On the Optimal Certification of Von Neumann Measurements

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OPEN On the optimal certifcation of von Neumann measurements Paulina Lewandowska1, Aleksandra Krawiec1, Ryszard Kukulski1, Łukasz Pawela1* & Zbigniew Puchała1,2

In this report we study certifcation of quantum measurements, which can be viewed as the extension of quantum hypotheses testing. This extension involves also the study of the input state and the measurement procedure. Here, we will be interested in two-point (binary) certifcation scheme in which the null and alternative hypotheses are single element sets. Our goal is to minimize the probability of the type II error given some fxed statistical signifcance. In this report, we begin with studying the two-point certifcation of pure quantum states and unitary channels to later use them to prove our main result, which is the certifcation of von Neumann measurements in single- shot and parallel scenarios. From our main result follow the conditions when two pure states, unitary operations and von Neumann measurements cannot be distinguished perfectly but still can be certifed with a given statistical signifcance. Moreover, we show the connection between the certifcation of quantum channels or von Neumann measurements and the notion of q-numerical range.

Te validation of sources producing quantum states and measurement devices, which are involved in quantum computation workfows, is a necessary step of quantum technology1–3. Te search for practical and reliable tools for validation of quantum architecture has attracted a lot of attention in recent years4–8. Rapid technology devel- opment and increasing interest in quantum computers paved the way towards creating more and more efcient validation methods of Noisy Intermediate-Scale Quantum devices (NISQ)9,10. Such a growth comes along with ever-increasing requirements for the precision of the components of quantum devices. Te tasks of ensuring the correctness of quantum devices are referred to as validation. Let us begin with sketching the problem of validation of quantum architectures. Imagine you are given a black box and are promised two things. First, it contains a pure quantum state (or a unitary matrix or a von Neumann POVM), and second, it contains one of two possible choices of these objects. Te owner of the box, Eve, tells you which of the two possibilities is contained within the box. Yet, for some reason, you do not completely trust her and decide to perform some kind of hypothesis testing scheme on the black box. You decide to take Eve’s promise as the null hypothesis, H0 , for this scheme and the second of the possibilities as the alternative hypothesis, H1 . Since now you own the box and are free to proceed as you want, you need to prepare some input into the box and perform a measurement on the output. A particular input state and fnal measurement (or only the measurement, for the case when the box contains a quantum state) will be called a certifcation strategy. Of course, just like in classical hypothesis testing, in our certifcation scheme we have two possible types of errors. Te type I error happens if we reject the null hypothesis when it was in reality true. Te type II error happens if we accept the null hypothesis when we should have rejected it. Te main aim of certifcation is fnding the optimal strategy which minimizes one type of error when the other one is fxed. In this work we are interested in the minimization of the type II error given a fxed type I error. Tis approach will be called certifcation. Certifcation of quantum objects is closely related with the other well-know method of validation, that is the problem of discrimination of those objects. Intuitively, in the discrimination problem we are given one of two quantum objects sampled according to a given a priori probability distribution. Hence, the probability of making an error in the discrimination task is equal to the average of the type I and type II errors over the assumed probability distribution. Terefore, the discrimination problem can be seen as symmetric distinguishability, as opposed to certifcation, that is asymmetric distinguishability. In other words, the main diference between both approaches is that the main task of discrimination is the minimization over the average of both types of possible errors while the certifcation concerns the minimization over one type of error when the bound of the other one is assumed. While in the basic version of the aforementioned scenario we focus on the case when the validated quantum object can be used exactly once, one can consider also the situation in which this object can be utilized multiple

1Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, ul. Bałtycka 5, 44‑100 Gliwice, Poland. 2Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, ul. Łojasiewicza 11, 30‑348 Kraków, Poland. *email: [email protected]

