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arXiv:0707.3970 · Physical Review A

Minimum-error discrimination between mixed quantum states

Daowen Qiu

Published Jan 23, 2008• 66 citations
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Abstract

We derive a general lower bound on the minimum-error probability for ambiguous discrimination between arbitrary $m$ mixed quantum states with given prior probabilities. When $m=2$, 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 $m$ mixed quantum states with the optimal failure probability for unambiguously discriminating the same states.

DOI
10.1103/physreva.77.012328

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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.
We consider two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is ambiguous discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i.e., error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum-error probability achievable in ambiguous discrimination, and the minimum failure probability that can be reached in unambiguous discrimination of the same two states. The latter turns out to be at least twice as large as the former for any two given states. As an example, we treat the case where the state of the quantum system is known to be, with arbitrary prior probability, either a given pure state, or a uniform statistical mixture of any number of mutually orthogonal states. For this case we derive an analytical result for the minimum probability of error and perform a quantitative comparison with the minimum failure probability.
For the optimal success probability under minimum-error discrimination between $r\ensuremath{\ge}2$ arbitrary quantum states prepared with any a priori probabilities, we find new general analytical lower and upper bounds and specify the relations between these new general bounds and the known general bounds, lower and upper. We also present the example where the values of the new general lower and upper bounds on the optimal success probability are tighter than the values of most of the general analytical bounds known in the literature. The new upper bound on the optimal success probability explicitly generalizes to r>2r>2r>2 the form of the Helstrom bound. For r=2r=2r=2, each of our new bounds, lower and upper, reduces to the Helstrom bound.
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 ${Q}_{U}\ensuremath{\geqslant}2{Q}_{E}$ 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 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 ${Q}_{U}\ensuremath{\geqslant}\frac{m}{m\ensuremath{-}1}{Q}_{E}$.
For the optimal success probability under minimum-error discrimination between r≥2r\geq2r≥2 arbitrary quantum states prepared with any a priori probabilities, we find new general analytical lower and upper bounds and specify the relations between these new general bounds and the general bounds known in the literature. We also present the example where the new general analytical bounds, lower and upper, on the optimal success probability are tighter than most of the general analytical bounds known in the literature. The new upper bound on the optimal success probability explicitly generalizes to r>2r>2r>2 the form of the Helstrom bound. For r=2r=2r=2, each of our new bounds, lower and upper, reduces to the Helstrom bound.