# Multiple-shot and unambiguous discrimination of von Neumann measurements

Zbigniew Puchała1,2 ,  Łukasz Pawela1 ,  Aleksandra Krawiec1,3 ,  Ryszard
Kukulski1,4 and  Michał Oszmaniec5,6 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
3Institute of Mathematics, Silesian University of Technology, ul. Kaszubska
23, 44-100 Gliwice, Poland  4Institute of Mathematics, University of Silesia,
ul. Bankowa 14, 40-007 Katowice, Poland  5Institute of Theoretical Physics and
Astrophysics, National Quantum Information Centre, Faculty of Mathematics,
Physics and Informatics, University of Gdańsk, ul. Wita Stwosza 57, 80-308
Gdańsk, Poland  6Center for Theoretical Physics, Polish Academy of Sciences
Al. Lotników 32/46, 02-668 Warszawa, Poland

###### Abstract.

We present an in-depth study of the problem of discrimination of von Neumann
measurements in finite-dimensional Hilbert spaces. Specifically, we consider
two scenarios: multiple-shot and unambiguous discrimination. In the case of
multiple-shot discrimination, we focus on discrimination of measurements with
the assistance of entanglement. Interestingly, we prove that in this setting,
all pairs of distinct von Neumann measurements can be distinguished perfectly
(i.e. with the unit success probability) using only a finite number of
queries. We also show that in this scenario queering the measurements _in
parallel_ gives the optimal strategy, and hence any possible adaptive methods
do not offer any advantage over the parallel scheme. In the second scenario,
we give the general expressions for the optimal discrimination probabilities
with and without the assistance of entanglement. Finally, we show that typical
pairs of Haar-random von Neumann measurements can be perfectly distinguished
with only two queries.

##  1\. Introduction

With the recent technological progress, quantum information science is not
merely a collection of purely theoretical ideas anymore. Indeed, quantum
protocols of increasing degree of complexity are currently being implemented
on more and more complicated quantum devices [1, 2] and are expected to soon
yield practical solutions to some real-world problems [3]. This situation
motivates the need for certification and benchmarking of various building-
blocks of quantum devices [4, 5, 6] (see [7] for a recent review).
Discrimination or quantum hypothesis testing constitute one of the paradigms
for assessing the quality of parts of quantum protocols [8, 9, 10, 11, 12]. In
this work, we present a comprehensive study of various scenarios of
discrimination of von Neumann measurements on a finite-dimensional Hilbert
space. Here, by von Neumann measurements, we understand fine-grained
projective measurements. The general problem of quantum channel discrimination
has attracted a lot of attention in recent years. One of the first results was
the study of discrimination of unitary operators [13, 14]. Later, this has
been extended to various settings, such as multipartite unitary operations
[15]. The follow-up paper [16] formulated necessary and sufficient conditions
under which quantum channels can be perfectly discriminated. Further works
investigated the adaptive [17, 18, 19, 20] and parallel [21] schemes for
discrimination of channels. Finally, some asymptotic results on discrimination
of typical quantum channels in large dimensions were obtained in [22].
Discrimination of quantum measurements, being a subset of quantum channels, is
thus of particular interest. Some of the earliest results on this topic
involve condition on perfect discrimination of two measurements [23, 24, 25,
26, 27]

Here we are interested in the following problem. Imagine we have an unknown
device hidden in a black box. We know it performs one of the two possible von
Neumann measurements, either $\mathcal{P}_{1}$ or $\mathcal{P}_{2}$.
Generally, whenever a quantum state is sent through the box, the box produces,
with probabilities predicted by quantum mechanics, classical labels
corresponding to the measurement outcomes. Our goal is to find schemes that
attain the optimal success probability for discrimination of measurements. The
results contained in this work concern the following two scenarios:

_Multiple-shot discrimination—_ In this setting, we are allowed to use the
black box containing von Neumann measurement many times. Furthermore, we can
prepare any input state with an arbitrarily large quantum memory (i.e., we can
use ancillas of arbitrarily large dimension), and we can perform any channels
between usages of the black box. This allows us to implement both parallel
(see Fig. 1) as well as adaptive discrimination strategies (see Fig. 2). We
focus on the case of entanglement-assisted discrimination. Our main finding is
that in the multiple-shot scenario, adaptive strategies do not offer any
advantage over parallel queries. Moreover, we derive an explicit dependence of
the diamond norm distance in the multiple-shot scenario as a function of the
diamond norm in the single-shot setting. As a consequence we recover a known
result [23] stating that given sufficiently many queries, every pair of
different von Neumann measurements can be distinguished perfectly (i.e. with
zero error probability).

Figure 1. A schematic representation of parallel discrimination scheme of
quantum measurements that uses two applications of a measurement.  Figure 2. A
schematic representation of adaptive discrimination scheme of quantum
measurements that uses two applications of a measurement. In this case
adaptive scheme amounts to application of the unitary channel $\Phi_{V_{i}}$
which depends on the result $i$ of the first measurement.

_Unambiguous discrimination—_ This scenario is an analogue to the well-known
scheme of unambiguous state discrimination [28]. Namely, for every query to
the black box, the decision procedure outputs $\mathcal{P}_{1}$,
$\mathcal{P}_{2}$, or the inconclusive answer. The latter means that the user
cannot decide which measurement was contained in the black box. Importantly,
we require that the procedure cannot wrongly identify the measurement (see
Fig. 3). Our main contribution to this problem is the derivation of the
general schemes, which attain the optimal success probability both with and
without the assistance of entanglement. Interestingly, we find that optimal
success probability $p_{u}$ for unambiguous discrimination of projective
measurements $\mathcal{P}_{1},\mathcal{P}_{2}$ with the assistance of
entanglement is functionally related to the diamond norm distance
$\|\mathcal{P}_{1}-\mathcal{P}_{2}\|_{\diamond}$, which quantifies
distinguishability of $\mathcal{P}_{1},\mathcal{P}_{2}$ with the assistance of
entanglement but without requiring unambiguous results. The specific formula
is given by

(1)
$p_{u}=1-\sqrt{1-\frac{1}{4}\left\|\mathcal{P}_{1}-\mathcal{P}_{2}\right\|_{\diamond}^{2}}$

and its geometric interpretation is presented in Fig. 4. Finally, we also
present simple formulas for the optimal discrimination probability of von
Neumann measurements for qubits.

Figure 3. A schematic representation of the setting of unambiguous measurement
discrimination scheme. The figure on the left shows an umbiguous dscrimination
without the assistance of entanglement while the figure on right shows
entanglement-assisted unambiguous discrimination. Figure 4.  Geometric
interpretation of the relationship between the diamond norm
$\|\mathcal{P}_{1}-\mathcal{P}_{2}\|_{\diamond}$ and the probability of
unambiguous discrimination $p_{\mathrm{u}}(\mathcal{P}_{1},\mathcal{P}_{2})$ .

To conclude this introduction, let us contrast our results with
transformations mapping probability distributions to probability
distributions, that is classical channels. In this setting, we know that if
such channels cannot be perfectly distinguished in one shot, they can not be
distinguished perfectly in any finite number of uses. What we can do is to
study the asymptotic behavior of error probability when the number of
applications of the channel tends to infinity. The error probability,
formulated in the language of the hypothesis testing of two distributions,
decays exponentially, and the optimal exponential error rate, depending on a
formulation, is given by the Stein bound, the Chernoff bound, the Hoeffding
bound, and the Han-Kobayashi bound, see [29] and references therein. In the
case of distinct von Neumann measurements and entanglement-assisted
discrimination, we show that one can perform perfect discrimination with the
use of a finite number of queries. Therefore, in contrary to the classical
channels, we do not have to consider the exponential error rate, as the error
probability drops to zero in a finite number of tries.

The rest of the paper is organized as follows. First, in Section 2, we give a
survey of the main concepts and notation used throughout this work (including
the basic background on discrimination of quantum channels and measurements).
Then, in Section 3, we present our results for the scenario of multiple-shot
measurement discrimination. The following Section 4 contains the results
concerning the unambiguous discrimination of quantum measurements. We obtain
an upper bound on the probability of success and show that the parallel scheme
is again optimal. Lastly, in Section 5 we summarize our results and give some
directions for future research.