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times in various confgurations. In the parallel scheme the validated object can be used many times, but no processing can be performed between the usages of this object. In the adaptive scenario however, we are allowed to perform any processing we want between the uses of the validated quantum object. Te problem of discrimination of quantum states and channels was solved analytically by Helstrom a few decades ago in11,12. Te multiple-shot scenario of the discrimination of quantum states was studied in13,14 whereas the discrimination of quantum channels in the multiple-shot scenario was investigated for example in15–19. Examples of channels which cannot be discriminated perfectly in the parallel scheme, but nonetheless can be discriminated perfectly using the adaptive approach, were discussed in20,21. Te work 22 paved the way for studying the discrimination of quantum measurements. Terefore, this work can be seen as a natural extension of our works 23,24, where we studied the discrimination of von Neumann measurements in single and multiple-shot scenarios, respectively. Nevertheless, one can also consider a scenario in which we are allowed to obtain an incon- clusive answer. Terefore, we arrive at the unambiguous discrimination of quantum operations discussed in24,25. Te problem of certifcation of quantum states and channels has not been studied as exhaustively as their discrimination. Te certifcation of quantum objects was frst studied by Helstrom in11, where the problem of pure state certifcation was considered. Further, certifcation schemes were established to the case of mixed states in26. A natural extension of quantum state certifcation is the certifcation of unitary operations. Tis problem was solved in 27. Considerations about multiple-shot scenario of certifcation of quantum states and unitary channels were investigated in 27. All the above-mentioned approaches towards the certifcation were considered in a fnite number of steps. Another common approach involves studying certifcation of quantum objects in the asymptotic regime28–30 which assumes that the number of copies of the given quantum object goes to infnity. It focuses on studying the convergence of the probability of making one type of error while a bound on the second one is assumed. Tis task is strictly related with the term of relative entropy and its asymptotic behavior31. For a more general overview of quantum certifcation we refer the reader to32,33. Tis work will begin with recalling one-shot certifcation scenario which will later be extended to the multiple-shot case. For this purpose, we will ofen make use of the terms of numerical range and q-numerical range as essential tools in the proofs 34–37. More specifcally, one of our results presented in this work is a geometric interpretation of the formula for minimized probability of the type II error in the problem of certifcation of unitary channels, which is strictly connected with the notion of q-numerical range. Later, basing on the results on the certifcation of unitary channels we will extend these considerations to the problem of certifcation of von Neumann measurements. It will turn out that the formula for minimized probability of the type II error can also be connected with the notion of q-numerical range. On top of that, we will show that entanglement can signifcantly improve the certifcation of von Neumann measurements. Eventually, we will prove that the parallel certifcation scheme is optimal. Tis work is organized as follows. We begin with preliminaries in “Preliminaries” section. Ten, in “Two- point certifcation of pure states” section we present the two-point certifcation of pure quantum states. Certifca- tion of unitary channels is discussed in “Certifcation of unitary channels” section. Afer presenting the known results we introduce geometrical interpretation of the problem of certifcation of unitary channels, expressed in terms of q-numerical range. Te certifcation of von Neumann measurements is studied in “Two-point certifcation of von Neumann measurements” section and our main result is stated therein as Teorem 3. “Parallel multiple-shot certifcation” section generalizes the results on certifcation to the multiple-shot scenario and the optimality of the parallel scheme for certifcation of von Neumann measurements is presented as Teorem 5. Preliminaries Let M be the set of all matrices of dimension d1 d2 over the feld C . For the sake of simplicity, square matri- d1,d2 × ces will be denoted by Md . Te set of quantum states, that is positive semidefnite operators having trace equal to one, will be denoted D . By default, when we write ψ , ϕ , we mean normalized pure states, unless we mention d | � | � otherwise. Te subset of Md consisting of unitary matrices will be denoted by Ud , while its subgroup of diagonal unitary operators will be denoted by DUd . Let U Ud be a unitary matrix. A unitary channel U is defned as † ∈ �U ( ) U U . A general quantum measurement, that is a positive operator valued measure (POVM) P is a · = · m collection of positive semidefnite operators E1, ..., Em called efects, which sum up to identity, i.e. Ei 1 . { } i 1 = If all the efects are rank-one projection operators, then such a measurement is called von Neumann= measurement. Every von Neumann measurement can be parameterized by a unitary matrix and hence we will use the notation PU for a von Neumann measurement with efects u1 u1 , ..., u u , where ui is the i-th column {| �� | | d�� d|} | � of the unitary matrix U. Te action of quantum measurement PU on some state ρ D can be expressed as the ∈ d action of a quantum channel d PU ρ ui ρ ui i i . (1) : → � | | �| �� | i 1 = As mentioned in the “Introduction” section, in this work we focus on two-point hypothesis testing of quantum objects. Te starting point towards the certifcation of quantum objects is the hypothesis testing of quantum states. Let H0 be a null hypothesis which states that the obtained state was ψ , while the alternative hypothesis, | � H1 , states that the obtained state was ϕ . Te certifcation is performed by the use of a binary measurement | � , 1 , where the efect corresponds to accepting the null hypothesis and 1 accepts the alternative { − } − hypothesis. In this work we will be considering only POVMs with two efects of this form. Terefore the efect uniquely determines the POVM and hence we will be using the words measurement and efect interchangeably.

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Assume we have a fxed measurement . We introduce the probability of the type I error, pI(�) , that is the probability of rejecting the null hypothesis when in fact it was true, as p (�) tr ((1 �) ψ ψ ) 1 tr (� ψ ψ ). I = − | �� | = − | �� | (2) Te type II error, pII(�) , that is the probability of accepting the null hypothesis H0 when in reality H1 occurred, is defned as p (�) tr (� ϕ ϕ ). II = | �� | (3) In the remainder of this work we will assume the statistical signifcance δ 0, 1 , that is the probability of the ∈[ ] type I error will be upper-bounded by δ . Our goal will be to fnd the most powerful test, that is to minimize the probability of the type II error by fnding the optimal measurement, which we will denote as 0 . Such 0 , which minimizes pII(�) while assuming the statistical signifcance δ , will be called an optimal measurement. Te minimized probability of type II error will be denoted by

pII min pII(�). := � p (�) δ (4) : I ≤ While certifying quantum channels and von Neumann measurements, we will also need to minimize over input states. Let a channel 0 correspond to hypothesis H0 and 1 correspond to hypothesis H1 . We defne ψ p| �(�) tr ((1 �)� ( ψ ψ )), I = − 0 | �� | ψ (5) p| �(�) tr (�� ( ψ ψ )). II = 1 | �� | Naturally, for each input state we can consider minimized probability of type II error, that is ψ ψ p| � min p| �(�). II ψ II (6) = � p| �(�) δ : I ≤ Finally, we will be interested in calculating optimized probability of type II error over all input states. Tis will be denoted as ψ pII min p| �. := ψ II (7) | � Note that the symbol pII is used in two contexts. In the problem of certifcation of states the minimization is performed only over measurements , while in the problem of certifcation of unitary channels and von Neumann measurements the minimization is over both measurements and input states ψ . In other words, pII is equal | � to the optimized probability of the type II error in certain certifcation problem. Te input state which minimizes pII will be called an optimal state. We will use the term optimal strategy to denote both the optimal state and the optimal measurement. Now, we introduce a basic toolbox for studying the certifcation of quantum objects which is strictly related with the problem of discrimination of quantum channels. First, we will be using the notion of the diamond norm. Te diamond norm of a superoperator is defned as

� max (� 1)(X) 1. � �⋄ := X 1 1 � ⊗ � (8) � � = Te celebrated theorem of Helstrom11 gives a lower bound on the probability of making an error in distinction in the scenario of symmetric discrimination of quantum channels. Te probability of incorrect symmetric discrimination between quantum channels and is bounded as follows 1 1 pe . (9) ≥ 2 − 4 � − �⋄ Moreover, our results will ofen make use of the terms of numerical range and q-numerical range37. Te numerical range is a subset of complex plane defned for a matrix X M as ∈ d W(X) ψ X ψ ψ ψ 1 := {� | | �:� | �= } (10) while the q-numerical range34–36 is defned for a matrix X M as ∈ d W (X) ψ X ϕ ψ ψ ϕ ϕ 1, ψ ϕ q, q C . q := {� | | �:� | �=� | �= � | �= ∈ } (11) Te standard numerical range is the special case of q-numerical range for q 1 , that is W(X) W1(X) . We = = will use the notation ν (X) min x x W (X) q := {| |: ∈ q } (12) to denote the distance on a complex plane from q-numerical range to zero. In the case when q 1 , we will simply = write ν(X) . Te main properties of q-numerical range are its convexity and compactness 36. Te detailed shape of q-numerical range is described in35. Te properties of q-numerical range15 that will be used throughout this paper are

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q′ W Wq for q q′, q, q′ R, q′ ⊆ q ≤ ∈ (13)

and W (X 1) W (X), q R. q ⊗ = q ∈ (14) From the above it is easy to see that ν (X 1) ν (X), q R. q ⊗ = q ∈ (15) In the Supplementary Materials we provide an animation of q-numerical range of unitary matrix U U3 with πi 2πi ∈ eigenvalues 1, e 3 and e 3 for all parameters q 0, 1 . ∈[ ] Two‑point certifcation of pure states In this section we recall the results concerning the certifcation of pure quantum states. We state the optimized probability of the type II error for the quantum hypothesis testing problem as well as the form of the optimal measurement which should be used for the certifcation. Although these results may seem quite technical, they will lay the groundwork for studying the certifcation of unitary channels and von Neumann measurements in further sections.

