization over one type of error when the other one is fixed.The problem of discrimination of quantum states and channels was solved analyticallyby Helstrom [15] a few decades ago. Sequential discrimination of quantum states wasstudied in [16, 17] and the discrimination of quantum channels was further studied forexample in [18–23]. The work [24] paved the way for studying the discrimination ofquantum measurements. Therefore, this work can be seen as a natural extension of ourworks [25] and [26], where we studied the discrimination of von Neumann measurementsin single and multiple-shot scenarios, respectively. Nevertheless, one can also consider ascenario in which we are allowed to obtain an inconclusive answer. Therefore, we arriveat the unambiguous discrimination of quantum operations discussed in [26, 27].All the above-mentioned approaches towards the certification were considered in afinite number of steps. Another common approach involves studying certification ofquantum objects in the asymptotic regime [28–30] which assumes that the number ofcopies of the given quantum object goes to infinity. It focuses on studying the con-vergence of the probability of making one type of error while a bound on the secondone is assumed. This task is strictly related with the term of relative entropy and itsasymptotic behavior [31]. For a more general overview of quantum certification we referthe reader to [32, 33].This work is organized as follows. We begin with preliminaries in Section 2. Then, inSection 3 we study the two-point certification of pure quantum states. The Theorem 1therein states the optimal probability of the type II error while Corollary 1 presents theoptimal strategy for certification. Certification of unitary channels is discussed in Sec-tion 4. The optimal probability of type II error is stated in Theorem 2, while the optimalstrategy is presented in Corollary 3. The certification of von Neumann measurements ERTIFICATION OF QUANTUM MEASUREMENTS 3 is studied in Section 5 and our main result is stated therein as Theorem 3. Section 6generalizes the results on certification to the multiple-shot scenario.2.
Preliminaries
Let M d ,d be the set of all matrices of dimension d × d over the field C . For the sakeof simplicity, square matrices will be denoted by M d . The set of quantum states, that ispositive semidefinite operators having trace equal one, will be denoted D d . By default,when we write | ψ (cid:105) , | φ (cid:105) , we mean normalized pure states, unless we mention otherwise.The subset of M d consisting of unitary matrices will be denoted by U d , while its subgroupof diagonal unitary operators will be denoted by DU d . Let U ∈ U d be a unitary matrix. Aunitary channel Φ U is defined as Φ U ( · ) = U · U † . A general quantum measurement, thatis a positive operator valued measure (POVM) P is a collection of positive semidefiniteoperators { E , . . . , E m } called effects , which sum up to identity, i.e. (cid:80) mi =1 E i = 1l. Ifall the effects are rank-one projection operators, then such a measurement is called vonNeumann measurement. Every von Neumann measurement can be parameterized by aunitary matrix and hence we will use the notation P U for a von Neumann measurementwith effects {| u (cid:105)(cid:104) u | , . . . , | u d (cid:105)(cid:104) u d |} , where | u i (cid:105) is the i -th column of the unitary matrix U . The action of quantum measurement P U on some state ρ ∈ D d can be expressed asthe action of a quantum channel P U : ρ → d (cid:88) i =1 (cid:104) u i | ρ | u i (cid:105)| i (cid:105)(cid:104) i | . (1)As mentioned in the Introduction, in this work we focus on two-point hypothesistesting of quantum objects. The starting point towards the certification of quantumobjects is the hypothesis testing of quantum states. Let H be a null hypothesis whichstates that the obtained state was | ψ (cid:105) , while the alternative hypothesis H states that theobtained state was | ϕ (cid:105) . The certification is performed by the use of a binary measurement { Ω , − Ω } , where the effect Ω corresponds to accepting the null hypothesis and 1l − Ωaccepts the alternative hypothesis. In this work we will be considering only POVMswith two effects of this form. Therefore the effect Ω uniquely determines the POVM andhence we will be using the words measurement and effect interchangeably.Assume we have a fixed measurement Ω. We introduce the probability of the type Ierror, p I (Ω), that is the probability of rejecting the null hypothesis when in fact it wastrue, as p I (Ω) = tr ((1l − Ω) | ψ (cid:105)(cid:104) ψ | ) = 1 − tr (Ω | ψ (cid:105)(cid:104) ψ | ) . (2)The type II error, p II (Ω), that is the probability of accepting the alternative hypothesiswhen H was correct, is defined as p II (Ω) = tr (Ω | ϕ (cid:105)(cid:104) ϕ | ) . (3)In the remainder of this work we will assume the statistical significance δ ∈ [0 ,
1] that isthe probability of the type I error will be upper-bounded by δ . Our goal will be to finda most powerful test, that is to minimize the probability of the type II error by findingthe optimal measurement, which we will denote as Ω . Such Ω , which minimizes p II (Ω) CERTIFICATION OF QUANTUM MEASUREMENTS while assuming the statistical significance δ , will be called an optimal measurement . Theminimized probability of type II error will be denoted by p II := min Ω: p I (Ω) ≤ δ p II (Ω) . (4)While certifying quantum channels and von Neumann measurements, we will alsoneed to minimize over input states. Let a channel Φ correspond to H hypothesis andΦ correspond to H hypothesis. We define p | ψ (cid:105) I (Ω) = tr ((1l − Ω)Φ ( | ψ (cid:105)(cid:104) ψ | )) p | ψ (cid:105) II (Ω) = tr (ΩΦ ( | ψ (cid:105)(cid:104) ψ | )) . (5)Naturally, for each input state we can consider minimized probability of type II error,that is p | ψ (cid:105) II = min Ω: p | ψ (cid:105) I (Ω) ≤ δ p | ψ (cid:105) II (Ω) . (6)Finally, we will be interested in calculating optimized probability of type II error overall input states. This will be denoted as p II := min | ψ (cid:105) p | ψ (cid:105) II . (7)Note that the symbol p II is used in two contexts. In the problem of certification ofstates the minimization is performed only over measurements Ω, while in the problemof certification of unitary channels and von Neumann measurements the minimizationis over both measurements Ω and input states | ψ (cid:105) . In other words, p II is equal to theoptimized probability of the type II error in certain certification problem.The input state which minimizes p II will be called an optimal state . We will use theterm optimal strategy to denote both the optimal state and the optimal measurement.Now, we introduce a basic toolbox for