##  2\. Preliminaries and main concepts

By $\mathcal{D}(\mathbb{C}^{d})$ we will denote the set of quantum states on
$\mathbb{C}^{d}$. Let $\mathrm{L}(\mathbb{C}^{d_{1}},\mathbb{C}^{d_{2}})$
denote the set of all linear operators acting from $\mathbb{C}^{d_{1}}$ to
$\mathbb{C}^{d_{2}}$. For brevity we will put
$\mathrm{L}(\mathbb{C}^{d},\mathbb{C}^{d})\equiv\mathrm{L}(\mathbb{C}^{d})$. A
quantum channel is a linear mapping
$\Phi:\mathrm{L}(\mathbb{C}^{d_{1}})\to\mathrm{L}(\mathbb{C}^{d_{2}})$ which
is completely positive and trace-preserving. The former means that for every
$\mathrm{L}(\mathbb{C}^{d_{1}})\otimes\mathrm{L}(\mathbb{C}^{s})\ni\rho\geq 0$
we have
$\mathrm{L}(C^{d_{2}})\otimes\mathrm{L}(\mathbb{C}^{s})\ni(\Phi\otimes{\rm
1l})(\rho)\geq 0$, while the latter means
$\operatorname{Tr}(\Phi(X))=\operatorname{Tr}X$ for every
$X\in\mathrm{L}(\mathbb{C}^{d_{1}})$. A set of generalized measurements
(POVMs) $\mathbb{C}^{d}$ will be denoted by $\mathrm{POVM}(\mathbb{C}^{d})$. A
general quantum measurement $\mathcal{M}$ on $\mathbb{C}^{d}$ is a tuple of
positive semidefinite operators111In this paper we restrict our attention to
measurements with a finite number of outcomes. on $\mathbb{C}^{d}$ that add up
to identity on $\mathbb{C}^{d}$ i.e. $\mathcal{M}=(M_{1},\ldots,M_{n})$ with
$M_{i}\geq 0$ and $\sum_{i}M_{i}={\rm 1l}$. If a quantum state $\sigma$ is
measured by a measurement $\mathcal{M}$, then the outcome $i$ is obtained with
the probability $p(i|\sigma,\mathcal{M})=\operatorname{tr}(\sigma M_{i})$
(Born rule). Therefore, a quantum measurement $\mathcal{M}$ can be uniquely
identified with a quantum channel

(2)
$\Psi_{\mathcal{M}}(\sigma)=\sum_{i=1}^{n}\operatorname{tr}(M_{i}\sigma)|i\rangle\langle
i|,$

where states $|i\rangle\langle i|$ are perfectly distinguishable (orthogonal)
pure states that can be regarded as states describing the state of a classical
register. In what follows we will abuse the notation and simply treat quantum
measurements (denoted by symbols $\mathcal{M},\mathcal{N},\mathcal{P}$, …) as
quantum channels having the classical outputs. Using this interpretation one
can readily use the results concerning the discrimination of quantum channels
for generalized measurements. In particular, for entanglement-assisted
discrimination of quantum channels we have a classic result due to Helstrom
[30]. It states that the probability of correct discrimination
$p_{\mathrm{disc}}(\Phi,\Psi)$ between two quantum channels $\Phi$ and $\Psi$
is given by

(3)
$p_{\mathrm{disc}}(\Phi,\Psi)=\frac{1}{2}+\frac{1}{4}\left\|\Phi-\Psi\right\|_{\diamond},$

where
$\left\|\mathbf{S}\right\|_{\diamond}=\max_{\|X\|_{1}=1}\left\|(\mathbf{S}\otimes{\rm
1l})(X)\right\|_{1}$ denotes the diamond norm of the linear map $\mathbf{S}$.
Any optimal $X$ is called a discriminator. Thus, if the value of the diamond
norm of the difference of two channels is strictly smaller than two, then the
two channels cannot be distinguished perfectly in a single-shot scenario.

In this work we will be concerned with von Neumann measurements i.e.
projective and fine-grained measurements on a Hilbert space of a given
dimension $d$. Von Neumann measurements $\mathcal{P}$ in $\mathbb{C}^{d}$ are
tuples of orthogonal projectors on vectors forming an orthonormal basis
$\\{|\psi_{i}\rangle\\}_{i=1}^{d}$ in $\mathbb{C}^{d}$ i.e.

(4)
$\mathcal{P}=\left(|\psi_{1}\rangle\langle\psi_{1}|,|\psi_{2}\rangle\langle\psi_{2}|,\ldots,|\psi_{d}\rangle\langle\psi_{d}|\right).$

In what follows we will use $\mathcal{P}_{{\rm 1l}}$ to denote the measurement
in the standard computational basis. We will also use $\mathcal{P}_{U}$ to
denote the von Neumann measurement in the basis $|\psi_{i}\rangle=U|i\rangle$
for a unitary $d\times d$ matrix $U\in\mathcal{U}_{d}$. In other words,
vectors $|\psi_{i}\rangle$ from Eq.(4) are columns of the matrix $U$. We also
specify the subset $\mathcal{DU}_{d}\subset\mathcal{U}_{d}$ of diagonal,
unitary matrices. Consider now a general task of discriminating between two
projective measurements $\mathcal{P}_{U_{1}}$, $\mathcal{P}_{U_{2}}$, and let
$p_{\mathrm{succ}}(\mathcal{P}_{U_{1}},\mathcal{P}_{U_{2}})$ be the optimal
probability for discriminating between these measurements (we do not specify
what kind of discrimination task we have in mind). Then, due to the unitary
invariance of the discrimination problem and identity
$\mathcal{P}_{U}(\cdot)=\mathcal{P}_{{\rm 1l}}\circ(U^{\dagger}\cdot U)$ we
obtain that
$p_{\mathrm{succ}}(\mathcal{P}_{U_{1}},\mathcal{P}_{U_{2}})=p_{\mathrm{succ}}(\mathcal{P}_{\rm
1l},\mathcal{P}_{U_{1}^{\dagger}U_{2}})$. Therefore, for any discrimination of
projective measurements, without loss of generality, we can limit ourselves to
considering the problem of distinguishing between the measurement in the
standard basis $\mathcal{P}_{\rm 1l}$ and another projective measurement
$\mathcal{P}_{U}$. It is important to note that definition of measurement
$\mathcal{P}_{U}$ distinguishes projective measurements differing only by
ordering of elements of the basis. Moreover, a set of unitary matrices
$\\{UE|\ E\in\mathcal{DU}_{d}\\}$ uniquely specifies the same projective
measurement, ie. $\mathcal{P}_{U}=\mathcal{P}_{UE}$ for all
$E\in\mathcal{DU}_{d}$.

Distinguishability of quantum measurements is strictly related to the
distinguishability of unitary channels. The prominent result [31, 32] gives an
expression which makes calculating the diamond norm of the difference of
unitary channels $\Phi_{U}$, $\Phi_{\rm 1l}$ substantially easier. It says
that the for a unitary matrix $U$ we have

(5) $\left\|\Phi_{U}-\Phi_{\rm 1l}\right\|_{\diamond}=2\sqrt{1-\nu^{2}},$

where $\nu=\min_{x\in W(U)}|x|$ and $W(X)$ denotes the numerical range of the
operator $X$. Building on this result, in [26] the following characterization
of the diamond norm of the distance between measurements was obtained

(6) $\left\|\mathcal{P}_{U}-\mathcal{P}_{\rm
1l}\right\|_{\diamond}=\min_{E\in\mathcal{DU}_{d}}\left\|\Phi_{UE}-\Phi_{\rm
1l}\right\|_{\diamond}.$

Therefore the distance between two von Neumann measurements is the minimal
value of the diamond norm of the difference between optimally coherified
channels [33]. Moreover, there is a simple condition which lets us check
whether two measurements are perfectly distinguishable [26, 21, 23]. It holds
that $\left\|\mathcal{P}_{U}-\mathcal{P}_{\rm 1l}\right\|_{\diamond}=2$ if and
only if there exists a state $\rho$ such that

(7) $\operatorname{diag}\left(\rho U\right)=0.$

Equation (6) will be of significant importance throughout this work. We will
also make use of the dephasing channel denoted
$\Delta(\rho)=\sum_{i}|i\rangle\langle i|\rho|i\rangle\langle i|$. Finally,
when talking about the eigenvalues of a unitary matrix $U\in\mathcal{U}_{d}$,
we will follow convention that
$\lambda_{1}=\mathrm{e}^{\mathrm{i}\alpha_{1}},\ldots,\lambda_{d}=\mathrm{e}^{\mathrm{i}\alpha_{d}}$
are ordered by their phases, if $\alpha_{1}\leq\ldots\leq\alpha_{d}$ for
$\alpha_{1},\ldots,\alpha_{d}\in[0,2\pi)$. The angle of the shortest arc
containing all eigenvalues will be denoted by $\Theta(U)$.

##  3\. Multiple-shot discrimination

Multiple-shot discrimination is a natural generalization of the single-shot
scheme studied in [26]. In this scenario we have access to multiple queries to
quantum measurements which is mathematically equivalent to the problem of
single-shot discrimination of channels $\mathcal{P}_{U^{\otimes
N}},\mathcal{P}_{{\rm 1l}^{\otimes N}}$ acting on states defined on multiple-
component Hilbert space $(\mathbb{C}^{d})^{\otimes N}$ (see Fig. 5). In the
following sections will write shortly $\mathcal{P}_{{\rm 1l}}$ instead of
$\mathcal{P}_{{\rm 1l}^{\otimes N}}$.

Figure 5. A schematic representation of the parallel discrimination scheme of
quantum measurements.

