Optimized maximum-confidence discrimination of N mixed quantum states and application to symmetric states

We study an optimized measurement which discriminates N mixed quantum states occurring with given prior probabilities. The measurement yields the maximum achievable confidence for each of the N conclusive outcomes, thereby keeping the overall probability of inconclusive outcomes as small as possible. It corresponds to optimum unambiguous discrimination when for each outcome the confidence is equal to unity. Necessary and sufficient optimality conditions are derived and general properties of the optimum measurement are obtained. The results are applied to the optimized maximum-confidence discrimination of N equiprobable symmetric mixed states. Analytical solutions are presented for a number of examples, including the discrimination of N symmetric pure states spanning a d-dimensional Hilbert space (d <= N) and the discrimination of N symmetric mixed qubit states.