Certifcation scheme. Assume we are given one of two known quantum states either ψ or ϕ . Te hypoth- | � | � esis H0 corresponds to the state ψ , while the alternative hypothesis H1 corresponds to the state ϕ . In other | � | � words, our goal is to decide whether the given state was ψ or ϕ . To make a decision, we need to measure the | � | � given state and we are allowed to use any POVM. We will use a quantum measurement with efects , 1 , { − } where the frst efect accepts the hypothesis H0 and the second efect 1 accepts H1 . Hence, the probability − of obtaining the type I error is given by p (�) ψ (1 �) ψ . I =� | − | � (16) Te probability of obtaining the type II error to be minimized yields

pII min ϕ � ϕ ϕ �0 ϕ , = � p (�) δ� | | � =: � | | � (17) : I ≤ where the minimization is performed by fnding the optimal measurement 0. Tis problem was explored in11. However, to keep this work self-consistent we present in Online Appendix A in the Supplementary Materials an alternative version of the proof.

Theorem 1. Consider the problem of two-point certifcation of pure quantum states with hypotheses given by

H0 ψ , H :|ϕ �. (18) 1 :|� and statistical signifcance δ 0, 1 . Ten, for the most powerful test, the probability of the type II error (17) yields ∈[ ] 0 if ψ ϕ √δ, pII 2 |� | �| ≤ = ψ ϕ √1 δ 1 ψ ϕ 2√δ if ψ ϕ > √δ. (19) |� | �| − − − |� | �| |� | �| Te proof of the above theorem is presented in Online Appendix A in the Supplementary Materials. Tis proof gives a construction of the optimal measurement which minimizes the probability of the type II error. Te exact form of such an optimal measurement is stated as the following corollary.

Corollary 1. Te optimal strategy for two-point certifcation of pure quantum states ψ and ϕ , with statistical | � | � signifcance δ yields

ω (1) if ψ ϕ √δ , then the optimal measurement is given by �0 ω ω , where ω | � , |� | �| ≤ =| �� | | �= ω ω ψ ϕ ψ ϕ ; ||| �|| | �=| �−� | �| � (2) if ψ ϕ > √δ , then the optimal measurement is given by �0 ω ω for ω √1 δ ψ √δ ψ , |� | �| =| �� | | �= − | �− | ⊥� ψ⊥ ψ ⊥ | � , where ψ⊥ ϕ ψ ϕ ψ . | �= ψ⊥ | �=| �−� | �| � ||| �|| Certifcation of unitary channels In this section we will be interested in certifcation of two unitary channels U and V for U, V U . Without ∈ d loss of generality we can assume that one of these unitary matrices is the identity matrix and then our task reduces to certifcation of channels 1 and U . In the most general case, we are allowed to use entanglement by adding an additional system. Hence, the null hypothesis H0 yields that the unknown channel is 1 1 and the ⊗ alternative hypothesis H1 yields that the unknown channel is U 1. ⊗

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Certifcation scheme. Te idea behind the scheme of certifcation of unitary channels is to reduce this problem to certifcation of quantum states discussed in the previous section. We prepare some (possibly entangled) input state ψ and perform the unknown channel on it. Te resulting state is either (1 1) ψ or | � ⊗ | � (U 1) ψ . Ten, we perform the measurement , 1 and make a decision whether the given channel ⊗ | � { − } was 1 1 or U 1 . Te efect corresponds to accepting H0 hypothesis while 1 corresponds to the ⊗ ⊗ − alternative hypothesis H1. Te results of minimization of the probability of the type II error over input states ψ and measurements | � are summarized as the following theorem. Tis reasoning is based on the results from Teorem 1, while a related study of this problem can be found in 27.

Theorem 2. Consider the problem of two-point certifcation of unitary channels with hypotheses

H0 1 1, : ⊗ (20) H 1. 1 : U ⊗ and statistical signifcance δ 0, 1 . Ten, for the most powerful test, the probability of the type II error yields ∈[ ] 0 if ψ0 U ψ0 √δ, pII 2 |� | | �| ≤ = ψ U ψ √1 δ 1 ψ U ψ 2√δ if ψ U ψ > √δ, (21) |� 0| | 0�| − − − |� 0| | 0�| |� 0| | 0�| where ψ0 arg min ψ ψ U ψ . | �∈ | � |� | | �| Proof. Let us frst introduce the hypotheses conditioned by the input state ψ | � ψ H| � ψ , 0 :| � ψ (22) H| � (U 1) ψ . 1 : ⊗ | � We do not make any assumptions on the dimension of the auxiliary system for the time being. It will turn out, however, that it sufces if its dimension equals one. Te hypotheses in (22) correspond to output states afer the application of the extended unitary channel on the state ψ . For these hypotheses we consider the statistical | � signifcance δ 0, 1 , that is ∈[ ] ψ p| �(�) tr ((1 �)(�1 1)( ψ ψ )) δ. (23) I = − ⊗ | �� | ≤ Our goal will be to calculate the minimized probability of the type II error

pII min min tr (�(�U 1)( ψ ψ )) tr (�0(�U 1)( ψ0 ψ0 )), ψ ψ = � p| �(�) δ ⊗ | �� | =: ⊗ | �� | (24) | � : I ≤ ψ0 where naturally, for the optimal strategy ψ0 and 0 it holds that p| �(�0) δ. | � I ≤ Now we will show that the use of entanglement is unnecessary. From Teorem 1 we know that the probability of the type II error, pII , depends on the minimization of the inner product min ψ ψ U 1 ψ . Directly from | � |� | ⊗ | �| the defnition of numerical range we can see that ψ U 1 ψ W(U 1) . From the property of numerical � | ⊗ | �∈ ⊗ range given in Eq. (14) and using the notation introduced in Eq. (12) we have ν(U 1) ν(U). ⊗ = (25) Let ψ0 be the considered optimal input state, i.e. ψ0 arg min ψ ψ U ψ . Terefore we can reformulate | � | �∈ | � |� | | �| our hypotheses as ψ H| 0� ψ , 0 :| 0� ψ (26) H| 0� U ψ . 1 : | 0� Tese hypotheses, when taking ϕ U ψ0 , were the subject of interest in Teorem 1. | � := | � Te next corollary follows directly from the above proof.