As we described in the preceding section, the problem of distinguishing
quantum measurements is intimately related to distinguishing unitary channels
[26]. In what follows we leverage this result to prove a number of results
regarding multiple-shot discrimination of von Neumann measurements.

###  3.1. Parallel discrimination

The first step in our study is to extend Eq. (6) to the parallel setting. Here
we study the form of the optimal matrix $E$ in the parallel scheme. The
following theorem states that it has a tensor product form.

######  Theorem 1.

Let $U\in U_{d}$ and let $\mathcal{P}_{U}$ be the corresponding von Neumann
measurement on $\mathbb{C}^{d}$. Then we have the following equality

(8) $\left\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{{\rm
1l}}\right\|_{\diamond}=\min_{E\in\mathcal{DU}_{d}}\left\|\Phi_{U^{\otimes
N}E^{\otimes N}}-\Phi_{{\rm 1l}}\right\|_{\diamond}.$

###### Proof.

In the first step of the proof assume that $\mathcal{P}_{U}$ and
$\mathcal{P}_{\rm 1l}$ are perfectly distinguishable in a single-shot scenario
i.e.

(9) $\left\|\mathcal{P}_{U}-\mathcal{P}_{\rm
1l}\right\|_{\diamond}=\min_{E\in\mathcal{DU}_{d}}\left\|\Phi_{UE}-\Phi_{\rm
1l}\right\|_{\diamond}=2.$

This trivially implies that for each $N\in\mathbb{N}$ the measurements
$\mathcal{P}_{U^{\otimes N}}$ and $\mathcal{P}_{{\rm 1l}}$ are perfectly
distinguishable and it holds that

(10) $\left\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{{\rm
1l}}\right\|_{\diamond}=\min_{F\in\mathcal{DU}_{d^{N}}}\left\|\Phi_{(U^{\otimes
N})F}-\Phi_{{\rm
1l}}\right\|_{\diamond}\leq\min_{E\in\mathcal{DU}_{d}}\left\|\Phi_{U^{\otimes
N}E^{\otimes N}}-\Phi_{{\rm 1l}}\right\|_{\diamond}\leq 2,$

which proves the thesis of the theorem in this case.

Now, consider the second case when $\mathcal{P}_{U}$ and $\mathcal{P}_{\rm
1l}$ are not perfectly distinguishable using a single query. Then, according
to equality (6) there exists an optimal matrix $E_{0}\in\mathcal{DU}_{d}$ such
that $\left\|\mathcal{P}_{U}-\mathcal{P}_{\rm
1l}\right\|_{\diamond}=\left\|\Phi_{UE_{0}}-\Phi_{\rm 1l}\right\|_{\diamond}$
and $0\not\in W(UE_{0})$.

The general proof strategy is to utilize the Lemma 2 stated in the Appendix A
in order to construct optimal discriminator $\rho_{0}$ for measurements
$\mathcal{P}_{U^{\otimes N}}$ and $\mathcal{P}_{{\rm 1l}}$. Hence we start by
establishing that the assumptions of this lemma are fulfilled. We will do so
utilizing Lemma 1 stated in the Appendix A. According to it the function
$(\rho,E)\mapsto|\operatorname{Tr}(\rho UE)|$ has a saddle point. Due to
equality $\min_{E\in\mathcal{DU}_{d}}\left\|\Phi_{UE}-\Phi_{{\rm
1l}}\right\|_{\diamond}=\left\|\Phi_{UE_{0}}-\Phi_{\rm 1l}\right\|_{\diamond}$
we have

(11)
$\max_{E\in\mathcal{DU}_{d}}\min_{\rho\in\mathcal{D}(\mathbb{C}^{d})}|\operatorname{Tr}(\rho
UE)|=\min_{\rho\in\mathcal{D}(\mathbb{C}^{d})}|\operatorname{Tr}(\rho
UE_{0})|.$

Therefore the former assumptions of Lemma 2 are satisfied for the matrix
$E_{0}$ and hence it is possible to take states $\rho_{1}$, $\rho_{d}$ which
fulfill the latter conditions of this Lemma.

To complete the proof we will separately study two cases. In the first one, we
assume that POVMs $\mathcal{P}_{U}$ and $\mathcal{P}_{{\rm 1l}}$ are not
perfectly distinguishable using $N$ queries i.e. $0\not\in W\left(U^{\otimes
N}E_{0}^{\otimes N}\right)$. Hence, as
$\operatorname{diag}(\rho_{1})=\operatorname{diag}(\rho_{d})$, then

(12) $\operatorname{diag}(\rho_{1}^{\otimes
N})=\operatorname{diag}(\rho_{d}^{\otimes N})$

and $\rho_{1}^{\otimes N},\rho_{d}^{\otimes N}$ lie on the subspaces spanned
by the eigenvectors of the matrix $U^{\otimes N}E_{0}^{\otimes N}$ eigenvalues
$\lambda_{1}^{N}$ and $\lambda_{d}^{N}$, respectively. Consequently, we
fulfilled the latter assumptions of Lemma 2 and the reverse implication of
this Lemma states that the unitary matrix $E_{0}^{\otimes N}$ is optimal and
for $\rho_{0}=\frac{1}{2}\rho_{1}^{\otimes N}+\frac{1}{2}\rho_{d}^{\otimes N}$
it holds that

(13)
$\min_{\rho\in\mathcal{D}(\mathbb{C}^{d})}\left|\operatorname{tr}\left(\rho(UE_{0})^{\otimes
N}\right)\right|=\left|\operatorname{tr}\left(\rho_{0}(UE_{0})^{\otimes
N}\right)\right|=\max_{F\in\mathcal{DU}_{d^{N}}}\min_{\rho\in\mathcal{D}(\mathbb{C}^{d^{N}})}\left|\operatorname{tr}\left(\rho
U^{\otimes N}F\right)\right|.$

Hence

(14) $\left\|\Phi_{U^{\otimes N}E_{0}^{\otimes N}}-\Phi_{{\rm
1l}}\right\|_{\diamond}=\min_{F\in\mathcal{DU}_{d^{N}}}\left\|\Phi_{U^{\otimes
N}F}-\Phi_{{\rm
1l}}\right\|_{\diamond}\leq\min_{E\in\mathcal{DU}_{d}}\left\|\Phi_{U^{\otimes
N}E^{\otimes N}}-\Phi_{{\rm 1l}}\right\|_{\diamond}$

and eventually

(15) $\left\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{{\rm
1l}}\right\|_{\diamond}=\min_{E\in\mathcal{DU}_{d}}\left\|\Phi_{U^{\otimes
N}E^{\otimes N}}-\Phi_{{\rm 1l}}\right\|_{\diamond}.$

In the second case, let $0\in W\left(U^{\otimes N}E_{0}^{\otimes N}\right)$.
Let us consider the situation when $N$ is the first index for which this
happens, which means that $0\not\in W\left(U^{\otimes N-1}E_{0}^{\otimes
N-1}\right)$.

If the measurements in question can be perfectly distinguished with $N$
queries, then we have
$0\in\textrm{conv}(\lambda_{1}^{N},\lambda_{1}\lambda_{d}^{N-1},\lambda_{d}^{N})$
and there exists a probability vector $p=(p_{1},p_{2},p_{3})$ such that

(16)
$p_{1}\lambda_{1}^{N}+p_{2}\lambda_{1}\lambda_{d}^{N-1}+p_{3}\lambda_{d}^{N}=0.$

Define a state

(17) $\rho=p_{1}\rho_{1}^{\otimes
N}+p_{2}\left(\rho_{1}\otimes\rho_{d}^{\otimes
N-1}\right)+p_{3}\rho_{d}^{\otimes N}.$

We will show that $\operatorname{diag}\left(\rho U^{\otimes N}\right)=0$.
Indeed

(18) $\begin{split}&\operatorname{diag}\left(\rho U^{\otimes N}E_{0}^{\otimes
N}\right)\\\ =&\operatorname{diag}\left(p_{1}\lambda_{1}^{N}\rho_{1}^{\otimes
N}+p_{2}\lambda_{1}\lambda_{d}^{N-1}\left(\rho_{1}\otimes\rho_{d}^{\otimes
N-1}\right)+p_{3}\lambda_{d}^{N}\rho_{d}^{\otimes N}\right)\\\
=&p_{1}\lambda_{1}^{N}\operatorname{diag}(\rho_{1}^{\otimes
N})+p_{2}\lambda_{1}\lambda_{d}^{N-1}\operatorname{diag}\left(\rho_{1}\otimes\rho_{d}^{\otimes
N-1}\right)+p_{3}\lambda_{d}^{N}\operatorname{diag}(\rho_{d}^{\otimes N})\\\
=&\left(p_{1}\lambda_{1}^{N}+p_{2}\lambda_{1}\lambda_{d}^{N-1}+p_{3}\lambda_{d}^{N}\right)\operatorname{diag}(\rho_{1}^{\otimes
N})=0.\end{split}$