studying the certification of quantum objectswhich is strictly related with the problem of discrimination of quantum channels. First,we will be using the notion of the diamond norm. The diamond norm of a superoperatorΨ is defined as (cid:107) Ψ (cid:107) (cid:5) := max (cid:107) X (cid:107) =1 (cid:107) (Ψ ⊗ X ) (cid:107) . (8)The celebrated theorem of Helstrom [15] gives a lower bound on the probability of makingan error in distinction in the scenario of symmetric discrimination of quantum channels.The probability of incorrect symmetric discrimination between quantum channels Φ andΨ is bounded as follows p e ≥ − (cid:107) Φ − Ψ (cid:107) (cid:5) . (9)Moreover, our results will often make use of the terms of numerical range and q -numerical range [14]. The numerical range is a subset of complex plane defined for amatrix X ∈ M d as W ( X ) := {(cid:104) ψ | X | ψ (cid:105) : (cid:104) ψ | ψ (cid:105) = 1 } (10)while the q -numerical range [11–13] is defined for a matrix X ∈ M d as W q ( X ) := {(cid:104) ψ | X | ϕ (cid:105) : (cid:104) ψ | ψ (cid:105) = (cid:104) ϕ | ϕ (cid:105) = 1 , (cid:104) ψ | ϕ (cid:105) = q, q ∈ C } . (11) ERTIFICATION OF QUANTUM MEASUREMENTS 5
The standard numerical range is the special case of q -numerical range for q = 1, that is W ( X ) = W ( X ). We will use the notation ν q ( X ) := min {| x | : x ∈ W q ( X ) } (12)to denote the distance on a complex plane from q -numerical range to zero. In the casewhen q = 1, we will simply write ν ( X ). The main properties of q -numerical range are itsconvexity and compactness [13]. The detailed shape of q -numerical range is describedin [12]. The properties of q -numerical range [18] that will be used throughout this paperare W q (cid:48) ⊆ q (cid:48) q W q for q ≤ q (cid:48) , q, q (cid:48) ∈ R (13)and W q ( X ⊗ W q ( X ) , q ∈ R . (14)From the above it easy to see that ν q ( X ⊗ ν q ( X ) , q ∈ R . (15)In the Supplementary Material we provide an animation of q -numerical range of unitarymatrix U ∈ U with eigenvalues 1 , e π i3 and e π i3 for all parameters q ∈ [0 , Two-point certification of pure states
In this section we present our results concerning the certification of pure quantumstates. We state the optimized probability of the type II error for the quantum hypothesistesting problem as well as the form of the optimal measurement which should be usedfor the certification. Although these results may seem quite technical, they will laythe groundwork for studying the certification of unitary channels and von Neumannmeasurements in further sections.Assume we are given one of two known quantum states either | ψ (cid:105) or | ϕ (cid:105) . The H hypothesis corresponds to the state | ψ (cid:105) , while the alternative H hypothesis correspondsto the state | ϕ (cid:105) .In other words, our goal is to decide whether the given state was | ψ (cid:105) or | ϕ (cid:105) . To makea decision we need to measure the given state and we are allowed to use any POVM. Wewill use a quantum measurement with effects { Ω , − Ω } , where the first effect Ω acceptsthe hypothesis H and the second effect 1l − Ω accepts H . Hence, the probability ofobtaining the type I error is given by p I (Ω) = (cid:104) ψ | (1l − Ω) | ψ (cid:105) . (16)The probability of obtaining the type II error to be minimized yields p II = min Ω: p I (Ω) ≤ δ (cid:104) ϕ | Ω | ϕ (cid:105) =: (cid:104) ϕ | Ω | ϕ (cid:105) , (17)where the minimization is performed by finding the optimal measurement Ω .Similar problem was explored in [34, Proposition 3.2.]. For general mixed states, theoptimal value of the probability p II was presented as an optimization problem over onereal parameter space. The following theorem states the solution of this optimization forpure states. available as auxiliary gif file in the arXiv submission CERTIFICATION OF QUANTUM MEASUREMENTS
Theorem 1.
Consider the problem of two-point certification of pure quantum states withhypotheses given by H : | ψ (cid:105) ,H : | ϕ (cid:105) . (18) and statistical significance δ ∈ [0 , . Then, for the most powerful test, the probability ofthe type II error (17) yields p II = (cid:40) if |(cid:104) ψ | ϕ (cid:105)| ≤ √ δ, (cid:16) |(cid:104) ψ | ϕ (cid:105)|√ − δ − (cid:112) − |(cid:104) ψ | ϕ (cid:105)| √ δ (cid:17) if |(cid:104) ψ | ϕ (cid:105)| > √ δ. (19)The proof of the above theorem is presented in Appendix A. This proof gives a con-struction of the optimal measurement which minimizes the probability of the type IIerror. The exact form of such an optimal measurement is stated as the following corol-lary. Corollary 1.
The optimal strategy for two-point certification of pure quantum states | ψ (cid:105) and | ϕ (cid:105) , with statistical significance δ yields(1) if |(cid:104) ψ | ϕ (cid:105)| ≤ √ δ , then the optimal measurement is given by Ω = | ω (cid:105)(cid:104) ω | , where | ω (cid:105) = | (cid:101) ω (cid:105)||| (cid:101) ω (cid:105)|| , | (cid:101) ω (cid:105) = | ψ (cid:105) − (cid:104) ϕ | ψ (cid:105)| ϕ (cid:105) ;(2) if |(cid:104) ψ | ϕ (cid:105)| > √ δ , then the optimal measurement is given by Ω = | ω (cid:105)(cid:104) ω | for | ω (cid:105) = √ − δ | ψ (cid:105) − √ δ | ψ ⊥ (cid:105) , | ψ ⊥ (cid:105) = | (cid:102) ψ ⊥ (cid:105)||| (cid:102) ψ ⊥ (cid:105)|| , where | (cid:102) ψ ⊥ (cid:105) = | ϕ (cid:105) − (cid:104) ψ | ϕ (cid:105)| ψ (cid:105) . Certification of unitary channels
In this section we will be interested in certification of two unitary channels Φ U andΦ V for U, V ∈ U d . Without loss of generality we can assume that one of these unitarymatrices is the identity matrix and then our task reduces to certification of channels Φ and Φ U . In the most general case, we are allowed to use entanglement by adding anadditional system. Hence, the null hypothesis H yields that the unknown channel isΦ ⊗
1l and the alternative H hypothesis yields that the unknown channel is Φ U ⊗ Certification scheme.
The idea behind the scheme of certification of unitarychannels is to reduce this problem to certification of quantum states discussed in theprevious section.We prepare some (possibly entangled) input state | ψ (cid:105) and perform the unknown chan-nel on it. The resulting state is either (1l ⊗ | ψ (cid:105) or ( U ⊗ | ψ (cid:105) . Then, we perform themeasurement { Ω , − Ω } and make a decision whether the given channel was Φ ⊗
1l orΦ U ⊗ H hypothesis while 1l − Ω correspondsto the alternative hypothesis H .The results of minimization of the probability of the type II error over input states | ψ (cid:105) and measurements Ω are summarized as the following theorem. This reasoning is basedon the results from Theorem 1. Related study of this problem can be found in [35]. ERTIFICATION OF QUANTUM MEASUREMENTS 7
Theorem 2.
