Generalized quantum state discrimination problems

Abstract

We address a broad class of optimization problems of finding quantum measurements, which includes the problems of finding an optimal measurement in the Bayes criterion and a measurement maximizing the average success probability with a fixed rate of inconclusive results. Our approach can deal with any problem in which each of the objective and constraint functions is formulated by the sum of the traces of the multiplication of a Hermitian operator and a detection operator. We first derive dual problems and necessary and sufficient conditions for an optimal measurement. We also consider the minimax version of these problems and provide necessary and sufficient conditions for a minimax solution. Finally, for optimization problem having a certain symmetry, there exists an optimal solution with the same symmetry. Examples are shown to illustrate how our results can be used

Mutual Information and Quantum Discord in Quantum State Discrimination with a Fixed Rate of Inconclusive Outcomes

“…This is a hard mathematical problem in which only the results for some lower and upper bounds [ 11 , 17 , 18 ] are known. The Bayes cost problem is associated with the minimization of an average cost function [ 19 ], for example, to find the minimum probability of error in a scheme of quantum state discrimination [ 20 ]. In particular, if Alice uses two non-orthogonal pure states, it is known that the minimum error discrimination implemented by Bob maximizes the mutual information and minimizes the error probability [ 11 , 16 , 21 ].…”

“…If the set of states used by Alice, in the quantum discrimination scenario, contains two or more non-orthogonal states, then it is impossible for Bob to distinguish them deterministically [ 33 ]. In the literature, there are several strategies that can be used to distinguish or discriminate between these states [ 16 , 20 ]. The strategy we need to use will depend on the particular application.…”

Quasi-Bell entangled coherent states and its quantum discrimination problem in the presence of thermal noise

“…1,2 Above all, our interest is focused on the so-called quasi-Bell entangled coherent states in this paper. The quasi-Bell entangled coherent states are defined as a set of the following four superpositions of two-mode coherent states: |α, ±α 12 + |−α, ∓α 12 and |α, ±α 12 − |−α, ∓α 12 . 3 In an ideal situation, it has been clarified that the quasi-Bell entangled coherent states are applicable to several applications.…”

Structure of Optimal State Discrimination in Generalized Probabilistic Theories

“…All these follow from the fact that the state space is convex. For comparison with quantum cases, the formulation for minimum-error discrimination has been shown in [14], and see also its applications to various forms of figures of merit in [15].…”