Thus from Eq. (7) we obtain that $\mathcal{P}_{U^{\otimes N}}$ and
$\mathcal{P}_{{\rm 1l}}$ are perfectly distinguishable, that is
$\left\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{{\rm
1l}}\right\|_{\diamond}=2$ and hence

(19) $\left\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{{\rm
1l}}\right\|_{\diamond}=\min_{E\in\mathcal{DU}_{d}}\left\|\Phi_{U^{\otimes
N}E^{\otimes N}}-\Phi_{{\rm 1l}}\right\|_{\diamond}.$

When $M$ is the first index for which $0\in W(U^{\otimes M}E_{0}^{\otimes M})$
and $N>M$, then the equality $\left\|\mathcal{P}_{U^{\otimes
M}}-\mathcal{P}_{{\rm 1l}}\right\|_{\diamond}=2$ implies that
$\left\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{{\rm
1l}}\right\|_{\diamond}=2$. Therefore,

(20) $\left\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{{\rm
1l}}\right\|_{\diamond}=\min_{E\in\mathcal{DU}_{d}}\left\|\Phi_{U^{\otimes
N}E^{\otimes N}}-\Phi_{{\rm 1l}}\right\|_{\diamond},$

which completes the proof. ∎

Now we are interested in calculating the number of usages of the black-box
required for perfect discrimination. Let us recall here that in the case of
distinguishing unitary operations this can always be achieved in a finite
number of steps $N=\lceil\frac{\pi}{\Theta(U)}\rceil$ [13].

The similar result is achievable in the case of distinguishing von Neumann
measurements. Using a formalism of the optimal unitary matrix $E_{0}$ i.e.

(21) $\left\|\mathcal{P}_{U}-\mathcal{P}_{{\rm
1l}}\right\|_{\diamond}=\left\|\Phi_{UE_{0}}-\Phi_{{\rm
1l}}\right\|_{\diamond}$

the number of queries needed for perfect discrimination can easily be
computed. We have already proven in Theorem 1 that if the matrix $UE_{0}$ is
in the optimal form, then the matrix $(UE_{0})^{\otimes N}$ will also be
optimal. It means that in order to calculate the diamond norm
$\left\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{{\rm 1l}}\right\|_{\diamond}$
it is sufficient to determine the value $\Theta((UE_{0})^{\otimes N})$, which
grows with $N$ like $\Theta((UE_{0})^{\otimes N})=N\Theta(UE_{0})$ until $0\in
W\left((UE_{0})^{\otimes N}\right)$. The first time zero enters the numerical
range $W\left((UE_{0})^{\otimes N}\right)$ is therefore equal to
$N=\lceil\frac{\pi}{\Upsilon(U)}\rceil$, where $\Upsilon(U)$ is an optimized
version of $\Theta(U)$ _i.e._

(22) $\Upsilon(U)=\min_{E\in\mathcal{DU}_{d}}\Theta(UE).$

We summarize the above discussion in the Corollary below providing the value
of $\left\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{{\rm
1l}}\right\|_{\diamond}$ in terms of $\Theta(U)$.

######  Corollary 1.

Let $N\in\mathbb{N}$, $U\in\mathcal{U}_{d}$. The following holds

  1. (i)

if $N\Upsilon(U)\geq\pi$, then $\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{\rm
1l}\|_{\diamond}=2$;

  2. (ii)

if $N\Upsilon(U)<\pi$, then $\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{\rm
1l}\|_{\diamond}=2\sin(\frac{N}{2}\Upsilon(U))$.

Another interesting property resulting from Theorem 1 and its proof is the
amount of the discriminator’s entanglement with environment. Note that the
minimal dimension of an auxiliary system needed for optimal discrimination is
equal to the rank of the state $\rho_{0}$. One special case involves the
situation $\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{\rm 1l}\|_{\diamond}<2$,
where $\rho_{0}=\frac{1}{2}\rho_{1}^{\otimes N}+\frac{1}{2}\rho_{d}^{\otimes
N}$. In the best case, when eigenvalues $\lambda_{1}$ and $\lambda_{d}$ are
not degenerated, the states $\rho_{1},\rho_{d}$ are pure, hence
$\text{rank}(\rho_{0})=2$. On the other hand, when the spectrum of $UE_{0}$
contains only $\lambda_{1}$ and $\lambda_{d}$, then the rank of $\rho_{0}$ can
be roughly upper-bounded by $(d-1)^{N}+1$.

The situation $\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{\rm
1l}\|_{\diamond}=2$ is more complicated to analyze, due to lack of an analytic
form of the discriminator in a general case. However, finding a pair of not
degenerated eigenvalues $\lambda_{1},\lambda_{d}$ saturating the latter
assumptions of Lemma 2 will lead to $\text{rank}(\rho_{0})=3$, where
$\rho_{0}$ is defined as in Eq. (17) and we check its optimality in the same
manner as in Eq. (18).

The structural characterization of multiple-shot discrimination of von Neumann
measurements given above allows us to draw strong conclusions about
distinguishability of generic pairs of von Neumann measurements. In this work
we restrict our attention to pairs of measurements distributed independently
according to the natural distribution coming from the Haar measure
$\mu(\mathcal{U}_{d})$ [34].

######  Theorem 2.

Consider two independently distributed Haar-random von Naumann measurements on
$\mathbb{C}^{d}$, i.e. $\mathcal{P}_{U},\mathcal{P}_{V}$, where
$U\sim\mu(\mathcal{U}_{d}),\ V\sim\mu(\mathcal{U}_{d})$. Let
$p_{\mathrm{opt}}(\mathcal{P}_{U^{\otimes 2}},\mathcal{P}_{V^{\otimes 2}})$ be
the optimal probability of discrimination measurements $\mathcal{P}_{U}$ and
$\mathcal{P}_{V}$ using two queries and assistance of entanglement. Then, we
have the following bound

(23)
$\underset{U,V\sim\mu(\mathcal{U}_{d})}{\mathrm{Pr}}\left(p_{\mathrm{opt}}(\mathcal{P}_{U^{\otimes
2}},\mathcal{P}_{V^{\otimes 2}})<1\right)\leq\frac{1}{2^{d-1}}.$

In other words, in the limit of large dimensions $d$, typical Haar-random von
Neumann measurements are perfectly distinguishable with the usage of two
queries and assistance of entanglement (the probability that they cannot be
perfectly distinguished is exponentially suppressed as a function of $d$).

###### Proof.

From the unitary invariance of the Haar measure and the symmetry of the
problem of measurement discrimination it follows that the distribution of the
random variable $p_{\mathrm{opt}}(\mathcal{P}_{{\rm 1l}^{\otimes
2}},\mathcal{P}_{U^{\otimes 2}})$ is identical to that of
$p_{\mathrm{opt}}(\mathcal{P}_{U^{\otimes 2}},\mathcal{P}_{V^{\otimes 2}})$.
Consequently, we have

(24)
$\underset{U,V\sim\mu(\mathcal{U}_{d})}{\mathrm{Pr}}\left(p_{\mathrm{opt}}(\mathcal{P}_{U^{\otimes
2}},\mathcal{P}_{V^{\otimes
2}})<1\right)=\underset{U\sim\mu(\mathcal{U}_{d})}{\mathrm{Pr}}\left(p_{\mathrm{opt}}(\mathcal{P}_{{\rm
1l}^{\otimes 2}},\mathcal{P}_{U^{\otimes 2}})<1\right).$

From Corollary 1 it follows that the condition $\|\mathcal{P}_{U}-P_{\rm
1l}\|_{\diamond}\geq\sqrt{2}$ implies $\|\mathcal{P}_{U^{\otimes 2}}-P_{{\rm
1l}^{\otimes 2}}\|_{\diamond}=2$ and consequently we have also perfect
discrimination of two copies of measurements:
$p_{\mathrm{opt}}(\mathcal{P}_{U^{\otimes 2}},\mathcal{P}_{{\rm 1l}^{\otimes
2}})=1$. Therefore we have

(25)
$\underset{U\sim\mu(\mathcal{U}_{d})}{\mathrm{Pr}}\left(p_{\mathrm{opt}}(\mathcal{P}^{\otimes
2}_{U},\mathcal{P}^{\otimes 2}_{\rm
1l})<1\right)\leq\underset{U\sim\mu(\mathcal{U}_{d})}{\mathrm{Pr}}\left(\|\mathcal{P}_{U}-\mathcal{P}_{\rm
1l}\|_{\diamond}\leq\sqrt{2}\right).$

Using now the characterization given in Eq.(6) in conjunction with the formula
in Eq.(5) we obtain $\|\mathcal{P}_{U}-P_{\rm 1l}\|_{\diamond}\geq
2\sqrt{1-|U_{11}|^{2}}$ (note that $U_{11}=\operatorname{tr}(|1\rangle\langle
1|U)\in W(U)$). Using this and simple algebra we get

