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Physical Review A

Relation between minimum-error discrimination and optimum unambiguous discrimination

Daowen Qiu, Lvjun Li

Published Sep 29, 2010• 8 citations
Publisher

Abstract

In this paper, we investigate the relationship between the minimum-error probability ${Q}_{E}$ of ambiguous discrimination and the optimal inconclusive probability ${Q}_{U}$ of unambiguous discrimination. It is known that for discriminating two states, the inequality ${Q}_{U}\ensuremath{\geqslant}2{Q}_{E}$ has been proved in the literature. The main technical results are as follows: (1) We show that, for discriminating more than two states, ${Q}_{U}\ensuremath{\geqslant}2{Q}_{E}$ may not hold again, but the infimum of ${Q}_{U}/{Q}_{E}$ is $1,$ and there is no supremum of ${Q}_{U}/{Q}_{E}$, which implies that the failure probabilities of the two schemes for discriminating some states may be narrowly or widely gapped. (2) We derive two concrete formulas of the minimum-error probability ${Q}_{E}$ and the optimal inconclusive probability ${Q}_{U}$, respectively, for ambiguous discrimination and unambiguous discrimination among arbitrary $m$ simultaneously diagonalizable mixed quantum states with given prior probabilities. In addition, we show that ${Q}_{E}$ and ${Q}_{U}$ satisfy the relationship that ${Q}_{U}\ensuremath{\geqslant}\frac{m}{m\ensuremath{-}1}{Q}_{E}$.