Consider the problem of two-point certification of unitary channels withhypotheses H : Φ ⊗ ,H : Φ U ⊗ . (20) and statistical significance δ ∈ [0 , . Then, for the most powerful test, the probability ofthe type II error yields p II = (cid:40) if |(cid:104) ψ | U | ψ (cid:105)| ≤ √ δ, (cid:16) |(cid:104) ψ | U | ψ (cid:105)|√ − δ − (cid:112) − |(cid:104) ψ | U | ψ (cid:105)| √ δ (cid:17) if |(cid:104) ψ | U | ψ (cid:105)| > √ δ, (21) where | ψ (cid:105) ∈ arg min | ψ (cid:105) |(cid:104) ψ | U | ψ (cid:105)| .Proof. Let us first introduce the hypotheses conditioned by the input state | ψ (cid:105) H | ψ (cid:105) : | ψ (cid:105) ,H | ψ (cid:105) : ( U ⊗ | ψ (cid:105) . (22)We do not make any assumptions on the dimension of the auxiliary system for the timebeing. It will appear however that it suffices if its dimension equals one. The hypothesesin (22) correspond to output states after the application of the extended unitary channelon the state | ψ (cid:105) . For these hypotheses we consider the statistical significance δ ∈ [0 , p | ψ (cid:105) I (Ω) = tr ((1l − Ω)(Φ ⊗ | ψ (cid:105)(cid:104) ψ | )) ≤ δ. (23)Our goal will be to calculate the minimized probability of the type II error p II = min | ψ (cid:105) min Ω: p | ψ (cid:105) I (Ω) ≤ δ tr(Ω(Φ U ⊗ | ψ (cid:105)(cid:104) ψ | )) =: tr(Ω (Φ U ⊗ | ψ (cid:105)(cid:104) ψ | )) , (24)where naturally, for the optimal strategy | ψ (cid:105) and Ω it holds that p | ψ (cid:105) I (Ω ) ≤ δ .Now we will show that the use of entanglement is unnecessary. From Theorem 1 weknow that the probability of the type II error, p II , depends on the minimization of theinner product min | ψ (cid:105) |(cid:104) ψ | U ⊗ | ψ (cid:105)| . Directly from the definition of numerical range wecan see that (cid:104) ψ | U ⊗ | ψ (cid:105) ∈ W ( U ⊗ ν ( U ⊗ ν ( U ) . (25)Let | ψ (cid:105) be the considered optimal input state, i.e. | ψ (cid:105) ∈ arg min | ψ (cid:105) |(cid:104) ψ | U | ψ (cid:105)| . Thereforewe can reformulate our hypotheses as H | ψ (cid:105) : | ψ (cid:105) ,H | ψ (cid:105) : U | ψ (cid:105) . (26)These hypotheses, when taking | ϕ (cid:105) := U | ψ (cid:105) , were the subject of interest in Theorem 1. (cid:3) The next corollary follows directly from the above proof.
Corollary 2.
Entanglement is not needed for the certification of unitary channels.
CERTIFICATION OF QUANTUM MEASUREMENTS
Finally, we present a short observation concerning the sufficiency of the use of purestates in the problem of certification of unitary channels.
Remark 1.
Without loss of generality, we can consider only pure input states. Tosee this, for any mixed state ρ consider its purification | ψ (cid:105) , that is a state satisfying Tr ( | ψ (cid:105)(cid:104) ψ | ) = ρ . Then tr ((1l − Ω)Φ ( ρ )) = tr (((1l − Ω) ⊗ ⊗ | ψ (cid:105)(cid:104) ψ | )) (27) and tr (ΩΦ U ( ρ )) = tr ((Ω ⊗ U ⊗ | ψ (cid:105)(cid:104) ψ | )) . (28)4.2. Connection with q -numerical range. There exists a close relationship betweenthe above results and the definition of numerical range, which can be seen from theproof of Theorem 2. It the work [18] the authors show the connection between thediscrimination of quantum channels and q -numerical range. In this section we show theconnection between certification of unitary channels and q -numerical range. Recall thedefinition of q -numerical range. W q ( X ) := {(cid:104) ψ | X | ϕ (cid:105) : (cid:104) ψ | ϕ (cid:105) = q } . (29)Using this notion and the notation introduced in Eq. (12) we can rewrite our results forthe probability of the type II error from Theorem 2 as p II = ν √ − δ ( U ⊗ ν √ − δ ( U ) . (30)An independent derivation of the above formula is presented in Appendix B.Let Θ be the angle between two most distant eigenvalues of a unitary matrix U . Then,from the above discussion we can draw a conclusion that for any statistical significance δ ∈ (0 , (cid:16) √ δ (cid:17) ≤ Θ < π , then although Φ U and Φ cannot be distinguishedperfectly, they can be certified with p II = 0. In other words, the numerical range W ( U )does not contain zero but √ − δ -numerical range, W √ − δ ( U ), does contain zero. Thesituation changes when 2 arccos (cid:16) √ δ (cid:17) > Θ. Then, both numerical range W ( U ) and √ − δ -numerical range W √ − δ ( U ) do not contain zero. This is presented in Fig. 1.Now we will work towards the construction of the optimal strategy, which will bestated as a corollary. Besides finding the optimal measurement which was shown inprevious section we will show a closed-form expression of the optimal input state. Forthis purpose we will make use of the spectral decomposition of a unitary matrix U givenby U = d (cid:88) i =1 λ i | x i (cid:105)(cid:104) x i | . (31)Let λ , λ d be a pair of the most distant eigenvalues of U . The following corollary isanalogous to the corollary from the previous section as it presents the optimal strategyfor the certification of unitary channels. Corollary 3. By | ψ (cid:105) we will denote the optimal state for two-point certification ofunitary channels and let | ϕ (cid:105) := U | ψ (cid:105) . Then, the optimal strategy yields(1) If ∈ W √ − δ ( U ) , then we have two cases ERTIFICATION OF QUANTUM MEASUREMENTS 9 p e √ p II Θ2 Figure 1.
Numerical range W ( U ) (red triangle) and √ − δ -numericalrange W √ − δ ( U ) (blue oval) of U ∈ U with eigenvalues 1 , e π i3 and e π i3 with statistical significance δ = 0 .
05. The value p e is the probability ofincorrect symmetric discrimination of channels Φ and Φ U . • if (cid:54)∈ W ( U ) , then we can take | ψ (cid:105) = 1 √ | x (cid:105) + 1 √ | x d (cid:105) (32) where | x (cid:105) , | x d (cid:105) are eigenvectors corresponding to the pair of the most distanteigenvalues λ , λ d of U . The optimal measurement is given by Ω = | ω (cid:105)(cid:104) ω | ,where | ω (cid:105) = | (cid:101) ω (cid:105)||| (cid:101) ω (cid:105)|| , | (cid:101) ω (cid:105) = | ψ (cid:105) − (cid:104) ϕ | ψ (cid:105)| ϕ (cid:105) , • if ∈ W ( U ) , then we have perfect symmetric distinguishability. Moreover,there exists the probability vector p such that (cid:80) di =1 λ i p i = 0 and we obtainthat | ψ (cid:105) = d (cid:88) i =1 √ p i | x i (cid:105) . (33) Analogously, we choose the optimal measurement given by Ω = | ω (cid:105)(cid:104) ω | ,where | ω (cid:105) = | (cid:101) ω (cid:105)||| (cid:101) ω (cid:105)|| , | (cid:101) ω (cid:105) = | ψ (cid:105) − (cid:104) ϕ | ψ (cid:105)| ϕ (cid:105) . It easy to see that in this casewe have Ω = | ψ (cid:105)(cid:104) ψ | .(2) If (cid:54)∈ W √ − δ ( U ) , then the discriminator is given by Eq. (32) , whereas theoptimal measurement is can be expressed as Ω = | ω (cid:105)(cid:104) ω | for | ω (cid:105) = √ − δ | ψ (cid:105) −√ δ | ψ ⊥ (cid:105) , | ψ ⊥ (cid:105) = | (cid:102) ψ ⊥ (cid:105)||| (cid:102) ψ ⊥ (cid:105)|| , where | (cid:102) ψ ⊥ (cid:105) = | ϕ (cid:105) − (cid:104) ψ | ϕ (cid:105)| ψ (cid:105) . Remark 2.