Corollary 2. Entanglement is not needed for the certifcation of unitary channels.

Te following remark states that while considering the input state to the certifcation scheme, we can restrict our attention to pure states only.

Remark 1. Without loss of generality, we can consider only pure input states. Te minimal value of linear objective function ρ tr (�� (ρ)) → U (27) over a convex set ρ D tr (��1(ρ)) 1 δ is achieved on the extremal points. { ∈ d : ≥ − }

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√pII

Θ 2

2pe

√1 δ ( ) U3 Figure 1. Numericalπ irange 2Wπi (U) (red triangle) and -numerical range W√1 δ U (blue oval) of U − − ∈ with eigenvalues 1, e 3 and e 3 with statistical signifcance δ 0.05 . Te value pe is the probability of incorrect = symmetric discrimination of channels 1 and U.

Connection with q‑numerical range. Tere exists a close relationship between the above results and the defnition of numerical range, which can be seen from the proof of Teorem 2. It the work 15 the authors show the connection between the discrimination of quantum channels and q-numerical range. In this section we show the connection between certifcation of unitary channels and q-numerical range. Recall the defnition of q-numerical range. W (X) ψ X ϕ ψ ϕ q . q := {� | | �:� | �= } (28) Using this notion and the notation introduced in Eq. (12) we can rewrite our results for the probability of the type II error from Teorem 2 as ν2 ( 1) ν2 ( ) pII √1 δ U √1 δ U . (29) = − ⊗ = − An independent derivation of the above formula is presented in Online Appendix B in the Supplementary Materials. Let be the angle between two most distant eigenvalues of a unitary matrix U. Ten, from the above discus- sion we can draw a conclusion that for any statistical signifcance δ (0, 1 , if 2 arccos √δ �<π , then ∈ ] ≤ although U and 1 cannot be distinguished perfectly, they can be certifed with pII 0 . In other words, the = numerical range W(U) does not contain zero but √1 δ-numerical range, W√1 δ(U) , does contain zero. Te − − situation changes when 2 arccos √δ >� . Ten, both numerical range W(U) and √1 δ-numerical range − W√1 δ(U) do not contain zero. Tis is presented in Fig. 1. Now− we will work towards the construction of the optimal strategy, which will be stated as a corollary. Besides fnding the optimal measurement which was shown in previous section we will show a closed-form expression of the optimal input state. For this purpose we will make use of the spectral decomposition of a unitary matrix U given by d U i xi xi . (30) = | �� | i 1 = Let 1, d be a pair of the most distant eigenvalues of U. Te following corollary is analogous to the corollary from the previous section as it presents the optimal strategy for the certifcation of unitary channels.

Corollary 3. By ψ0 we will denote the optimal state for two-point certifcation of unitary channels and let | � ϕ U ψ0 . Ten, the optimal strategy yields | � := | �

(1) If 0 W√1 δ(U) , then we have two cases ∈ − • if 0 W(U) , then we can take ∈

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1 1 ψ0 x1 xd , (31) | �= √2 | �+ √2 | �

where x1 , xd are eigenvectors corresponding to the pair of the most distant eigenvalues 1 , d of U. Te | � | � ω optimal measurement is given by �0 ω ω , where ω | � , ω ψ0 ϕ ψ0 ϕ , =| �� | | �= ω | �=| �−� | �| � • if 0 W(U) , then we have perfect symmetric distinguishability.||| Moreover,�|| there exists the probability vector ∈ d p such that i 1 ipi 0 and we obtain that = = d ψ0 √pi xi . (32) | �= | � i 1 = ω Analogously, we choose the optimal measurement given by �0 ω ω , where ω | � , =| �� | | �= ω ω ψ0 ϕ ψ0 ϕ . It easy to see that in this case we have �0 ψ0 ψ0 . ||| �|| | �=| �−� | �| � =| �� | (2) If 0 W√1 δ(U) , then the discriminator is given by Eq. (31), whereas the optimal measurement is can ∈ − ψ0⊥ be expressed as �0 ω ω for ω √1 δ ψ0 √δ ψ0⊥ , ψ0⊥ | � , whe re =| �� | | �= − | �− | � | �= ψ ||| 0⊥�|| ψ0⊥ ϕ ψ0 ϕ ψ0 . | �=| �−� | �| �

Remark 2. Observe that the optimal input state ψ0 does not depend on δ , while the optimal measurement 0 | � 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 the problem of unitary channel discrimination. Two‑point certifcation of von Neumann measurements In this section we will focus on the certifcation of von Neumann measurements. Recall that every quantum measurement can be associated with a measure-and-prepare quantum channel. Terefore, while studying the certifcation of quantum measurements we will ofen take advantage of the certifcation of quantum channels discussed in the previous section, where we assumed that one of the unitaries was the identity. Similarly, also in the case of certifcation of von Neumann measurements we will assume that one of the measurements is in the computational basis. Hence, we will be certifying the measurement P1 under the alternative hypothesis PU. While certifying quantum channels, the most general scenario allows for the use of entanglement by adding an additional system. Hence, in our case of certifcation of von Neumann measurements, the hypothesis H0 yields that the unknown measurement is P1 1 whereas for the alternative hypothesis yields that the measurement ⊗ is PU 1. ⊗ Now we recall some technical tools which will be used to prove the main result of this work. It was shown 23 in (Teorem 1) that the diamond norm distance between von Neumann measurements PU and P1 is given by

PU P1 min UE 1 , || − ||⋄ = E DU || − ||⋄ (33) ∈ d where DUd 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 of discrimination of unitary channels. From12 we know that the diamond norm distance between two unitary channels U and 1 is expressed as

2 �U �1 2 1 ν (U), (34) || − ||⋄ = − where ν(U) min x x W(U) . = {| |: ∈ } Certifcation scheme. Te scenario of certifcation of von Neumann measurements is as follows. We prepare some (possibly entangled) input state ψ and, as previously, we perform the unknown von Neumann meas- | � urement on one part of it. Ten, afer performing the measurement, the null hypothesis H0 corresponds to the state (P1 1)( ψ ψ ) , while the alternative hypothesis H1 corresponds to the state (PU 1)( ψ ψ ) . Our ⊗ | �� | ⊗ | �� | goal is to fnd an optimal input state and a measurement for which the probability of the type II error is saturated, while the statistical signifcance δ is assumed. Te results of minimization are stated as the following theorem.