(26)
$\underset{U\sim\mu(\mathcal{U}_{d})}{\mathrm{Pr}}\left(\mathcal{P}_{U^{\otimes
2}},\mathcal{P}_{{\rm 1l}^{\otimes
2}})<1\right)\leq\underset{U\sim\mu(\mathcal{U}_{d})}{\mathrm{Pr}}\left(|U_{11}|^{2}\geq\frac{1}{2}\right).$

The right-hand side of the above inequality can be computed exactly using the
property that for Haar-distributed $U$ the random variable $X=|U_{11}|^{2}$ is
distributed according to the beta distribution $p(X)=(d-1)(1-X)^{d-2}$ (see
for instance Eq. (9) in [35]). The simple integration gives $(1/2)^{d-1}$,
which together with Eq.(24) gives the claimed result. ∎

###  3.2. Optimality of the parallel scheme

In this subsection we will prove a theorem which thesis is rendered in the
spirit of the results obtained in [36] for discrimination of unitary channels.
The authors of [36] showed that parallel discrimination scheme is optimal
among all possible architectures for the case of discrimination of unitary
channels. Nevertheless, this is no longer true when styduing the
discrimination of general rank-one POVMs [20]. In the case of discrimination
of POVMs with rank-one effects one may need to use adaptive discrimination
scheme to obtain perfect distinguishability. While considering the
discrimination of von Neumann measurements, one could expect that we can
improve the discrimination by performing some processing based on the obtained
measurement’s labels. It appears that such processing will not improve the
discrimination. The following theorem states that while distinguishing two
distinct von Neumann measurement, the use of parallel discrimination scheme is
optimal.

######  Theorem 3.

Let $U\in\mathcal{U}_{d}$. Consider the distinguishability of general quantum
network with $N$ uses of the black box in which there is one of two
measurements - either $\mathcal{P}_{U}$ or $\mathcal{P}_{\rm 1l}$. Then the
probability of correct distinction cannot be better then in the parallel
scenario.

###### Proof.

Without loss of generality we may assume that the processing is performed
using only unitary operations. Indeed, using Stinespring dilation theorem, any
channel might be represented via a unitary channel on a larger system followed
by the partial trace operation. What is left to observe is that
$\left\|\operatorname{tr}_{B}(X_{AB})\right\|_{1}\leq\left\|X_{AB}\right\|_{1}$
for arbitrary bipartite matrix $X_{AB}$.

Figure 6. Schematic depiction of the sequential scheme. In the $k$th step,
after obtaining the label $i_{k}$, we utilize all labels $i_{1},\ldots,i_{k}$
to modify the remaining parts of the state in the hope of improving
distinguishability.

The sequential scheme of disrimination of von Neumann measurements is shown in
Fig. 6 and can be expressed as a channel

(27) $\Psi_{U}=\left(\Delta_{1,\ldots,N}\otimes{\rm 1l}\right)\Phi_{A_{U}},$

associated with a matrix $A_{U}$. Here $\Delta_{1,\ldots,N}$ is the dephasing
channel on subsystems $1,\ldots,N$. The channel $\Phi_{A_{U}}$ is shown in
Fig. 7 and the exact form of this transformation can be found in Appendix B.

Figure 7. Visualization of the channel $\Phi_{A_{U}}$. Here
$V^{(k)}=\sum_{i_{1},\ldots,i_{k}}|i_{1},\ldots,i_{k}\rangle\langle
i_{1},\ldots,i_{k}|\otimes V^{(k)}_{i_{1},\ldots,i_{k}}$.

Assuming that matrix $U$ is chosen in the optimal form as in (22) _i.e._
$\Upsilon(U)=\Theta(U)$ we may calculate the distance between $\Psi_{U}$ and
$\Psi_{\rm 1l}$ as

(28) $\begin{split}\max_{\rho}\left\|\left(\Psi_{U}-\Psi_{\rm
1l}\right)(\rho)\right\|_{1}&=\max_{\rho}\left\|\left[\left(\Delta_{1,\ldots,N}\otimes{\rm
1l}\right)\left(\Phi_{A_{U}}-\Phi_{A_{\rm
1l}}\right)\right](\rho)\right\|_{1}\\\
&\leq\max_{\rho}\left\|\left(\Phi_{A_{U}}-\Phi_{A_{\rm
1l}}\right)(\rho)\right\|_{1}\leq\max_{\rho}\left\|\left(\Phi_{U^{\otimes
N}\otimes{\rm 1l}}-\Phi_{{\rm 1l}}\right)(\rho)\right\|_{1}\\\
&=\left\|\Phi_{U^{\otimes N}}-\Phi_{{\rm
1l}}\right\|_{\diamond}=\left\|\mathcal{P}_{U^{\otimes N}}-\mathcal{P}_{\rm
1l}\right\|_{\diamond},\end{split}$

where we maximize over states $\rho$ of appropriate dimensions. The induced
trace norm is monotonically decreasing under the action of channels and this
gives us first inequality. The second one follows from the optimality of the
parallel scheme of distinguishing unitary channels [36]. Therefore, the
adaptive scenario does not give any advantage over the parallel scheme. ∎

##  4\. Unambiguous discrimination

The unambiguous discrimination of measurements $\mathcal{P}_{\rm 1l}$ and
$\mathcal{P}_{U}$ can be understood as unambiguous discrimination [28] of
states generated by the corresponding channel. Specifically, for a fixed input
state $\sigma$, the output states $\mathcal{P}_{\rm
1l}(\sigma),\mathcal{P}_{U}(\sigma)$ can be unambiguously discriminated using
the measurement strategy $\mathcal{M}=(M_{\rm 1l},M_{U},M_{?})$, where the
first two effects represent conclusive answers and the last one corresponds to
the inconclusive output of the procedure. For equal a priori probabilities of
occurrence of $\mathcal{P}_{\rm 1l}$ and $\mathcal{P}_{U}$, as well as fixed
$\sigma$ and $\mathcal{M}$, the success probability is given by

(29) $p_{\mathrm{u}}\left(\mathcal{P}_{\rm
1l},\mathcal{P}_{U};\sigma,\mathcal{M}\right)=\frac{1}{2}\left(\operatorname{tr}(M_{\rm
1l}\mathcal{P}_{\rm
1l}(\sigma))+\operatorname{tr}(M_{U}\mathcal{P}_{U}(\sigma))\right),$

where additionally the unambiguity condition has to be satisfied:

(30) $\operatorname{tr}(M_{U}\mathcal{P}_{\rm
1l}(\sigma))=\operatorname{tr}(M_{\rm 1l}\mathcal{P}_{U}(\sigma))=0.$

The optimal success probability of unambiguous discrimination of measurements
$\mathcal{P}_{\rm 1l},\mathcal{P}_{U}$ _without the assistance of
entanglement_ can be now defined as the maximum of (29) over all strategies
that do not use entanglement. Formally, we have

(31) $\tilde{p}_{\mathrm{u}}\left(\mathcal{P}_{\rm
1l},\mathcal{P}_{U}\right)\mathrel{:=}\max_{\sigma\in\mathcal{D}(\mathbb{C}^{d})}\max_{\mathcal{M}\in\mathrm{POVM}(\mathbb{C}^{d})}p_{u}\left(\mathcal{P}_{\rm
1l},\mathcal{P}_{U};\sigma,\mathcal{M}\right),$

where $\mathcal{M}\in\mathrm{POVM}(\mathbb{C}^{d})$ is a three-outcome
measurement on $\mathbb{C}^{d}$ that additionally satisfies constrains (30).
Likewise, the optimal _entanglement-assisted_ unambiguous discrimination
probability is given by

(32) $p_{\mathrm{u}}\left(\mathcal{P}_{\rm
1l},\mathcal{P}_{U}\right)\mathrel{:=}\max_{\sigma\in\mathcal{D}(\mathbb{C}^{d}\otimes\mathbb{C}^{d^{\prime}})}\max_{\mathcal{M}\in\mathrm{POVM}(\mathbb{C}^{d}\otimes\mathbb{C}^{d^{\prime}})}p_{\mathrm{u}}\left(\mathcal{P}_{\rm
1l},\mathcal{P}_{U};\sigma,\mathcal{M}\right),$

where this time $\sigma$ is a (possibly entangled) state on the extended
Hilbert space $\mathbb{C}^{d}\otimes\mathbb{C}^{d^{\prime}}$. In what follows
we will show that without loss of generality it is enough to consider the
dimension $d^{\prime}$ of ancilla equal to the dimension of the system $d$.

In the remainder of this section we will present our main result which gives
the probability of unambiguous discrimination with the use of entanglement,
presented in Theorem 5. Aside from giving a simple expression for this
probability, this result reduces the problem of unambiguous measurement
discrimination to a convex optimization task and gives a simple relationship
between the diamond norm and the probability of unambiguous discrimination.

Additionally we provide a result for unambiguous discrimination without the
utilization of entangled states, stated as Theorem 4. The downside of this
approach is that we arrive at complicated optimization task.