Observe that the optimal input state | ψ (cid:105) does not depend on δ , while theoptimal measurement Ω does depend on the parameter δ in each case. It is also worth noting that the optimal state in quantum hypothesis testing is of the same form as in theproblem of unitary channel discrimination. Two-point certification of Von Neumann measurements
In this section we will focus on the certification of von Neumann measurements. Re-call that every quantum measurement can be associated with a measure-and-preparequantum channel. Therefore, while studying the certification of quantum measurementswe will often take advantage of the certification of quantum channels discussed in theprevious section. Following [26], we will assume that one of the measurements is inthe computational basis. Hence, we will be certifying the measurement P under thealternative hypothesis P U .While certifying quantum channels, the most general scenario allows for the use ofentanglement by adding an additional system. Hence, in our case of certification of vonNeumann measurements, the H hypothesis yields that the unknown measurement is P ⊗
1l whereas for the alternative hypothesis yields that the measurement is P U ⊗ P U and P is given by ||P U − P || (cid:5) = min E ∈DU d || Φ UE − Φ || (cid:5) , (34)where DU d is the subgroup of diagonal unitary matrices of dimension d . As we can see,the problem of discrimination of von Neumann measurements reduces to the problem ofdiscrimination of unitary channels. From [36] we know that the diamond norm distancebetween two unitary channels Φ U and Φ is expressed as || Φ U − Φ || (cid:5) = 2 (cid:112) − ν ( U ) , (35)where ν ( U ) = min {| x | : x ∈ W ( U ) } .5.1. Certification scheme.
The scenario of certification of von Neumann measure-ments is as follows. We prepare some (possibly entangled) input state | ψ (cid:105) and, aspreviously, we perform the unknown von Neumann measurement on one part of it.Then, after performing the measurement, the null hypothesis H corresponds to thestate ( P ⊗ | ψ (cid:105)(cid:104) ψ | ) while the alternative hypothesis H corresponds to the state( P U ⊗ | ψ (cid:105)(cid:104) ψ | ). Our goal is to find an optimal input state and measurement forwhich the probability of the type II error is saturated, while the statistical significance δ is assumed. The results of minimization are summarized as a theorem, which proof ispresented in Appendix C. Theorem 3.
Consider the problem of two-point certification of von Neumann measure-ments with hypotheses H : P ⊗ H : P U ⊗ . (36) and statistical significance δ ∈ [0 , . Then, for the most powerful test, the probability ofthe type II error yields p II = max E ∈DU d ν √ − δ ( U E ) . (37) ERTIFICATION OF QUANTUM MEASUREMENTS 11
It is worth mentioning that we do not make any assumptions on the dimension of theauxiliary system, however its dimension is obviously upper-bounded by the dimension ofthe input states. Additionally, the dimension of the auxiliary system can be reduced tothe Schmidt rank of the input state | ψ (cid:105) [25, Proposition 4]. The difference between thecertification of unitary channels and certification of von Neumann measurements is thatin the latter case the entanglement indeed can significantly improve the certification.However, in contrast to the certification of unitary channels, the output states ( P ⊗ | ψ (cid:105)(cid:104) ψ | ) and ( P U ⊗ | ψ (cid:105)(cid:104) ψ | ) are not necessarily pure. Hence the proof of the Theorem3 requires more advanced techniques. Luckily, we still can make use of the calculationsfrom Section 4, due to the fact that formally mixed states ( P ⊗ | ψ (cid:105)(cid:104) ψ | ) and ( P U ⊗ | ψ (cid:105)(cid:104) ψ | ), conditioned by obtaining the label i ∈ { , . . . , d } , are pure. This observationwill be crucial in the proof of the above theorem presented in Appendix C. Remark 3.
The optimal strategy is described in the proof of Theorem 3. Constructionof such a strategy depends whether the value of ||P U − P || (cid:5) is smaller or equal two.Similarly to the case of unitary channel certification, the optimal input state | ψ (cid:105) doesnot depend on δ , while the optimal measurement Ω does depend on δ . Moreover, theoptimal state has the same form as in the problem of discrimination of von Neumannmeasurements. Parallel multiple-shot certification
In this section we focus on the scenario in which we have access to N copies ofquantum objects. It is worth noting that copies of a given channel can be used in manyconfigurations. One possibility is the parallel scheme, which is described by the tensorproduct, or the sequential scheme, that is the compositions of channels. Nevertheless, allthese schemes are the special cases of the most general adaptive scheme which uses theformalism of quantum combs, sometimes called quantum networks [37]. In this paper,we will restrict our attention only to the parallel case which is optimal for two-pointhypothesis testing.More technically, in each case the copies of quantum objects are described by thetensor product. We can clearly see that tensor product of pure states is again a purestate, tensor product of unitary channels is a unitary channel and tensor product ofvon Neumann measurements is a von Neumann measurement. Therefore, we are able toapply our results from previous sections.Let us begin with the certification of pure states. Such certification can be understoodas certifying states | ψ (cid:105) ⊗ N and | ϕ (cid:105) ⊗ N . The following corollary generalizes the results fromTheorem 1. Corollary 4.
In the case of certification of pure states with statistical significance δ ∈ [0 , , the minimized probability of the type II error yields p ( N ) II = (cid:40) |(cid:104) ψ | ϕ (cid:105)| N ≤ √ δ (cid:16) |(cid:104) ψ | ϕ (cid:105)| N √ − δ − (cid:112) − |(cid:104) ψ | ϕ (cid:105)| N √ δ (cid:17) |(cid:104) ψ | ϕ (cid:105)| N > √ δ (38) where N is the number of uses of the pure state. One can note that for a given statistical significance δ , by taking N ≥ log √ δ log |(cid:104) ψ | ϕ (cid:105)| we obtain p II = 0. This is not in contradiction with the statement that if one cannotdistinguish states perfectly in one step, then they cannot by distinguished perfectly in anyfinite number of tries, because the error is hidden in p I . This error decays exponentially,and the optimal exponential error rate, depending on a formulation, can be stated as theStein bound, the Chernoff bound, the Hoeffding bound, and the Han-Kobayashi bound,see [38] and references therein.Secondly, we focus on the certification of unitary channels. The scenario of parallelcertification can be seen as certifying channels Φ ⊗ N and Φ U ⊗ N . Hence for the parallelcertification of such unitary channels we have the following corollary generalizing theresults from Theorem 2. Corollary 5.
In the case of parallel certification of unitary channels with statisticalsignificance δ ∈ [0 , , the minimized probability of the type II error yields p ( N ) II = ν √ − δ (cid:0) U ⊗ N (cid:1) (39) where N is the number of uses of the unitary channel. From the above it follows that if 0 ∈ W √ − δ ( U ⊗ N ), then the channels Φ ⊗ N andΦ ⊗ NU can be certified with p II = 0. Let Θ be the angle between a pair of two mostdistant eigenvalues of a unitary matrix U . The perfect certification can be achievedby taking N = (cid:100) √ δ Θ (cid:101) . Observe that in the special case δ = 0, we recover thewell-known formula N = (cid:100) π Θ (cid:101) being the number of unitary channels required for perfectdiscrimination in the scheme of symmetric distinguishability of unitary channels [20].The dependence between the number N of used unitary channels and the shape of W √ − δ ( U ⊗ N ) is presented in Fig. 2.Finally, we will consider the certification of von Neumann measurements P and P U .Similarly to unitary channels, we consider only the parallel scheme and therefore thiscan be understood as certifying von Neumann measurements P U ⊗ N and P ⊗ N . Thisleads us to the following generalization of Theorem 3. Corollary 6.