Theorem 3. Consider the problem of two-point certifcation of von Neumann measurements with hypotheses

H0 P1 1, : ⊗ (35) H P 1. 1 : U ⊗ and statistical signifcance δ 0, 1 . Ten, for the most powerful test, the probability of the type II error yields ∈[ ] 2 pII max ν (UE). = E DU √1 δ (36) ∈ d − It is worth mentioning that we do not make any assumptions on the dimension of the auxiliary system, however its dimension is obviously upper-bounded by the dimension of the input states. Additionally, the dimension of the auxiliary system can be reduced to the Schmidt rank of the input state ψ 23 (Proposition 4). It is worth | � mentioning here that in the certifcation of von Neumann measurements entanglement can signifcantly improve the outcome of the protocol while in the case of unitary channel certifcation it provides no beneft.

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In contrast to the certifcation of unitary channels, the output states (P1 1)( ψ ψ ) and (PU 1)( ψ ψ ) ⊗ | �� | ⊗ | �� | are not necessarily pure. Hence the proof of the Teorem 3 requires more advanced techniques. Luckily, we still can make use of the calculations from “Certifcation of unitary channels” section, due to the fact that formally mixed states (P1 1)( ψ ψ ) and (PU 1)( ψ ψ ) , conditioned by obtaining the label i 1, ..., d , are pure. ⊗ | �� | ⊗ | �� | ∈{ } Proof of Theorem 3. In the scheme of certifcation of von Neumann measurements the optimized probability of type II error can be expressed as

pII min min tr (�(PU 1)( ψ ψ )). ψ ψ := � p| �(�) δ ⊗ | �� | (37) | � : I ≤ Our goal is to prove that 2 pII max ν (UE). = E DU √1 δ (38) ∈ d − Te proof is divided into two parts. In the frst part we will utilize data processing inequality presented in Lemma 1 in Online Appendix C in the Supplementary Materials. Tanks to that, we will show the lower bound for pII . In the second part we will use some technical lemmas presented in Online Appendix C in the Supplementary 23 Materials and we will utilize the results from to show the upper bound for pII.

Te lower bound. Tis part of the proof will mostly be based on data processing inequality. To show that 2 pII max ν (UE) ≥ E DU √1 δ (39) ∈ d − let us begin with an observation that every quantum von Neumann measurement PU can be rewritten as � � † , where denotes the completely dephasing channel and E DU . Terefore, utilizing data process- ◦ (UE) ∈ d ing inequality in Lemma 1 in Online Appendix C, along with the certifcation scheme of unitary channels in Teorem 2, the optimized probability of the type II error is lower-bounded by 1 2 † 2 pII min min tr (�(�(UE)† )( ψ ψ )) ν (UE) ν (UE) ≥ ψ ψ ⊗ | �� | = √1 δ = √1 δ (40) | � � pI| �(�) δ − − : ≤ which holds for each E DU . Hence, maximizing the value of ν2 (UE) over E DU leads to the lower ∈ d √1 δ ∈ d bound of the form − 2 pII max ν (UE). ≥ E DU √1 δ (41) ∈ d −

Te upper bound. Now we proceed to proving the upper bound. Te proof of the inequality 2 pII max ν (UE) ≤ E DU √1 δ (42) ∈ d − will be divided into two cases depending on diamond norm distance between considered measurements PU and P1 . In either case we will construct a strategy, that is choose a state ψ0 and a measurement 0 . As for every | � choice of ψ and it holds that | � p tr (�(P 1)( ψ ψ )), II ≤ U ⊗ | �� | (43) we will show that for some fxed ψ0 and 0 it holds that | � 2 tr (�0(PU 1)( ψ0 ψ0 )) max ν (UE). ⊗ | �� | = E DU √1 δ (44) ∈ d − First we focus on the case when PU P1 2 . We take a state ψ0 for which it holds that � − �⋄ = | � 1 PU P1 ((PU P1) )( ψ0 ψ0 ) 1. (45) � − �⋄ =� − ⊗ | �� | � Ten, the output states (PU 1)( ψ0 ψ0 ) and (P1 1)( ψ0 ψ0 ) are orthogonal and by taking the measure- ⊗ | �� | ⊗ | �� | ment 0 as the projection onto the support of (P1 1)( ψ0 ψ0 ) we obtain ⊗ | �� | tr (� (P 1)( ψ ψ )) 0. 0 U ⊗ | 0�� 0| = (46) Let us recall that

2 �U �1 2 1 ν (U), (47) || − ||⋄ = − where ν(U) min x x W(U) and it holds that 23 = {| |: ∈ } PU P1 min UE 1 , || − ||⋄ = E DU || − ||⋄ (48) ∈ d where DUd is the subgroup of diagonal unitary matrices of dimension d.