###  4.1. Unambiguous discrimination without assistance of entanglement

Let us fix the input state $\sigma\in\mathcal{D}(\mathbb{C}^{d})$. Then, for
every Hermitian operator $X$ on $\mathbb{C}^{d}$ we have

(33) $\operatorname{tr}(\mathcal{P}_{\rm
1l}(\sigma)X)=\operatorname{tr}(\mathcal{P}_{\rm 1l}(\sigma)\tilde{X})\ ,\
\operatorname{tr}(\mathcal{P}_{U}(\sigma)X)=\operatorname{tr}(\mathcal{P}_{U}(\sigma)\tilde{X}),$

where $\tilde{X}=\operatorname{diag}(X)$ is a dephased version of the operator
$X$. Let now $\mathcal{M}=(M_{\rm 1l},M_{U},M_{?})$ be a measurement on
$\mathbb{C}^{d}$ satisfying Eq.(30). Then, by the virtue of (33) we have

(34) $\tilde{p}_{\mathrm{u}}\left(\mathcal{P}_{\rm
1l},\mathcal{P}_{U};\sigma,\mathcal{M}\right)=\tilde{p}_{\mathrm{u}}\left(\mathcal{P}_{\rm
1l},\mathcal{P}_{U};\sigma,\tilde{\mathcal{M}}\right),$

where $\tilde{\mathcal{M}}\mathrel{:=}\left(\operatorname{diag}(M_{\rm
1l}),\operatorname{diag}(M_{U}),\operatorname{diag}(M_{?})\right)$ (it can be
easily checked that $\tilde{\mathcal{M}}$ is a POVM on $\mathbb{C}^{d}$ since
$\operatorname{diag}(\cdot)$ is completely positive and unital). Consequently,
without loss of generality, for any fixed input state $\sigma$, we can
restrict our attention to measurements $\mathcal{M}$ with diagonal effects.

From the unambiguity condition Eq.(30) we obtain that $M_{\rm
1l}\perp\mathrm{supp}(\mathcal{P}_{U}(\sigma))$ and
$M_{U}\perp\mathrm{supp}(\mathcal{P}_{\rm 1l}(\sigma))$. Therefore, the
optimal measurements can be always chosen as projectors onto disjoint subsets
$\Gamma,\Delta$ of $\\{1,\ldots,d\\}$. For any subset
$A\subset\\{1,2,\dots,d\\}$ we define projectors

(35) $\Pi_{A}\mathrel{:=}\sum_{i\in A}|i\rangle\langle i|\ ,\
\Theta_{A}\mathrel{:=}U\ \Pi_{A}\ U^{\dagger}.$

Now, if the result of the final measurement is from the set $\Gamma$, we know
with certainty that the unknown measurement was $\mathcal{P}_{\rm 1l}$.
Analogously, if the result was from the set $\Delta$, then we can be sure that
there was the second measurement in the unknown device. The formula for the
success probability reads

(36) $\tilde{p}_{\mathrm{u}}\left(\mathcal{P}_{\rm
1l},\mathcal{P}_{U};\sigma,\Gamma,\Delta\right)=\frac{1}{2}\operatorname{tr}(\Pi_{\Gamma}\sigma)+\frac{1}{2}\operatorname{tr}(\Theta_{\Delta}\sigma).$

Importantly, the input state $\sigma$ satisfies $\sigma\perp\Pi_{\Delta}$ and
$\sigma\perp\Theta_{\Gamma}$. For fixed subsets $\Delta,\Gamma$, due to
linearity, the maximum over $\sigma$ equals
$\|\mathbb{P}_{\Gamma,\Delta}(\Pi_{\Gamma}+\Theta_{\Delta})\mathbb{P}_{\Gamma,\Delta}\|$,
where $\|\cdot\|$ denotes the operator norm and $\mathbb{P}_{\Gamma,\Delta}$
is the orthogonal projector onto
$\textrm{Span}\left(\\{U|i\rangle\\}_{i\in\Gamma^{c}}\right)\cap\textrm{Span}\left(\\{|j\rangle\\}_{j\in\Delta^{c}}\right)$,
where $A^{c}$ denotes the complement of $A$. By optimizing over disjoint
subsets $\Delta,\Gamma\subset\\{1,\ldots,d\\}$ we obtain the following result.

######  Theorem 4.

The optimal success probability of unambiguous discrimination, without the use
of entanglement, between von Neumann measurements $\mathcal{P}_{\rm 1l}$ and
$\mathcal{P}_{U}$ is given by

(37) $\tilde{p}_{\mathrm{u}}\left(\mathcal{P}_{\rm
1l},\mathcal{P}_{U}\right)=\frac{1}{2}\max_{\Gamma,\Delta\subset\\{1,2,\dots,d\\}:\Gamma\cap\Delta=\emptyset}\Big{\|}\mathbb{P}_{\Gamma,\Delta}(\Pi_{\Gamma}+\Theta_{\Delta})\mathbb{P}_{\Gamma,\Delta}\Big{\|}\,$

with $\mathbb{P}_{\Gamma,\Delta}$ defined as above.

######  Remark.

The projector $\mathbb{P}_{\Gamma,\Delta}$ projects onto the intersection of
supports of $\Pi_{\Delta^{c}}$ and $\Theta_{\Gamma^{c}}$. By the use of
Theorem 4 from [37], we can express the optimal probability of unambiguous
discrimination as

(38) $\begin{split}\tilde{p}_{\mathrm{u}}\left(\mathcal{P}_{\rm
1l},\mathcal{P}_{U}\right)=2\max_{\Gamma,\Delta\subset\\{1,2,\dots,d\\}:\Gamma\cap\Delta=\emptyset}\Big{\|}&\Pi_{\Delta^{c}}(\Pi_{\Delta^{c}}+\Theta_{\Gamma^{c}})^{-1}\Theta_{\Delta}(\Pi_{\Delta^{c}}+\Theta_{\Gamma^{c}})^{-1}\
\Pi_{\Delta^{c}}\\\
&+\Theta_{\Gamma^{c}}(\Pi_{\Delta^{c}}+\Theta_{\Gamma^{c}})^{-1}\Pi_{\Gamma}(\Pi_{\Delta^{c}}+\Theta_{\Gamma^{c}})^{-1}\
\Theta_{\Gamma^{c}}\Big{\|},\end{split}$

where $(\cdot)^{-1}$ denotes Moore-Penrose pseudo inverse [38]. Moreover, the
optimal input state is the one which gives the above norm.

######  Corollary 2.

In the case of qubit measurements optimal probability of unambiguous
discrimination of $\mathcal{P}_{\rm 1l}$ and $\mathcal{P}_{U}$ is is given by
the following(discontinuous) function

(39) $\tilde{p}_{\mathrm{u}}\left(\mathcal{P}_{\rm
1l},\mathcal{P}_{U}\right)=\begin{cases}1&\text{ if }\ \ |U_{1,2}|^{2}=1\\\
\frac{1}{2}|U_{1,2}|^{2}&\text{ if }\ \ |U_{1,2}|^{2}<1\end{cases}\ .$

In both cases the optimal input state can be chosen to be $|1\rangle\langle
1|$.

######  Remark.

The above considerations can be extended to unambiguous discrimination of
multiple copies of von Neumann measurements applied in parallel. To this end,
if we have access to $N$ parallel queries to a black box measurement, it
suffices to replace unitaries ${\rm 1l}$ by ${\rm 1l}^{\otimes N}$ and $U$ by
$U^{\otimes N}$ in the above computations. Interestingly, in the contrast to
unambiguous discrimination of quantum states [28], having access to two copies
of black box measurements, sometimes allows attaining perfect discrimination.
Specifically, consider the problem of discriminating between $\mathcal{P}_{\rm
1l}$ and $\mathcal{P}_{H}$, where $H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}$. Explicit computation shows that by taking the input state
as $|\Psi\rangle=\frac{1}{\sqrt{2}}(|1\rangle|1\rangle-|2\rangle|2\rangle)$
allows us to perfectly distinguish between $\mathcal{P}^{\otimes 2}_{\rm 1l}$
and $\mathcal{P}^{\otimes 2}_{H}$.