In the case of certification of von Neumann measurements with statisticalsignificance δ ∈ [0 , , the minimized probability of the type II error yields p ( N ) II = max E ∈DU dN ν √ − δ (cid:0) U ⊗ N E (cid:1) (40) where N is the number of uses of the von Neumann measurements. Moreover, by theTheorem in [26] we can establish the product form of E . It implies that the optimalprobability of the type II error is given by p ( N ) II = max E ∈DU d ν √ − δ (cid:0) U ⊗ N E ⊗ N (cid:1) (41)It is known that the parallel scheme is optimal for the discrimination of unitarychannels [39] and von Neumann measurements [26]. In what follows we will state that asimilar situation appears also in the case of certification. Remark 4.
The parallel scheme is optimal for certification of unitary channels andvon Neumann measurements. More formally, for any adaptive certification scheme, theprobability of the type II error cannot be smaller than in the parallel scheme.
ERTIFICATION OF QUANTUM MEASUREMENTS 13 N = 1 N = 2 N = 3 N = 4 Figure 2.
Numerical ranges W ( U ⊗ N ) (polytops) and √ − δ -numericalranges W √ − δ ( U ⊗ N ) (ovals) of U ∈ U with eigenvalues 1 and e π i3 , for N = 1 , , , δ = 0 . p II is lower-bounded by the probability of the type II error for unitary chan-nels. Then, optimality of the parallel scheme for von Neumann measurements followsfrom Eq (41) together with the optimality of the parallel scheme for unitary channels’certification. Conclusions
In this work we studied the two-point certification of quantum states, unitary channelsand von Neumann measurements. The problem of certification of quantum objects isinextricably related with quantum hypothesis testing. We were interested in minimizingthe probability of type II error (probability of accepting null hypothesis when it waswrong) given the upper bound on the probability of type I error (probability of rejectingthe correct null hypothesis).In the case of certification of pure states we found the minimized probability of thetype II error and the optimal measurements which should be used for the certificationprocedure. As for the certification of unitary channels, we also found the minimizedprobability of the type II error as well as the optimal input state and measurement. Ontop of that, we pointed out the connection of certification of unitary channels with thenotion of q -numerical range. Moreover, it turned out that the use of entanglement doesnot improve the certification of unitary channels. We were also considering the certifi-cation of von Neumann measurements and found a formula for minimized probability ofthe type II error and the optimal certification strategy. Remarkably, it appeared thatin the case of certification of von Neumann measurements the use of entangled inputstates can indeed significantly improve the certification. In each scenario, we providedescription of the optimal strategy, proving that the optimal input state does not dependon statistical significance level, while the optimal measurement does depend on δ .Finally, we focused on the certification of the aforementioned quantum objects inthe parallel scheme. More precisely, we generalized the above results for the situationwhen the quantum objects can be used N times in parallel. Remarkably, it turned outthat optimal certification of von Neumann measurements can be performed without anyprocessing in the parallel scheme. Acknowledgments
This work was supported by the Foundation for Polish Science (FNP) under grantnumber POIR.04.04.00-00-17C1/18-00.We would like to thank Bart(cid:32)lomiej Gardas for fruitful discussions.
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Appendix A. Certification of states
In this appendix we present the proof of Theorem 1.
Proof of Theorem 1.
Without loss of generality we can assume that | ϕ (cid:105) = α | ψ (cid:105) + β | ψ ⊥ (cid:105) ,for some α, β ≥ α + β = 1. For any effect (cid:101) Ω satisfying (cid:104) ψ | (cid:101) Ω | ψ (cid:105) ≥ − δ , theeffect Ω defined as Ω = Π (cid:101) ΩΠ, where Π = | ψ (cid:105)(cid:104) ψ | + | ψ ⊥ (cid:105)(cid:104) ψ ⊥ | , also satisfies the condition (cid:104) ψ | Ω | ψ (cid:105) ≥ − δ and simultaneously returns the same value of probability of type II error.Hence, we can assume that rank-2 operator Ω satisfies Ω = ΠΩΠ. From the above, letΩ = a Π + b | ω (cid:105)(cid:104) ω | , where | ω (cid:105) = c | ψ (cid:105) − d | ψ ⊥ (cid:105) , c ≥ d ∈ C , such that c + | d | = 1 and a, b ∈ [0 , a + b ≤
1. By the assumption on the value p I , we have p I (Ω) = (cid:104) ψ | Ω | ψ (cid:105) = a + bc ≥ − δ. (42)Let us calculate the probability p II : p II = min Ω: p I (Ω) ≤ δ (cid:104) ϕ | Ω | ϕ (cid:105) = min a,b,c,d ∈A (cid:0) α ( a + bc ) + β ( a + b | d | ) − αβbc (cid:60) ( d ) (cid:1) (43)where A := { a, b, c, d : a + b ≤ , a + bc ≥ − δ, c + | d | = 1 , a, b, c ∈ [0 , , d ∈ C } .Note that the above formula is minimized when d ∈ R is nonnegative. Hence (cid:104) ϕ | Ω | ϕ (cid:105) = a + b ( αc − βd ) . (44)Thus, our task reduces to minimizing the formula p II = min a,b,c ∈B a + b (cid:16) αc − β (cid:112) − c (cid:17) (45)where B := { a, b, c ∈ [0 , , a + b ≤ , a + bc ≥ − δ } . We consider two cases.(1) If α ≤ √ δ , then we take a = 0 , b = 1 , c = β, d = (cid:112) − β . In this case a, b, c, ∈ B and we obtain p II = 0. The optimal strategy is represented by effect Ω = | ω (cid:105)(cid:104) ω | ,where | ω (cid:105) = β | ψ (cid:105) − α | ψ ⊥ (cid:105) .(2) Let α > √ δ and take a = 0 , b = 1 , c = √ − δ, d = (cid:112) − √ − δ . Again a, b, c, ∈ B and p II = (cid:16) α √ − δ − β √ δ (cid:17) . The optimal strategy is representedby effect Ω = | ω (cid:105)(cid:104) ω | where | ω (cid:105) = √ − δ | ψ (cid:105) − √ δ | ψ ⊥ (cid:105) . The optimality of thisvalue can be checked by using standard constrained optimization techniques. ERTIFICATION OF QUANTUM MEASUREMENTS 17 (cid:3)