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2 Ten, utilizing Eqs. (47) and (48) we obtain that maxE DU ν (UE) 0 . Terefore, by the property that ∈ d = 0 W√1 δ(UE) whenever 0 W(UE) (see Online Appendix B), we have that ∈ − ∈ max ν2 (UE) 0. E DU √1 δ = (49) ∈ d − Secondly, we consider the situation when PU P1 < 2. Let � − �⋄ E0 arg max ν(UE). ∈ E DU (50) ∈ d Again, by referring to Eqs. (47) and (48) we obtain that ν(UE0) > 0 . Let 1, d be a pair of the most distant eigenvalues of UE0 . Note that the following relation holds

1 d ν(UE0) | + | . (51) = 2

As the assumptions of the Lemma 2 in Online Appendix C are saturated for the defned E0 , we consider the input state d ψ0 √ρ0 i i , (52) | �= | �⊗|� i 1 = where the existence of ρ0 together with its properties are described in Lemma 2 and Corollary 1 in Online Appendix C. Let us defne sets ρ ρ 1 1 √ 0 i i √ 0 Ci � 0 � , tr ( �) | �� | δ (53) := : ≤ ≤ − i ρ i ≤ � | 0| � for each i such that i ρ i 0 . Now we take the measurement 0 as � | | � �= d 0 i i ⊤, (54) = | �� |⊗ i i 1 = where i Ci is defned as ∈ † √ρ0U i i U √ρ0 �i arg min tr � | �� | (55) ∈ � C i ρ0 i ∈ i � | | � for each i 1, ..., d such that i ρ0 i 0 and i 0 otherwise. ∈{ } � | | � �= = Now we check that the statistical signifcance is satisfed, that is for the described strategy we have d ψ0 1 p| �(�0) 1 tr (�0(P1 )( ψ0 ψ0 )) 1 tr �i√ρ0 i i √ρ0 δ. (56) I = − ⊗ | �� | = − | �� | ≤ i 1 = Hence, it remains to show that for this setting 2 tr (�0(PU 1)( ψ0 ψ0 )) max ν (UE). ⊗ | �� | = E DU √1 δ (57) ∈ d − Direct calculations reveal that d tr (� (P 1)( ψ ψ )) tr � √ρ U i i U†√ρ 0 U ⊗ | 0�� 0| = i 0 | �� | 0 i 1 = (58) d † √ρ0U i i U √ρ0 i ρ0 i tr �i | �� | . = � | | � i ρ i i 1 0 = � | | � Let us defne † i √ρ0U i i U √ρ0 p| tr �i | �� | . (59) II = i ρ i � | 0| � Note that due to Corollary 1 in Online Appendix C the absolute value of the inner product between pure states √ρ i √ρ U i 0| � and 0 | � is the same for every i 1, ..., d i ρ i 0 . Terefore we can consider the certifcation √ρ0 i √ρ0 i ∈{ }:�| | � �= of� pure| �� states� conditioned| �� on the obtained label i with statistical signifcance δ . From the Teorem 1 we know i i j that pII| depends only on such an inner product between the certified states, hence pII| pII| for each i 1 d = i, j i ρ i , j ρ j 0 . Terefore, we have that the value of pII| will depend on +2 . Tus w.l.o.g. we can assume :�|1 | � � | | � �= that p| 0 and hence II �=

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d † i 1 √ρ0U 1 1 U √ρ0 i ρ0 i p| p| tr �1 | �� | (60) � | | � II = II = 1 ρ 1 i 1 0 = � | | � and in the remaining of the proof we will show that

1 2 pII| max ν (UE). = E DU √1 δ (61) ∈ d − It is sufcient to study two cases depending on the relation between √δ and the inner product 1 ρ U 1 � | 0 | � 1 + d . 1 ρ 1 = 2 (62) 0 � | | � 1 In the case when 1+ d √δ , then due to Teorem 1 we get p | 0 . On the other hand, we know that 2 ≤ II = 0 W (UE0) and hence also ∈ √1 δ − max ν2 (UE) 0. E DU √1 δ = (63) ∈ d − 1 d √ In the case when +2 > δ , then from Teorem 1 we know that 2 2 1 1 d 1 d p| + √1 δ 1 + √δ . (64) II =  2 − − − 2  � � � � � � � � � � � � �  On the other hand, for E0 DU satisfying� Eq. (� 50) we have � � ∈ d 2 2 2 1 d 1 d ν (UE0) + √1 δ 1 + √δ . (65) √1 δ =  2 − − − 2  − � � � � � � � � � � � � �  2 By the particular choice of E0 DUd , this� value is �equal to maxE DU� ν (UE� ) , hence combining the above ∈ ∈ d √1 δ equations we fnally obtain −

1 2 pII| max ν (UE). = E DU √1 δ (66) ∈ d − To sum up, we indicated strategies 0 and ψ0 for which the optimized probability of type II error was equal to | � max DU ν2 (UE) E d √1 δ . Combining this with the previously proven inequality ∈ − 2 pII max ν (UE) ≥ E DU √1 δ (67) ∈ d − gives us Eq. (38) and proves that the proposed strategy ψ0 , �0 is optimal. | �

Remark 3. Similarly to the case of unitary channel certifcation, the optimal input state ψ0 does not depend on | � δ , while the optimal measurement 0 does depend on δ . Moreover, the optimal state has the same form as in the problem of discrimination of von Neumann measurements.

Finally, in Algorithm 1 we present a protocol which describes the optimal certifcation strategy based on the proof of Teorem 3.

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Below, we provide a simple example of application of Algorithm 1 in certifcation of a measurement performed in the Hadamard basis.

Example 1. We will certify von Neumann measurements PH and P1 , where H is the Hadamard matrix.

1. We calculate the distance between P1 and PH . Using semidefinite programming 23,38 we obtain 1 1 i 0 PH P1 √2 . Observe that matrix E0 minimizing (48) is of the form E0 + , || − ||⋄ = = √2 0 1 i − − which means

PH P1 H 1 E0 . (68) || − ||⋄ = || − ||⋄ In order to construct ρ0 we use Lemma 2 in the Supplementary Materials. Tere exist states ρ1, ρ2 of the 1 1 i 1 1 i form ρ1 and ρ2 − . Tus, from Corollary 1 in the Supplementary Materials we have = 2 i 1 = 2 i 1 − 1 1 10 ρ0 (ρ1 ρ2) . (69) = 2 + = 2 01 Hence, the input state ψ0 has a form | � 2 1 ψ i i . | 0� := √ | �⊗|� (70) i 1 2 = 2. We perform P on the frst subsystem of ψ0 ψ0 , that is (P 1)( ψ0 ψ0 ). | �� | ⊗ | �� |

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3. We read the measurement label either i 1 or i 2. = = 4. We reduce our problem to certifcation between states ψi H i or ψi i . | �= | � | �=|� 5. According to the optimal strategy for two-point certifcation of pure quantum states (Corollary 1), we prepare conditional measurements Ŵi . For a fxed statistical signifcance δ and a given label i, the optimal measurement Ŵi is defned as