DOI
10.1103/physreva.82.032333

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In this paper, we investigate the relationship between the minimum-error probability QE{Q}_{E}QE​ of ambiguous discrimination and the optimal inconclusive probability QU{Q}_{U}QU​ of unambiguous discrimination. It is known that for discriminating two states, the inequality QU\ensuremath⩾2QE{Q}_{U}\ensuremath{\geqslant}2{Q}_{E}QU​\ensuremath⩾2QE​ has been proved in the literature. The main technical results are as follows: (1) We show that, for discriminating more than two states, QU\ensuremath⩾2QE{Q}_{U}\ensuremath{\geqslant}2{Q}_{E}QU​\ensuremath⩾2QE​ may not hold again, but the infimum of QU/QE{Q}_{U}/{Q}_{E}QU​/QE​ is 1,1,1, and there is no supremum of QU/QE{Q}_{U}/{Q}_{E}QU​/QE​, which implies that the failure probabilities of the two schemes for discriminating some states may be narrowly or widely gapped. (2) We derive two concrete formulas of the minimum-error probability QE{Q}_{E}QE​ and the optimal inconclusive probability QU{Q}_{U}QU​, respectively, for ambiguous discrimination and unambiguous discrimination among arbitrary mmm simultaneously diagonalizable mixed quantum states with given prior probabilities. In addition, we show that QE{Q}_{E}QE​ and QU{Q}_{U}QU​ satisfy the relationship that QU\ensuremath⩾mm\ensuremath−1QE{Q}_{U}\ensuremath{\geqslant}\frac{m}{m\ensuremath{-}1}{Q}_{E}QU​\ensuremath⩾m\ensuremath−1m​QE​.
The minimum-error probability of ambiguous discrimination for two quantum states is the well-known {\it Helstrom limit} presented in 1976. Since then, it has been thought of as an intractable problem to obtain the minimum-error probability for ambiguously discriminating arbitrary mmm quantum states. In this paper, we obtain a new lower bound on the minimum-error probability for ambiguous discrimination and compare this bound with six other bounds in the literature. Moreover, we show that the bound between ambiguous and unambiguous discrimination does not extend to ensembles of more than two states. Specifically, the main technical contributions are described as follows: (1) We derive a new lower bound on the minimum-error probability for ambiguous discrimination among arbitrary mmm mixed quantum states with given prior probabilities, and we present a necessary and sufficient condition to show that this lower bound is attainable. (2) We compare this new lower bound with six other bounds in the literature in detail, and, in some cases, this bound is optimal. (3) It is known that if m=2m=2m=2, the optimal inconclusive probability of unambiguous discrimination QUQ_{U}QU​ and the minimum-error probability of ambiguous discrimination QEQ_{E}QE​ between arbitrary given mmm mixed quantum states have the relationship QU≥2QEQ_{U}\geq 2Q_{E}QU​≥2QE​. In this paper, we show that, however, if m>2m>2m>2, the relationship QU≥2QEQ_{U}\geq 2Q_{E}QU​≥2QE​ may not hold again in general, and there may be no supremum of QU/QEQ_{U}/Q_{E}QU​/QE​ for more than two states, which may also reflect an essential difference between discrimination for two-states and multi-states. (4) A number of examples are constructed.
We present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between unambiguous and minimum error discrimination, the two standard approaches to quantum state discrimination. We introduce a very general method that enables us to obtain the solution in a wide range of cases and give a complete characterization of the minimum discrimination error as a function of the rate of inconclusive answers. A critical value of this rate is identified that coincides with the minimum failure probability in the cases where unambiguous discrimination is possible and provides a natural generalization of it when states cannot be unambiguously discriminated. The method is illustrated on two explicit examples: discrimination of two pure states with arbitrary prior probabilities and discrimination of trine states.
We derive a general lower bound on the minimum-error probability for ambiguous discrimination between arbitrary mmm mixed quantum states with given prior probabilities. When m=2m=2m=2, this bound is precisely the well-known Helstrom limit. Also, we give a general lower bound on the minimum-error probability for discriminating quantum operations. Then we further analyze how this lower bound is attainable for ambiguous discrimination of mixed quantum states by presenting necessary and sufficient conditions related to it. Furthermore, with a restricted condition, we work out an upper bound on the minimum-error probability for ambiguous discrimination of mixed quantum states. Therefore, some sufficient conditions are obtained for the minimum-error probability attaining this bound. Finally, under the condition of the minimum-error probability attaining this bound, we compare the minimum-error probability for ambiguously discriminating arbitrary mmm mixed quantum states with the optimal failure probability for unambiguously discriminating the same states.
We derive a new bound on the minimum-error probability for ambiguous discrimination and compare the new bound with six other bounds in the literature. Specifically, the main technical contributions are as follows. (1) We obtain a general lower bound on the minimum-error probability for ambiguous discrimination among arbitrary mmm mixed quantum states with given prior probabilities. We further analyze how this lower bound is attainable by presenting a necessary and sufficient condition related to it. (2) We compare this new lower bound with six other bounds in the literature, in detail. We show that this new bound improves the previous one in the literature and in the equiprobable case; the new bound also improves another one. Moreover, we give an example to show that this new bound is the tightest one for some cases.
Based on our previous publication [U. Herzog and J. A. Bergou, Phys. Rev. A 71 , 050301(R)(2005)] we investigate the optimum measurement for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. Unambiguous discrimination of nonorthogonal states is possible in a probabilistic way, at the expense of a nonzero probability of inconclusive results, where the measurement fails. Along with a discussion of the general problem, we give an example illustrating our method of solution. We also provide general inequalities for the minimum achievable failure probability and discuss in more detail the necessary conditions that must be fulfilled when its absolute lower bound, proportional to the fidelity of the states, can be reached.
We study an optimum measurement for quantum state discrimination, which maximizes the probability of correct results when the probability of inconclusive results is fixed at a given value. The measurement describes minimum-error discrimination if this value is zero, while under certain conditions it corresponds to optimized maximum-confidence discrimination, or to optimum unambiguous discrimination, respectively, when the fixed value reaches a definite minimum. Using operator conditions that determine the optimum measurement, we derive analytical solutions for the discrimination of two mixed qubit states, including the case of two pure states occurring with arbitrary prior probabilities, and for the discrimination of NNN symmetric states, both pure and mixed. We also consider a case where the given density operators resolve the identity operator, and we specify the optimality conditions for partially symmetric states. Moreover, we show that from the complete solution for arbitrary values of the fixed rate of inconclusive results one can always obtain the optimum measurement in another strategy where the error rate is fixed, and vice versa.