Optimal state discrimination with a fixed rate of inconclusive results: Analytical solutions and relation to state discrimination with a fixed error rate

摘要

We study an optimum measurement for quantum state discrimination, which maximizes the probability ofncorrect results when the probability of inconclusive results is fixed at a given value. The measurement describesnminimum-error discrimination if this value is zero, while under certain conditions it corresponds to optimizednmaximum-confidence discrimination, or to optimum unambiguous discrimination, respectively, when the fixednvalue reaches a definite minimum. Using operator conditions that determine the optimum measurement, wenderive analytical solutions for the discrimination of two mixed qubit states, including the case of two pure statesnoccurring with arbitrary prior probabilities, and for the discrimination of N symmetric states, both pure andnmixed. We also consider a case where the given density operators resolve the identity operator, and we specifynthe optimality conditions for partially symmetric states. Moreover, we show that from the complete solution fornarbitrary values of the fixed rate of inconclusive results one can always obtain the optimum measurement innanother strategy where the error rate is fixed, and vice versa.