1 (1) if δ , then Ŵi H i⊥ i⊥ H. ≥ 21 = | �� | (2) if δ< , then Ŵi γ γ for γ √1 δ i √δ i . 2 =| �� | | �= − | �− | ⊥�

6. We perform the measurement Ŵi, 1 Ŵi on ψi ψi . { − } | �� | 7. We read the measurement outcome k 1, 2 . ∈{ } 8. Finally, basing on value of k we make a decision whether we accept (for k 1 ) or reject (for k 2 ) the null = = hypothesis H0. Parallel multiple‑shot certifcation In this section we focus on the scenarios in which we have access to N copies of the certifed quantum objects. Te copies of a given object can be used in many confgurations. Te most general strategy, in the literature referred to as the adaptive scenario, assumes that we are allowed to perform any processing between uses of each provided copy. Te adaptive scenario can be described with the formalism of quantum networks 39. In this paper we restrict our attention only to the special case of adaptive strategy, when all copies are used in parallel. Tis approach, known as the parallel scenario, has a simplifed description based on the tensor product of the copies of a given quantum object. Henceforth, in our cases, the tensor product of pure states will be again a pure state, tensor product of unitary channels—a unitary channel and tensor product of von Neumann measurements—a von Neumann measurement. Terefore, we will be able to apply our results from previous sections. At this point, a natural question arises: can the parallel scenario be optimal? In the case of discrimination of unitary channels and von Neumann measurements, the positive answer was obtained 24,40. Nevertheless, in some special cases of quantum channels or quantum measurements by using an adaptive scenario we are able to improve the probability of correct discrimination of such objects20,21. In the case of certifcation of quantum objects, the optimality of parallel scenario was showed for unitary channels27. In this section, we prove the parallel approach for certifcation of von Neumann measurements is also optimal, see Teorem 5. Let us begin with the certifcation of pure states. Such certifcation can be understood as certifying states ψ N and ϕ N . Te following corollary generalizes the results from Teorem 1. | �⊗ | �⊗ Corollary 4. In the case of certifcation of pure states ψ N and ϕ N with statistical signifcance δ 0, 1 , the | �⊗ | �⊗ ∈[ ] minimized probability of the type II error yields ψ ϕ N √δ ( ) 0 p N 2 |� | �| ≤ II = ψ ϕ N √1 δ 1 ψ ϕ 2N √δ ψ ϕ N > √δ, (71) |� | �| − − − |� | �| |� | �| where N is the number of uses of the pure state.

log √δ One can note that for a given statistical signifcance δ , by taking N log ψ ϕ we obtain pII 0 . Tis is not ≥ |� | �| = in contradiction with the statement that if one cannot distinguish states perfectly in one step, then they cannot by distinguished perfectly in any fnite number of tries, because the error is hidden in pI . Tis error decays exponentially, and the optimal exponential error rate, depending on a formulation, can be stated as the Stein bound, the Chernof bound, the Hoefding bound, and the Han–Kobayashi bound, see41 and references therein. Secondly, we focus on the certifcation of unitary channels. Te scenario of parallel certifcation can be seen 1 N N as certifying channels ⊗ and U⊗ . Hence, for the parallel certifcation of such unitary channels we have the following corollary generalizing the results from Teorem 2.

Corollary 5. 1 N N In the case of parallel certifcation of unitary channels ⊗ and U⊗ with statistical signifcance δ 0, 1 , the minimized probability of the type II error yields ∈[ ] (N) ν2 N pII √1 δ U⊗ , (72) = − where N is the number of uses of the unitary channel.

N N N From the above it follows that if 0 W√1 δ(U⊗ ) , then the channels ⊗1 and U⊗ can be certifed with ∈ − pII 0 . Let be the angle between a pair of two most distant eigenvalues of a unitary matrix U. Te perfect = √ certifcation can be achieved by taking N 2 arccos δ . Observe that in the special case δ 0 , we recover the =⌈ � ⌉ = well-known formula N π being the number of unitary channels required for perfect discrimination in the =⌈� ⌉ scheme of symmetric distinguishability of unitary channels 17. Te dependence between the number N of used N unitary channels and the shape of W√1 δ(U⊗ ) is presented in Fig. 2. We have established similar results −for the case of certifying N copies of von Neumann measurements P1 and PU . We consider only the parallel scenario and therefore this can be understood as certifying von Neumann

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N =1 N =2

N =3 N =4

Figure 2. Numerical ranges W(U N ) (polytopes) and √1 δ-numerical ranges W (U N ) (ovals) of πi ⊗ − √1 δ ⊗ U U2 with eigenvalues 1 and e 3 , for N 1, 2, 3, 4 with statistical signifcance δ 0.7−. ∈ = =

P N P1 N measurements U⊗ and ⊗ . Tis issue is studied in Teorem 4. Moreover, it will turn out in Teorem 5 that the parallel scenario is optimal for the certifcation of von Neumann measurements.

Theorem 4. P N P1 N In the case of certifcation of von Neumann measurements U⊗ and ⊗ with statistical signifcance δ 0, 1 , the minimized probability of the type II error yields ∈[ ] (N) 2 N N pII max ν U⊗ E⊗ , = E DU √1 δ (73) ∈ d − where N is the number of uses of the von Neumann measurements.

Proof. P1 N P N Te von Neumann measurements ⊗ and U⊗ satisfy assumptions of Teorem 3, therefore we have (N) ν2 N pII max √1 δ U⊗ E . = E DU N (74) ∈ d − Whereas, the equality ν2 N ν2 N N max √1 δ U⊗ E max √1 δ U⊗ E⊗ E DU N = E DU (75) ∈ d − ∈ d − follows from24 (Teorem 1).

Finally, we state the theorem providing the optimality of parallel scenario for the certifcation of von Neu- mann measurements.

Theorem 5. Te parallel scenario for certifcation of von Neumann measurements is optimal. More formally, for any adaptive certifcation scenario, the probability of the type II error cannot be smaller than in the parallel scenario.

