Minimum-error discrimination between mixed quantum states

TL;DR

This paper establishes bounds and conditions for minimum-error discrimination between mixed quantum states, extending known limits and analyzing attainability and comparisons with unambiguous discrimination.

Contribution

It provides a general lower bound for ambiguous discrimination of mixed states, conditions for its attainability, and compares it with unambiguous discrimination.

Findings

Derived a lower bound matching Helstrom limit for two states

Provided conditions for the bound's attainability

Compared minimum-error and unambiguous discrimination probabilities

Abstract

We derive a general lower bound on the minimum-error probability for {\it 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 a 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 {\it ambiguously} discriminating arbitrary $m$ mixed quantum states with the optimal failure probability for {\it unambiguously} discriminating the same states.