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arXiv:2002.05452 · Quantum Information Processing

Discrimination of POVMs with rank-one effects

Aleksandra Krawiec, Łukasz Pawela, Zbigniew Puchała

Published Nov 24, 2020• 12 citations
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Abstract

Abstract The main goal of this work is to provide an insight into the problem of discrimination of positive operator-valued measures with rank-one effects. It is our intention to study multiple-shot discrimination of such measurements, that is the case when we are able to use to unknown measurement a given number of times. Furthermore, we are interested in comparing two possible discrimination schemes: the parallel and adaptive ones. To this end, we construct a pair of symmetric informationally complete positive operator-valued measures which can be perfectly discriminated in a two-shot adaptive scheme but cannot be distinguished in the parallel scheme. On top of this, we provide an explicit algorithm which allows us to find this adaptive scheme.

DOI
10.1007/s11128-020-02883-3

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Abstract The main goal of this work is to provide an insight into the problem of discrimination of positive operator-valued measures with rank-one effects. It is our intention to study multiple-shot discrimination of such measurements, that is the case when we are able to use to unknown measurement a given number of times. Furthermore, we are interested in comparing two possible discrimination schemes: the parallel and adaptive ones. To this end, we construct a pair of symmetric informationally complete positive operator-valued measures which can be perfectly discriminated in a two-shot adaptive scheme but cannot be distinguished in the parallel scheme. On top of this, we provide an explicit algorithm which allows us to find this adaptive scheme.
There has been much interest in so-called SIC-POVMs, i.e., rank 1 symmetric informationally complete positive operator valued measures. In this paper we discuss the larger class of POVMs that are symmetric and informationally complete, but not necessarily rank 1. This class of POVMs is of some independent interest. In particular it includes a POVM that is closely related to the discrete Wigner function. However, it is interesting mainly because of the light it casts on the problem of constructing rank 1, symmetric, informationally complete POVMs. In this connection we derive an extremal condition alternative to the one derived by Renes et al.
We present an in-depth study of the problem of multiple-shot discrimination of von Neumann measurements in finite-dimensional Hilbert spaces. Specifically, we consider two scenarios: minimum error and unambiguous discrimination. In the case of minimum error discrimination, we focus on discrimination of measurements with the assistance of entanglement. We provide an alternative proof of the fact that 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. Moreover, we analytically find the minimal number of queries needed for perfect discrimination. We also show that in this scenario querying 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 unambiguous discrimination 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.
We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension d^2-1 corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC-POVMs contains weak SIC-POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC-POVM.
We introduce a mathematical framework for the description of measurements of quantum processes. Using this framework, process estimation problems can be treated in a similar way as state estimation problems, only replacing the concept of positive-operator-valued measure (POVM) by the concept of process POVM (PPOVM). In particular, we will show that any measurement of qudit channels can be described by a collection of effects (positive operators) defined on a two-qudit system. However, the effects forming a PPOVM are not normalized in the usual sense. We will demonstrate the use of this formalism in discrimination problems by showing that perfect channel discrimination is equivalent to a specific unambiguous state discrimination.
We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC POVMs contains weak SIC POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC POVM.
Optimization of the mean efficiency for unambiguous (or error-free) discrimination among NNN given linearly independent nonorthogonal states should be realized in a way to keep the probabilistic quantum-mechanical interpretation. This imposes a condition on a certain matrix to be positive semidefinite. We reformulated this condition in such a way that the conditioned optimization problem for the mean efficiency was reduced to finding an unconditioned maximum of a function defined on a unit NNN sphere for equiprobable states and on an NNN ellipsoid if the states are given with different probabilities. We established that for equiprobable states a point on the sphere with equal values of Cartesian coordinates, which we call symmetric point, plays a special role. Sufficient conditions for a vector set are formulated for which the mean efficiency for equiprobable states takes its maximal value at the symmetric point. This set, in particular, includes previously studied symmetric states. A subset of symmetric states, for which the optimal measurement corresponds to a positive-operator-valued measure (POVM) requiring a one-dimensional ancilla space is constructed. We presented our constructions of a POVM suitable for the ancilla space dimension varying from 1 to NNN and the Neumark extension differing from the existing schemes by the property that it is straightforwardly applicable to the case when it is desirable to present the whole space system+ancilla\text{system}+\text{ancilla}system+ancilla as the tensor product of a two-dimensional ancilla space and the NNN-dimensional system space.