On the optimal certification of von Neumann measurements

In this report we study certification 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) certification 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 fixed statistical significance. In this report, we begin with studying the two-point certification of pure quantum states and unitary channels to later use them to prove our 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 pure states, unitary operations and von Neumann measurements cannot be distinguished perfectly but still can be certified with a given statistical significance. Moreover, we show the connection between the certification of quantum channels or von Neumann measurements and the notion of q-numerical range.

The optimal strategy is represented by effect The optimality of this value can be checked by using standard constrained optimization techniques.
Appendix B. q-numerical range and certification of unitary channels B.1. q-numerical range in the problem of two-point certification of unitary channels. In this appendix we will present an alternative derivation the result for the probability 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 I (Ω) = tr((1l − Ω)|ψ ψ|) ≤ δ. Let us consider Ω = |ω ω|. Hence, we have The probability of the type II error takes the form Let us recall that the q-numerical range is defined as and we use the notation Now from the definition of the q-numerical range for q = √ 1 − δ and its properties [1] and it easy to see that which will imply that Therefore, we conclude that the use of entanglement for the case of certification of unitary channels does not improve the certification. B.2. Distance of q-numerical range to zero. In this subsection we will focus on calculating the distance from the q-numerical range the to the origin of the coordinate system. Let us begin with the two-dimensional case when the unitary matrix U has two eigenvalues λ 1 and λ 2 . Without loss of generality we can assume λ 1 = 1. From [2] we know that the q-numerical range is an elliptical disc with eccentricity equal to q and foci qλ 1 and qλ 2 , see Fig 1. Let c denote the distance from the center of the ellipse to the focus and a be the distance from the center of the ellipse to its vertex. Using this notation the eccentricity yields q = c/a. Let b denote the distance from the center of the ellipse to its co-vertex, which it the point which saturates the minimum.  Figure 1. Schematic illustration of an ellipse and notation used in Appendix, where we use shortcut notation ν := ν q (U ).
First, we will calculate b. We note that From the properties of the ellipse and the form of the eccentricity q we have Hence On the other hand we have and therefore Now we need to show that the above expression for the distance ν q (U ) is valid also for higher dimensions. The boundary of q-numerical ranges for larger matrices is described in [2]. It consists of parts of a few ellipses obtained is an analogous way. Let λ 1 and λ d be the pair of the most distant eigenvalues of U . Let λ i and λ j bo some pair of eigenvalues such that i, j = 1, d. Let ν q (U ) be the distance from zero the ellipse built on λ i and λ j in the same way as above. Our goal is to prove that ν q (U ) > ν q (U ).
We note that λ 1 − λ 2 > λ i − λ j . Hence to prove that ν q (U ) > ν q (U ) it suffices to show that λ 1 + λ 2 < λ i + λ j . As all the eigenvalues lie on the unit circle, the from the parallelogram law we have λ 1 + λ 2 2 = 4 − λ 1 − λ 2 2 . Therefore and thus ν q (U ) > ν q (U ), from which it follows that holds for any dimension d. The above formula can be easily translated into trigonometric functions where Θ is the angle between λ 1 and λ d . Hence, we have Therefore, Appendix C. Certification of von Neumann measurements In this appendix we recall a few technical lemmas necessary to prove the main theorem in the paper. The first lemma is the data processing inequality. This inequality, along with its proof, can be found eg. in [3]. However, to keep this work self-consistent we present our modified version of them.
Proof. Let us consider two-point certification of two quantum states ρ and σ with statistical significance δ. To calculate the probability of the type II error, p II , we formulate the problem as min Ω: tr(Ωρ)≥1−δ tr(Ωσ).
Now, consider the scenario in which we use as processing the quantum channel Φ on states ρ and σ. We want to calculate min Ω:tr(ΩΦ(ρ))≥1−δ tr(ΩΦ(σ)) which is equivalent to It easy to see that Φ † (Ω) is also a measurement and Eventually, we obtain the data processing inequality given by The following lemma is proved in the work [4].
Lemma 2. (Lemma 5 from [4], direct implication) Assume that E 0 ∈ DU d satisfies the condition Let λ 1 , λ d be a pair of the most distant eigenvalues of U E 0 and Π 1 , Π d be the projectors onto the subspaces spanned by the eigenvectors corresponding to λ 1 and λ d , respectively. Then, there exist states ρ 1 , ρ d , satisfying the following conditions The next proposition follows directly from Lemma 2.
Corollary 1. Let ρ 0 = 1 2 ρ 1 + 1 2 ρ d be the state satisfying conditions given by Eq. (29). Then, for each i ∈ {1, . . . , d} we have Moreover, for each i ∈ {1, . . . , d} such that i|ρ 0 |i = 0 we get Appendix D. Animation of q-numerical range For an animation of the behavior of the q-numerical range of a unitary matrix U ∈ U 3 see the attached gif file.