Appendix B. q -numerical range and certification of unitary channels B.1. q -numerical range in the problem of two-point certification of unitarychannels. In this appendix we will present an alternative derivation the result for theprobability of the type II error in the certification of unitary channels given in Eq. (30).We would like to bound the probability of the type I error by δ , that is p | ψ (cid:105) I (Ω) =tr((1l − Ω) | ψ (cid:105)(cid:104) ψ | ) ≤ δ . Let us consider Ω = | ω (cid:105)(cid:104) ω | . Hence, we havetr (Ω | ψ (cid:105)(cid:104) ψ | ) = |(cid:104) ω | ψ (cid:105)| ≥ − δ. (46)The probability of the type II error takes the form p II = min | ψ (cid:105) min Ω: p | ψ (cid:105) I (Ω) ≤ δ tr (cid:16) Ω( U ⊗ | ψ (cid:105)(cid:104) ψ | ( U † ⊗ (cid:17) = min | ψ (cid:105) min | ω (cid:105) : p | ψ (cid:105) I ( | ω (cid:105)(cid:104) ω | ) ≤ δ (cid:104) ψ | ( U † ⊗ | ω (cid:105)(cid:104) ω | ( U ⊗ | ψ (cid:105) = min | ψ (cid:105) min | ω (cid:105) : p | ψ (cid:105) I ( | ω (cid:105)(cid:104) ω | ) ≤ δ |(cid:104) ψ | ( U ⊗ | ω (cid:105)| . (47)Let us recall that the q -numerical range is defined as W q ( A ) = {(cid:104) ξ | A | ξ (cid:105) : (cid:104) ξ | ξ (cid:105) = q } . (48)Now from the definition of the q -numerical range for q = √ − δ and its properties givenby Eq. (13) and (14) we obtain that p II = ν √ − δ ( U ⊗ ν √ − δ ( U ) (49)from which we conclude that the use of entanglement for the case of certification ofunitary channels does not improve the certification.B.2. Distance of q -numerical range to zero. In this subsection we will focus oncalculating the distance from the q -numerical range the to the origin of the coordinatesystem. Let us begin with the two-dimensional case when the unitary matrix U has twoeigenvalues λ and λ . Without loss of generality we can assume λ = 1. From [12]we know that the q -numerical range is an elliptical disc with eccentricity equal to q andfoci qλ and qλ , see Fig 3. Let c denote the distance from the center of the ellipse tothe focus and a be the distance from the center of the ellipse to its vertex. Using thisnotation the eccentricity yields q = c/a . Let b denote the distance from the center ofthe ellipse to its co-vertex, which it the point which saturates the minimum.First, we will calculate b . We note that c = 12 (cid:107) qλ − qλ (cid:107) = q (cid:107) λ − λ (cid:107) = √ − δ (cid:107) λ − λ (cid:107) . (50)From the properties of the ellipse and the form of the eccentricity q we have b = (cid:112) a − c = (cid:115) c q − c = c (cid:114) q − c (cid:114) − δ − c (cid:114) δ − δ . (51) ν qλ qλ b c a Figure 3.
Schematic illustration of an ellipse and notation used in Ap-pendix, where we use shortcut notation ν := ν q ( U ). BHence b = √ − δ (cid:107) λ − λ (cid:107) (cid:114) δ − δ = √ δ (cid:107) λ − λ (cid:107) . (52)On the other hand we have ν q ( U ) + b = (cid:13)(cid:13)(cid:13)(cid:13) qλ + qλ (cid:13)(cid:13)(cid:13)(cid:13) = q (cid:107) λ + λ (cid:107) = √ − δ (cid:107) λ + λ (cid:107) (53)and therefore ν q ( U ) = √ − δ (cid:107) λ + λ (cid:107) − √ δ (cid:107) λ − λ (cid:107) = 12 (cid:16) √ − δ (cid:107) λ + λ (cid:107) − √ δ (cid:107) λ − λ (cid:107) (cid:17) . (54)Now we need to show that the above expression for the distance ν q ( U ) is valid also forhigher dimensions. The boundary of q -numerical ranges for larger matrices is describedin [12]. It consists of parts of a few ellipses obtained is an analogous way. Let λ and λ d be the pair of the most distant eigenvalues of U . Let λ i and λ j bo some pair ofeigenvalues such that i, j (cid:54) = 1 , d . Let (cid:101) ν q ( U ) be the distance from zero the ellipse builton λ i and λ j in the same way as above. Our goal is to prove that (cid:101) ν q ( U ) > ν q ( U ).We note that (cid:107) λ − λ (cid:107) > (cid:107) λ i − λ j (cid:107) . Hence to prove that (cid:101) ν q ( U ) > ν q ( U ) it sufficesto show that (cid:107) λ + λ (cid:107) < (cid:107) λ i + λ j (cid:107) . As all the eigenvalues lie on the unit circle, thefrom the parallelogram law we have (cid:107) λ + λ (cid:107) = 4 − (cid:107) λ − λ (cid:107) . Therefore (cid:107) λ + λ (cid:107) = (cid:113) − (cid:107) λ − λ (cid:107) < (cid:113) − (cid:107) λ i − λ j (cid:107) = (cid:114) − (cid:16) − (cid:107) λ i + λ j (cid:107) (cid:17) = (cid:107) λ i + λ j (cid:107) . (55) ERTIFICATION OF QUANTUM MEASUREMENTS 19 and thus (cid:101) ν q ( U ) > ν q ( U ), from which it follows that ν √ − δ ( U ) = 12 (cid:16) √ − δ (cid:107) λ + λ d (cid:107) − √ δ (cid:107) λ − λ d (cid:107) (cid:17) (56)holds for any dimension d . The above formula can be easily translated into trigonometricfunctions where Θ is the angle between λ and λ d . Hence, we have ν √ − δ ( U ) = √ − δ cos (cid:18) Θ2 (cid:19) − √ δ sin (cid:18) Θ2 (cid:19) . (57)Therefore, p II = ν √ − δ ( U ⊗ ν √ − δ ( U ) = (cid:18) √ − δ cos (cid:18) Θ2 (cid:19) − √ δ sin (cid:18) Θ2 (cid:19)(cid:19) . (58) Appendix C. Certification of von Neumann measurements
In this Appendix we begin with proving a few technical lemmas and later present theproof of Theorem 3. The first lemma is the data processing inequality. This inequality,along with its proof, can be found eg. in [40]. However, to keep this work self-consistentwe present our modified version of them.
Lemma 1. (Data processing inequality) Let δ > and Ω be a positive semidefiniteoperator such that Ω ≤ . For any quantum channel Φ and quantum states ρ, σ thefollowing holds min Ω:tr(Ω ρ ) ≥ − δ tr(Ω σ ) ≤ min Ω:tr(ΩΦ( ρ )) ≥ − δ tr(ΩΦ( σ )) . (59) Proof.
Let us consider two-point certification of two quantum states ρ and σ with sta-tistical significance δ . To calculate the probability of the type II error, p II , we formulatethe problem as min Ω: tr(Ω ρ ) ≥ − δ tr(Ω σ ) . (60)Now, consider the scenario in which we use as processing the quantum channel Φ onstates ρ and σ . We want to calculatemin Ω:tr(ΩΦ( ρ )) ≥ − δ tr(ΩΦ( σ )) (61)which is equivalent to min Ω:tr(Φ † (Ω) ρ ) ≥ − δ tr(Φ † (Ω) σ ) . (62)It easy to see that Φ † (Ω) is also a measurement and { Φ † (Ω) : tr(Φ † (Ω) ρ ) ≥ − δ } ⊆ { Ω : tr(Ω ρ ) ≥ − δ } . (63)Eventually, we obtain the data processing inequality given bymin Ω:tr(Ω ρ ) ≥ − δ tr(Ω σ ) ≤ min Ω:tr(ΩΦ( ρ )) ≥ − δ tr(ΩΦ( σ )) . (64) (cid:3) The following lemma is proven in the work [25].