###  4.2. Unambiguous discrimination with assistance of entanglement

Knowing that for perfect discrimination of quantum measurements the use of
entanglement significantly improves the discrimination [26], we ask whether
the same is true for unambiguous discrimination. As explained earlier, we
treat the quantum measurement $\mathcal{M}$ with effects
$\\{M_{i}\\}_{i=1}^{n}$ as a measure and prepare channel with the classical
output (see Eq.(2)). Consider now the situation in which a black-box
measurement acts on a system extended by the ancilla space $\mathcal{H}_{B}$
(of some dimension $d_{1}$). Without loss of generality we assume that the
input state is pure i.e. $\sigma=|\psi_{AB}\rangle\langle\psi_{AB}|$. Let $X$
be a matrix such that
$|\psi_{AB}\rangle=\sum_{i,j=1}^{d,d_{1}}X_{ij}|i\rangle|j\rangle$. The action
of the channel $\mathcal{M}$ on $|\psi_{AB}\rangle\langle\psi_{AB}|$ can be
expressed as

(40) $\mathcal{M}\otimes{\rm
1l}_{B}(|\psi_{AB}\rangle\langle\psi_{AB}|)=\sum_{i=1}^{n}|i\rangle\langle
i|\otimes X^{T}M^{T}_{i}\overline{X}.$

Now, if we apply one of two von Neumann measurements $\mathcal{P}_{{\rm 1l}}$
or $\mathcal{P}_{U}$, we get the following states

(41) $\displaystyle\mathcal{P}_{\rm 1l}\otimes{\rm
1l}_{B}(|\psi_{AB}\rangle\langle\psi_{AB}|)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{d}|i\rangle\langle i|\otimes X^{T}|i\rangle\langle
i|\overline{X},$ (42) $\displaystyle\mathcal{P}_{U}\otimes{\rm
1l}_{B}(|\psi_{AB}\rangle\langle\psi_{AB}|)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{d}|i\rangle\langle i|\otimes
X^{T}\overline{U}|i\rangle\langle i|U^{T}\overline{X}.$

By using analogous reasoning to the one given in the preceding section it can
be shown that for a given input state $|\psi_{AB}\rangle$, the optimal
measurement unambiguously discriminating the above two states has the
following structure

(43) $\mathcal{N}=\sum_{i=1}^{d}|i\rangle\langle i|\otimes\mathcal{T}_{i},$

where $\mathcal{T}_{i}$ is a POVM on $\mathcal{H}_{B}$. In other words, the
optimal discrimination strategy amounts to first reading off a classical label
$i$ and then performing a measurement $\mathcal{T}_{i}$ on the auxiliary
register. From Eq.(41) we see that upon obtaining the label $i$, the state of
the auxiliary subsystem is either

(44) $|x_{i}\rangle\langle x_{i}|=p_{i}^{-1}X^{\top}|i\rangle\langle
i|\overline{X},$

when measurement $\mathcal{P}_{\rm 1l}$ was performed, or it is given by

(45) $|y_{i}\rangle\langle
y_{i}|=q_{i}^{-1}X^{\top}\overline{U}|i\rangle\langle i|U^{T}\overline{X}$

if $\mathcal{P}_{U}$ was implemented. In the above formulas $p_{i},q_{i}$ are
responsible for normalization. We assume that $p_{i}>0$ and $q_{i}>0$
(otherwise the specific outcome $i$ does not occur). We see that states
$|x_{i}\rangle\langle x_{i}|$, $|y_{i}\rangle\langle y_{i}|$ are pure and
therefore the optimal measurements
$\mathcal{T}_{i}=\\{T^{(i)}_{1},T^{(i)}_{2},T^{(i)}_{?}\\}$ will be simply
given by

(46) $\begin{split}T^{(i)}_{1}&=\gamma_{1}({\rm 1l}-|y_{i}\rangle\langle
y_{i}|),\\\ T^{(i)}_{2}&=\gamma_{2}({\rm 1l}-|x_{i}\rangle\langle x_{i}|),\\\
T^{(i)}_{?}&={\rm 1l}-T_{1}-T_{2},\end{split}$

for some choice of $\gamma_{1,2}$ which guarantees the non-negativity of
$T^{(i)}_{?}$.

The probability of success in unambiguous discrimination of pure states
$|x\rangle,|y\rangle$ with unequal a priori probabilities $\eta,1-\eta$ is
given by [39]

(47) $p^{u}_{succ}(x,y,\eta)=\begin{cases}1-\eta-(1-\eta)c^{2}&\text{for
}\eta<\frac{c^{2}}{1+c^{2}}\\\ 1-2c\sqrt{\eta(1-\eta)}&\text{for
}\frac{c^{2}}{1+c^{2}}\leq\eta\leq\frac{1}{1+c^{2}}\\\ 1-(1-\eta)-\eta
c^{2}&\text{for }\frac{1}{1+c^{2}}<\eta,\end{cases}$

where $c=|\langle{x}|{y}\rangle|$.

We will use the following upper bound

(48) $p^{u}_{succ}(x,y,\eta)\leq 1-2c\sqrt{\eta(1-\eta)},$

which can be verified directly by elementary calculations.

Let $\rho=XX^{\dagger}$. The overlap $c_{i}$ between states of the auxiliary
subsystem is given by

(49) $c_{i}=|\langle{x_{i}}|{y_{i}}\rangle|=|\langle
i|\overline{X}X^{\top}\overline{U}|i\rangle|/\sqrt{p_{i}q_{i}}=|\langle i|\rho
U|i\rangle|/\sqrt{p_{i}q_{i}},$

while a priori probabilities of $|x_{i}\rangle,|y_{i}\rangle$ upon obtaining
label $i$ are
$\eta_{i}=\frac{p_{i}}{p_{i}+q_{i}},1-\eta_{i}=\frac{q_{i}}{p_{i}+q_{i}}$.
Taking the above into account, we get that probability of success in
unambiguous measurement on auxiliary subsystem given that label $i$ was
observed can be bounded from above by

(50) $p^{u}_{succ}(x_{i},y_{i},\eta_{i})\leq
1-2c_{i}\frac{\sqrt{p_{i}q_{i}}}{p_{i}+q_{i}}=1-\frac{2|\langle i|\rho
U|i\rangle|}{p_{i}+q_{i}}.$

Therefore, the overall probability of success is bounded by

(51) $\begin{split}p_{\mathrm{u}}(\mathcal{P}_{{\rm
1l}},\mathcal{P}_{U})&=\max_{|\psi_{AB}\rangle}\sum_{i}P(label=i)p^{u}_{succ}(x_{i},y_{i},\eta_{i})\\\
&\leq\max_{\rho}\sum_{i}\frac{1}{2}(p_{i}+q_{i})\left(1-\frac{2|\langle i|\rho
U|i\rangle|}{p_{i}+q_{i}}\right)\\\ &=1-\min_{\rho}\sum_{i}|\langle i|\rho
U|i\rangle|.\end{split}$

The above bound is tight since the optimal state $\rho$ will give equal
probabilities $p_{i}$ and $q_{i}$ – this follows from Lemma 2. The above
considerations may be stated as the following theorem.

######  Theorem 5.

The optimal success probability of unambiguous discrimination between von
Neumann measurements $\mathcal{P}_{\rm 1l}$ and $\mathcal{P}_{U}$ is given by

(52) $p_{\mathrm{u}}(\mathcal{P}_{{\rm
1l}},\mathcal{P}_{U})=1-\min_{\rho}\sum_{i}|\langle i|\rho U|i\rangle|.$

Figure 8. Schematic depiction of the relationship between the diamond norm
$\|\mathcal{P}_{U}-\mathcal{P}_{\rm 1l}\|_{\diamond}$, the probability of
unambiguous discrimination $p_{\mathrm{u}}(\mathcal{P}_{{\rm
1l}},\mathcal{P}_{U})$ and $\Upsilon(U)$.

These results give a nice geometric interpretation for the relationship
between the diamond norm and the probability of unambiguous discrimination.
This is depicted in Fig. 8. We start with a von Neumann measurement in a basis
given by some unitary matrix $U$ and try to distinguish it from the
measurement in the computational basis. We denote $U$’s eigenvalues as
$\lambda_{1},\ldots,\lambda_{d}$ ordered according to their phases and put
$\measuredangle(\lambda_{1},\lambda_{d})=\Upsilon(U)$. The dependence of the
diamond norm and probability of unambiguous discrimination is clearly shown.

######  Remark 1.

The above calculations can be easily extended to the case of parallel
discrimination scheme. It suffices to substitute $U$ with $U^{\otimes N}$ and
then we obtain that

(53) $p_{\mathrm{u}}(\mathcal{P}_{U^{\otimes N}},\mathcal{P}_{\rm
1l})=1-\min_{\rho}\sum_{i}|\langle i|\rho U^{\otimes N}|i\rangle|.$

The following corollary states that in the qubit case the unambiguous
discrimination with the assistance of entanglement always outperforms the
unambiguous discrimination without the use of entanglement. On top of that,
the special cases for which the use of entanglement does not give any
advantage are described.

######  Corollary 3.

Let $\mathcal{P}_{\rm 1l}$ and $\mathcal{P}_{U}$ be two von Neumann
measurements on a qubit. If $|U_{1,1}|\not\in\\{0,1\\}$, then the probability
in the case of entanglement-assisted unambiguous discrimination is given by

(54) $p_{\mathrm{u}}^{e}=1-|U_{1,1}|$

and it is always greater then the probability without assistance of
entanglement

(55) $p_{u}=\begin{cases}1&\text{ if }\ \ |U_{1,2}|^{2}=1\\\
\frac{1}{2}|U_{1,2}|^{2}&\text{ if }\ \ |U_{1,2}|^{2}<1.\end{cases}$

Moreover, if $|U_{1,1}|\in\\{0,1\\}$, then
$p_{\mathrm{u}}^{e}=p_{\mathrm{u}}$.

