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Physical Review A

Bounds on unambiguous discrimination between quantum states

Lvjun Li

Published Sep 17, 2012• 6 citations
Publisher

Abstract

In this paper, we present a general upper bound on the success probability for unambiguous discrimination among arbitrary $m$ mixed quantum states with given a priori probability. We further analyze how this upper bound can be achievable by presenting a sufficient condition related to it, and we compare this upper bound with three other upper bounds. Moreover, for the issue of the unambiguous identification of $m$ unknown multicopy qudit pure states, we also get a bound.

DOI
10.1103/physreva.86.032320

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In this paper, we present a general upper bound on the success probability for unambiguous discrimination among arbitrary mmm mixed quantum states with given a priori probability. We further analyze how this upper bound can be achievable by presenting a sufficient condition related to it, and we compare this upper bound with three other upper bounds. Moreover, for the issue of the unambiguous identification of mmm unknown multicopy qudit pure states, we also get a bound.
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.
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 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.
Recently the problem of unambiguous state discrimination of mixed quantum states has attracted much attention. So far, bounds on the optimum success probability have been derived [T. Rudolph, R. W. Spekkens, and P. S. Turner, Phys. Rev. A 68, 010301(R) (2003)]. For two mixed states they are given in terms of the fidelity. Here we give tighter bounds as well as necessary and sufficient conditions for two mixed states to reach these bounds. Moreover we construct the corresponding optimal measurement strategies. With this result, we provide analytical solutions for unambiguous discrimination of a class of generic mixed states. This goes beyond known results which are all reducible to some pure state case. Additionally, we show that examples exist where the bounds cannot be reached.