Lemma 2. (Lemma 5 from [25], direct implication) Assume that E ∈ DU d satisfiesthe condition || Φ UE − Φ || (cid:5) = ||P U − P || (cid:5) < . (65) Let λ , λ d be a pair of the most distant eigenvalues of U E and Π , Π d be the projectorsonto the subspaces spanned by the eigenvectors corresponding to λ and λ d , respectively.Then, there exist states ρ , ρ d , satisfying the following conditions ρ = Π ρ Π ρ d = Π d ρ d Π d diag( ρ ) = diag( ρ d ) . (66)The next proposition follows directly from Lemma 2. Corollary 7.
Let ρ = ρ + ρ d be the state satisfying conditions given by Eq. (66) .Then, for each i ∈ { , . . . , d } we have tr ( √ ρ | i (cid:105)(cid:104) i |√ ρ ) = tr (cid:16) √ ρ U | i (cid:105)(cid:104) i | U † √ ρ (cid:17) . (67) Moreover, for each i ∈ { , . . . , d } such that (cid:104) i | ρ | i (cid:105) (cid:54) = 0 we get (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) i | ρ U | i (cid:105)(cid:104) i | ρ | i (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) λ + λ d (cid:12)(cid:12)(cid:12)(cid:12) . (68) Proof.
Let U = (cid:80) di =1 λ i Π i , where { Π i } di =1 is a set of orthogonal projectors. Thentr (cid:16) √ ρ U | i (cid:105)(cid:104) i | U † √ ρ (cid:17) = (cid:104) i | U † ρU | i (cid:105) = (cid:104) i | U † (cid:18) ρ + 12 ρ d (cid:19) U | i (cid:105) = (cid:104) i | U † (cid:18)
12 Π ρ Π + 12 Π d ρ d Π d (cid:19) U | i (cid:105) = (cid:104) i | (cid:32) d (cid:88) i =1 λ i Π † i (cid:33) (cid:18)
12 Π ρ Π + 12 Π d ρ d Π d (cid:19) (cid:32) d (cid:88) i =1 λ i Π i (cid:33) | i (cid:105) = (cid:104) i | (cid:18) ρ + 12 ρ d (cid:19) | i (cid:105) = tr ( √ ρ | i (cid:105)(cid:104) i |√ ρ ) . (69)where the third equality follows from Lemma 2.To prove the second part of the proposition we calculate (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) i | ρ U | i (cid:105)(cid:104) i | ρ | i (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) i | (cid:0) ρ + ρ d (cid:1) (cid:16)(cid:80) di =1 λ i Π i (cid:17) | i (cid:105)(cid:104) i | ρ | i (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) i | (cid:80) di =1 λ i (cid:0) Π ρ Π + Π d ρ d Π d (cid:1) Π i | i (cid:105)(cid:104) i | ρ | i (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) i | (cid:0) λ Π ρ Π + λ d Π d ρ d Π d (cid:1) | i (cid:105)(cid:104) i | ρ | i (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) i | (cid:0) λ ρ + λ d ρ d (cid:1) | i (cid:105)(cid:104) i | ρ | i (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) λ + λ d (cid:12)(cid:12)(cid:12)(cid:12) . (70) ERTIFICATION OF QUANTUM MEASUREMENTS 21 (cid:3)
Proof of Theorem 3.
In the scheme of certification of von Neumann measurements theoptimized probability of type II error can be expressed as p II = min | ψ (cid:105) min Ω: p | ψ (cid:105) I (Ω) ≤ δ tr (Ω ( P U ⊗ | ψ (cid:105)(cid:104) ψ | )) . (71)Our goal is to prove that p II = max E ∈DU d ν √ − δ ( U E ) . (72)The proof is divided into two parts. In the first part we will utilize data processinginequality (Lemma 1) to show the lower bound for p II . In the second part we will useCorollary 7 to show the upper bound for p II . The lower bound.
This part of the proof mostly will be based on data processing in-equality. To show that p II ≥ max E ∈DU d ν √ − δ ( U E ) (73)let us begin with an observation that every quantum von Neumann measurement P U can be rewritten as ∆ ◦ Φ UE , where ∆ denotes the completely dephasing channel and E ∈ DU d . Therefore, utilizing data processing inequality in Lemma 1, along with thecertification scheme of unitary channels in Theorem 2, the optimized probability of thetype II error is lower-bounded by p II ≥ min | ψ (cid:105) min Ω: p | ψ (cid:105) I (Ω) ≤ δ tr(Ω(Φ UE ⊗ | ψ (cid:105)(cid:104) ψ | )) = ν √ − δ ( U E ) (74)which holds for each E ∈ DU d . Hence, maximizing the value of ν √ − δ ( U E ) over E ∈DU d leads to the lower bound of the form p II ≥ max E ∈DU d ν √ − δ ( U E ) . (75) The upper bound.
Now we proceed to proving the upper bound. The proof of theinequality p II ≤ max E ∈DU d ν √ − δ ( U E ) (76)will be divided into two cases depending on diamond norm distance between consideredmeasurements P U and P . In either case we will construct a strategy, that is choose astate | ψ (cid:105) and a measurement Ω . As for every choice of | ψ (cid:105) and Ω it holds that p II ≤ tr (Ω( P U ⊗ | ψ (cid:105)(cid:104) ψ | )) , (77)we will show that for some fixed | ψ (cid:105) and Ω it holds thattr (Ω ( P U ⊗ | ψ (cid:105)(cid:104) ψ | )) = max E ∈DU d ν √ − δ ( U E ) . (78)First we focus on the case when (cid:107)P U − P (cid:107) (cid:5) = 2. We take a state | ψ (cid:105) for which itholds that (cid:107)P U − P (cid:107) (cid:5) = (cid:107) (( P U − P ) ⊗ | ψ (cid:105)(cid:104) ψ | ) (cid:107) . (79) Then, the output states ( P U ⊗ | ψ (cid:105)(cid:104) ψ | ) and ( P ⊗ | ψ (cid:105)(cid:104) ψ | ) are orthogonal and bytaking the measurement Ω as the projection onto the support of ( P ⊗ | ψ (cid:105)(cid:104) ψ | ) weobtain tr (Ω ( P U ⊗ | ψ (cid:105)(cid:104) ψ | )) = 0 . (80)On the other hand, utilizing Eq. (35) and (34) we obtain that max E ∈DU d ν ( U E ) = 0.Therefore, by the property that 0 ∈ W √ − δ ( U E ) whenever 0 ∈ W ( U E ) (see AppendixB), we have that max E ∈DU d ν √ − δ ( U E ) = 0 . (81)Secondly, we consider the situation when (cid:107)P U − P (cid:107) (cid:5) < E ∈ arg max E ∈DU d ν ( U E ) . (82)Again, by referring to Eq. (34) and (35) we obtain that ν ( U E ) >