Proof. Let us denote by �( ) an adaptive scenario, whose input are N copies of a given quantum operation and · (N) (N) (N) outputs a quantum state. Let p be the minimized probability of the type II error for states � P1 , � P . ˜II U Defne d1 d2 ... dN to be a non-decreasing sequence of natural numbers and assume that dN d′ d′′ ≤ ≤ ≤ = N N for d′ , d′′ N . Te numbers d1, ..., dN will denote sizes of auxiliary systems occurring in a construction of N N ∈ P(N) P(N) � 1 , � U . We assume that the last auxiliary system having dimension dN is a tensor product of two subsystems: one with dimension dN′ and second with dimension dN′′ , which will be traced out. Te general adap- 24 tive scenario for certifcation of N copies of von Neumann measurements PU , P1 can be represented as :

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k1 k1 ... k1 P k2 ... k2 P ... k3 ψ | 0 ...... Vk1 . . . Vk1,k2 ... kN P tr ... dN d1 d2 dN dN

Figure 3. Application of an adaptive scenario �( ) on N copies of a von Neumann measurement P where · P P1, PU . ∈{ }

Φ ... ∆ k1

Φ ... ∆ k2

... ∆ k3 ψ0 . W1 . W2 . . . | ......

... Φ ∆ kN tr ... dN d1 d2 dN dN

Figure 4. An equivalent description of Eq. (76), formalized as Eq. (78). Te scenario �( ) is applied to N copies · of unitary channel , where � �1, � † . As a last step we perform a completely dephasing channel . ∈{ (UE0) }

P(N) 1 P 1 1 P 1 � U ( dN 1 U d tr d ) �N 1 ... ( d U dN 2d ) = − ⊗ ⊗ N′ ⊗ N′′ ◦ − ◦ ◦ ⊗ ⊗ − 2 ◦ � (P 1 N 1 )( ψ ψ ), 1 U d − d1 0 0 ◦ ⊗ | �� | (76) P(N) 1 P 1 1 P 1 � 1 ( dN 1 1 d tr d ) �N 1 ... ( d 1 dN 2d ) = − ⊗ ⊗ N′ ⊗ N′′ ◦ − ◦ ◦ ⊗ ⊗ − 2 ◦ �1 (P1 1 N 1 )( ψ0 ψ0 ). ◦ ⊗ d − d1 | �� | † Te channels i are given by �i(X) WiXW , such that = i W k , k , ..., k k , k , ..., k V ... , i = | 1 2 i�� 1 2 i|⊗ k1,k2, ,ki (77) k1, k2,...,ki V M N i N i i 1, ..., N 1 where k1,k2,...,ki d − di,d − di 1 are isometry matrices for each . Te above scenario is pre- ∈ + ∈{ − } sented in Fig. 3. P � † Let us decompose a von Neumann measurement U as a composition of a unitary channel (UE0) and the P � � † completely dephasing channel , that is U (UE0) , where E0 is a matrix maximizing Eq. (73). Terefore, P=(N) ◦ P(N) one can rewrite the adaptive scenarios � U and � 1 as (see also Fig. 4) P(N) N 1 (N) � �⊗ d tr d � � † , U = ⊗ N′ ⊗ N′′ ◦ (UE0) (78) (N) N (N) � P1 �⊗ 1 tr � �1 , = ⊗ dN′ ⊗ dN′′ ◦ where

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(N) 1 1 1 1 � � † ( dN 1 �(UE )† dN ) �N 1 ... ( d �(UE )† dN 2d ) (UE0) = − ⊗ 0 ⊗ ◦ − ◦ ◦ ⊗ 0 ⊗ − 2 ◦ � (� † 1 N 1 )( ψ ψ ), 1 (UE0) d − d1 0 0 ◦ ⊗ | �� | (79) (N) 1 1 1 1 � �1 ( dN 1 �1 dN ) �N 1 ... ( d �1 dN 2d ) = − ⊗ ⊗ ◦ − ◦ ◦ ⊗ ⊗ − 2 ◦ �1 (�1 1 N 1 )( ψ0 ψ0 ). ◦ ⊗ d − d1 | �� |

(N) (N) Let us observe that the scenarios � � † , � �1 describe certifcation of N copies of unitary channels (UE0) � † 1 (UE0) and . Using the data processing inequality in Lemma 1 in Supplementary Materials, the probability of the (N) (N) (N) (N) type II error for certifcation between � P and � P1 is no smaller that for � � † and � �1 . U (UE0) 27 � † 1 Following , the minimized probability of the type II error for N-copies of unitary channels (UE 0) and (N) N is achieved in the parallel scenario, that is whenever � � † �⊗ † ( ψ0 ψ0 ) and (UE0) = (UE0) | �� | (N) N � �1 �⊗1 ( ψ0 ψ0 ) , where ψ0 is an optimal state. For this case, from Corollary 5 the probability of = | �� | | � 2 N N � † 1 ν U E the type II error for certifcation of N copies of unitary channels (UE0) and is equal √1 δ ⊗ 0⊗ . − Hence, we obtain (N) ν2 N N pII √1 δ U⊗ E0⊗ . (80) ˜ ≥ − From Teorem 4, we have

(N) ν2 N N pII √1 δ U⊗ E0⊗ , (81) = − (N) where pII is the minimized probability of type II error for the parallel certifcation of von Neumann measurements P1 and PU . Ten, we have (N) (N) p p (82) ˜II ≥ II which fnishes the proof.

Conclusions In this work we studied the two-point certifcation of quantum states, unitary channels and von Neumann measurements. Te problem of certifcation of quantum objects is inextricably related with quantum hypothesis testing. We were interested in minimizing the probability of type II error given the upper bound on the probability of type I error. Although the problems of certifcation of quantum states and unitary channels are well-studied, we pointed out the connection of certifcation of unitary channels with the notion of q-numerical range. Aferwards, we extended this approach to the certifcation of von Neumann measurements and found a formula for minimized probability of the type II error and the optimal certifcation strategy. It turned out that this formula can be also connected with the notion of q-numerical range. Remarkably, it appeared that in the case of certifcation of von Neumann measurements the use of entangled input state can signifcantly improve the certifcation. Finally, we focused on the certifcation of the von Neumann measurements in the parallel scenario. More precisely, we generalized the above results for the situation when the von Neumann measurements can be used N times in parallel. We showed that optimal certifcation of von Neumann measurements can be performed without any additional processing, i.e. in the parallel way.

Received: 24 September 2020; Accepted: 4 January 2021

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