Basing on Remark 1, we note that the angle $\Upsilon(U)$ increases in the
multiple-shot case with the number of queries. This is depicted in Fig. 9 for
two- and three-shots scenarios.

Figure 9. Figure similar to Fig. 8 which represents two- (left) and three-
(right) shots scenario.

###  4.3. Optimality of the parallel scheme for unambiguous discrimination

In the general scheme we are allowed to use conditional unitary
transformations $\\{X_{i}\\}$ after each measurement, so our setting for
discrimination is the same as in Fig. 6. Henceforth, we will assume that
matrix $U$ is chosen in the optimal form as in (22) _i.e._
$\Upsilon(U)=\Theta(U)$. Let us denote

(56) $\begin{split}&|\tilde{x}_{i}\rangle=\left(|i\rangle\langle i|\otimes{\rm
1l}_{N+1}\right)A_{{\rm 1l}}|\psi_{A,B}\rangle\\\
&|\tilde{y}_{i}\rangle=\left(|i\rangle\langle i|\otimes{\rm
1l}_{N+1}\right)A_{U}|\psi_{A,B}\rangle.\end{split}$

Using the same notation as was used in Section 3.2 and repeating the
calculation from the single-shot scenario we can upper-bound the probability
of success discrimination.

(57) $\begin{split}p_{\mathrm{u}}(\Psi_{U},\Psi_{\rm
1l})\leq&1-\sum_{i}|\langle{\tilde{x}_{i}}|{\tilde{y}_{i}}\rangle|\leq
1-\left|\sum_{i}\langle{\tilde{x}_{i}}|{\tilde{y}_{i}}\rangle\right|=1-\left|\langle\psi_{A,B}|A_{\rm
1l}^{\dagger}A_{U}|\psi_{A,B}\rangle\right|.\end{split}$

From the work [36] we know there exists a state $|\phi\rangle$ such that

$|\langle\psi_{A,B}|A_{\rm
1l}^{\dagger}A_{U}|\psi_{A,B}\rangle|\geq|\langle\phi|U^{\otimes
N}|\phi\rangle|.$

This implies that our probability is upper-bounded by
$1-\min_{\rho}\sum_{i}|\langle i|\rho U^{\otimes N}|i\rangle|$, which is
achievable in parallel scheme as in Remark 1.

##  5\. Conclusions

We have presented a comprehensive treatment of the problem of discrimination
of von Neumann measurements. First of all, we showed that for two measurements
$\mathcal{P}_{1}$ and $\mathcal{P}_{2}$, $\mathcal{P}_{1}\neq\mathcal{P}_{2}$,
there exists a finite number $N$ of uses of the black box which allows us to
achieve perfect discrimination. This is formally stated in Corollary 1. We
also proved that the parallel discrimination scheme is optimal in the scenario
of multiple-shot discrimination of von Neumann measurements (see Theorem 3).

Moreover, we studied unambiguous discrimination of von Neumann measurements.
Our main contribution to this problem was the derivation of the general
schemes that attain the optimal success probability both with (see Theorem 5)
and without (see Theorem 4) the assistance of entanglement. Interestingly, for
entanglement-assisted unambiguous discrimination the optimal success
probability is functionally related to the corresponding success probability
for minimal error discrimination. Finally, we show that the parallel scheme is
also optimal for unambiguous discrimination.

There are many interesting directions for further study that still remain to
be explored. Below we list the most important (in our opinion) open research
problems:

  * •

Generalization of our results from projective measurements to other classes of
measurements such as projective-simulable measurements [40], measurements with
limited number of outcomes [41] or general quantum measurements (POVMs).

  * •

Systematical study of the problem of unambiguous discrimination of projective
measurements in the multiple-shot regime.

  * •

Can typical pairs of Haar-random projective measurements on $\mathbb{C}^{d}$
be discriminated perfectly using only one query and the assistance of
entanglement as $d\rightarrow\infty$?

  * •

How much entanglement is needed to attain the optimal success probability of
multiple-shot discrimination of generic projective measurements on
$\mathbb{C}^{d}$? In the same scenario, is it necessary to adopt the final
measurement to the pair of measurements to be discriminated?

## Acknowledgements

We would like to thank Alessandro Bisio and Giulio Chiribella for fruitful
discussions.

This work was supported by the Polish National Science Centre under project
numbers 2016/22/E/ST6/00062 (ZP, AK, RK) and 2015/18/E/ST2/ 00327 (ŁP). MO
acknowledges the support of Homing programme of the Foundation for Polish
Science co-financed by the European Union under the European Regional
Development Fund.

##  Appendix A Imperfect discrimination of von Neumann measurements: the form
of a discriminator

In this appendix we will quote two lemmas from [26] which contribute the main
part of the proof of Theorem 1. The first lemma states that a function
$|\operatorname{Tr}(\rho UE)|$ has a saddle point. The second lemma gives an
equivalent condition to the existence of the saddle point and presents the
form of the optimal state.

######  Lemma 1 (Lemma 4 from [26]).

Let us fix a unitary matrix $U\in\mathcal{U}_{d}$. Then,

(58)
$\max_{E\in\mathcal{DU}_{d}}\min_{\rho\in\mathcal{D}(\mathbb{C}^{d})}|\operatorname{Tr}(\rho
UE)|=\min_{\rho\in\mathcal{D}(\mathbb{C}^{d})}\max_{E\in\mathcal{DU}_{d}}|\operatorname{Tr}(\rho
UE)|.$

######  Lemma 2 (Lemma 5 from [26]).

Let us fix a unitary matrix $U\in\mathcal{U}_{d}$ and

  * •

$E_{0}\in\mathcal{DU}_{d}$ and
$D(E)=\min_{\rho\in\mathcal{D}(\mathbb{C}^{d})}|\operatorname{Tr}\rho UE|$,

  * •

$D(E_{0})>0$,

  * •

$\lambda_{1},\lambda_{d}$ denote the most distant pair of eigenvalues of
$UE_{0}$

  * •

$P_{1}$, $P_{d}$ denote the projectors on the subspaces spanned by the
eigenvectors corresponding to $\lambda_{1}$, $\lambda_{d}$.

Then, the function
$\mathcal{D}(\mathbb{C}^{d})\times\mathcal{DU}_{d}\ni(\rho,E)\mapsto|\operatorname{Tr}(\rho
UE)|$ has a saddle point in $(\rho_{0},E_{0})$ i.e.

(59)
$\max_{E\in\mathcal{DU}_{d}}\min_{\rho\in\mathcal{D}(\mathbb{C}^{d})}|\operatorname{Tr}(\rho
UE)|=|\operatorname{Tr}(\rho_{0}UE_{0})|=\min_{\rho\in\mathcal{D}(\mathbb{C}^{d})}\max_{E\in\mathcal{DU}_{d}}|\operatorname{Tr}(\rho
UE)|$

if and only if there exist states $\rho_{1},\rho_{d}$ such that

  * •

$\rho_{1}=P_{1}\rho_{1}P_{1}$,

  * •

$\rho_{d}=P_{d}\rho_{d}P_{d}$,

  * •

$\operatorname{diag}(\rho_{1})=\operatorname{diag}(\rho_{d})$.

Moreover, if the above holds, then the state $\rho_{0}$ satisfying Eq. (59)
can be chosen as $\frac{1}{2}\rho_{1}+\frac{1}{2}\rho_{d}$.

##  Appendix B Explicit form of the matrix $A_{U}$

The matrix $A_{U}$ is a general operation which allows for adaptive
information processing in the sequential discrimination scenario of von
Neumann measurements. It consists of a sequence of unitary matrices $U$ acting
successively on given registers interlacing with classically controlled
unitary operations $V^{i}$. The explicit form of the matrix $A_{U}$ is given
by

(60) $\begin{split}A_{U}=&\left({\rm 1l}_{1,\ldots N-1}\otimes U\otimes{\rm
1l}_{N+1}\right)\\\
&\left(\sum_{i_{1},\ldots,i_{N-1}}|i_{1},\ldots,i_{N-1}\rangle\langle
i_{1},\ldots,i_{N-1}|\otimes V^{(N-1)}_{i_{1},\ldots,i_{N-1}}\right)\\\
&\left({\rm 1l}_{1,\ldots,N-2}\otimes U\otimes{\rm 1l}_{N,N+1}\right)\\\
&\left(\sum_{i_{1},\ldots,i_{N-2}}|i_{1},\ldots,i_{N-2}\rangle\langle
i_{1},\ldots,i_{N-2}|\otimes V^{(N-2)}_{i_{1},\ldots,i_{N-2}}\right)\\\
&\ldots\\\ &\left({\rm 1l}_{1}\otimes U\otimes{\rm 1l}_{3,\ldots
N+1}\right)\\\ &\left(\sum_{i_{1}}|i_{1}\rangle\langle i_{1}|\otimes
V^{(1)}_{i_{1}}\right)\\\ &\left(U\otimes{\rm 1l}_{2,\ldots
N+1}\right).\end{split}$

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