0. Let λ , λ d be apair of the most distant eigenvalues of U E . Note that the following relation holds ν ( U E ) = | λ + λ d | . (83)As the assumptions of the Lemma 2 are saturated for the defined E , we consider theinput state | ψ (cid:105) = d (cid:88) i =1 (cid:113) ρ (cid:62) | i (cid:105) ⊗ | i (cid:105) (84)where ρ is given by Lemma 2. Define sets C i := (cid:26) Ω : 0 ≤ Ω ≤ , tr (cid:18) (1l − Ω) √ ρ | i (cid:105)(cid:104) i |√ ρ (cid:104) i | ρ | i (cid:105) (cid:19) ≤ δ (cid:27) (85)for each i such that (cid:104) i | ρ | i (cid:105) (cid:54) = 0. Now we take the measurement Ω asΩ = d (cid:88) i =1 | i (cid:105)(cid:104) i | ⊗ Ω i (86)where Ω i ∈ C i is defined astr (cid:32) Ω i √ ρ U | i (cid:105)(cid:104) i | U † √ ρ (cid:104) i | ρ | i (cid:105) (cid:33) := min (cid:101) Ω ∈C i tr (cid:32)(cid:101) Ω √ ρ U | i (cid:105)(cid:104) i | U † √ ρ (cid:104) i | ρ | i (cid:105) (cid:33) (87)for each i ∈ { , . . . , d } such that (cid:104) i | ρ | i (cid:105) (cid:54) = 0 and Ω i = 0 otherwise.Now we check that the statistical significance is satisfied, that is for the describedstrategy we have p | ψ (cid:105) I (Ω ) = 1 − tr (Ω ( P ⊗ | ψ (cid:105)(cid:104) ψ | )) = 1 − d (cid:88) i =1 tr (Ω i √ ρ | i (cid:105)(cid:104) i |√ ρ ) ≤ δ. (88)Hence, it remains to show that for this settingtr (Ω ( P U ⊗ | ψ (cid:105)(cid:104) ψ | )) = max E ∈DU d ν √ − δ ( U E ) . (89) ERTIFICATION OF QUANTUM MEASUREMENTS 23
Direct calculations reveal thattr (Ω ( P U ⊗ | ψ (cid:105)(cid:104) ψ | )) = d (cid:88) i =1 tr (cid:16) Ω i √ ρ U | i (cid:105)(cid:104) i | U † √ ρ (cid:17) = d (cid:88) i =1 (cid:104) i | ρ | i (cid:105) tr (cid:32) Ω i √ ρ U | i (cid:105)(cid:104) i | U † √ ρ (cid:104) i | ρ | i (cid:105) (cid:33) . (90)Let us define p | i II = tr (cid:32) Ω i √ ρ U | i (cid:105)(cid:104) i | U † √ ρ (cid:104) i | ρ | i (cid:105) (cid:33) . (91)Note that due to Corollary 7 the states √ ρ | i (cid:105)(cid:107)√ ρ | i (cid:105)(cid:107) and √ ρ U | i (cid:105)(cid:107)√ ρ | i (cid:105)(cid:107) are both pure and for every i ∈ { , . . . , d } : (cid:104) i | ρ | i (cid:105) (cid:54) = 0 the inner product between them is the same. Therefore we canconsider the certification of pure states conditioned on the obtained label i with statisticalsignificance δ . From the Theorem 1 we know that p | i II depends only on such an innerproduct between the certified states, hence p | i II = p | j II for each i, j : (cid:104) i | ρ | i (cid:105) , (cid:104) j | ρ | j (cid:105) (cid:54) = 0.Therefore, from Corollary 7 we have that the value of p | i II will depend on (cid:12)(cid:12)(cid:12) λ + λ d (cid:12)(cid:12)(cid:12) . Thusw.l.o.g. we can assume that p | (cid:54) = 0 and hence d (cid:88) i =1 (cid:104) i | ρ | i (cid:105) p | i II = p | = tr (cid:32) Ω √ ρ U | (cid:105)(cid:104) | U † √ ρ (cid:104) | ρ | (cid:105) (cid:33) (92)and in the remaining of the proof we will show that p | = max E ∈DU d ν √ − δ ( U E ) . (93)It is sufficient to study two cases depending on the relation between √ δ and the innerproduct (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) | ρ U | (cid:105)(cid:104) | ρ | (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) λ + λ d (cid:12)(cid:12)(cid:12)(cid:12) . (94)In the case when (cid:12)(cid:12)(cid:12) λ + λ d (cid:12)(cid:12)(cid:12) ≤ √ δ , then due to Theorem 1 we get p | = 0. On the otherhand, from Section 4 we know that 0 ∈ W √ − δ ( U E ) and hence alsomax E ∈DU d ν √ − δ ( U E ) = 0 . (95)In the case when (cid:12)(cid:12)(cid:12) λ + λ d (cid:12)(cid:12)(cid:12) > √ δ , then from Theorem 1 we know that p | = (cid:12)(cid:12)(cid:12)(cid:12) λ + λ d (cid:12)(cid:12)(cid:12)(cid:12) √ − δ − (cid:115) − (cid:12)(cid:12)(cid:12)(cid:12) λ + λ d (cid:12)(cid:12)(cid:12)(cid:12) √ δ . (96)On the other hand, for E ∈ DU d satisfying Eq. (82) we have ν √ − δ ( U E ) = (cid:12)(cid:12)(cid:12)(cid:12) λ + λ d (cid:12)(cid:12)(cid:12)(cid:12) √ − δ − (cid:115) − (cid:12)(cid:12)(cid:12)(cid:12) λ + λ d (cid:12)(cid:12)(cid:12)(cid:12) √ δ . (97) By the particular choice of E ∈ DU d , this value is equal to max E ∈DU d ν √ − δ ( U E ),hence combining the above equations we finally obtain p | = max E ∈DU d ν √ − δ ( U E ) . (98)To sum up, we indicated strategies Ω and | ψ (cid:105) for which the optimized probability oftype II error was equal to max E ∈DU d ν √ − δ ( U E ). Combining this with the previouslyproven inequality p II ≥ max E ∈DU d ν √ − δ ( U E ) (99)gives us Eq. (72) and proves that the proposed strategy | ψ (cid:105) , Ω is optimal. (cid:3) Finally, we present a remark which describes the optimal strategy based on the proofof Theorem 3.
Remark 5.
Construction of an optimal strategy for the certification of von Neumannmeasurements depends on the value of ||P U − P || (cid:5) . In the case ||P U − P || (cid:5) = 2 , thenthe optimal state | ψ (cid:105) is the state for which (cid:107)P U − P (cid:107) (cid:5) = (cid:107) (( P U − P ) ⊗ | ψ (cid:105)(cid:104) ψ | ) (cid:107) (100) while the optimal measurement Ω is the projection onto the support of ( P ⊗ | ψ (cid:105)(cid:104) ψ | ) When ||P U −P || (cid:5) < , then the optimal input state is | ψ (cid:105) = (cid:80) di =1 (cid:113) ρ (cid:62) | i (cid:105)⊗| i (cid:105) , where ρ is given by Lemma 2. The optimal measurement has the form Ω = (cid:80) di =1 | i (cid:105)(cid:104) i | ⊗ Ω i where Ω i are defined in Eq. (87) .Similarly to the case of unitary channel certification, the optimal input state | ψ (cid:105) doesnot depend on δ , while the optimal measurement Ω does depend on δδ