We study an optimized measurement that discriminates two mixed quantum states
with maximum confidence for each conclusive result, thereby keeping the overall
probability of inconclusive results as small as possible. When the rank of the
detection operators associated with the two different conclusive outcomes does
not exceed unity we obtain a general solution. As an application, we consider
the discrimination of two mixed qubit states. Moreover, for the case of
higher-rank detection operators we give a solution for particular states. The
relation of the optimized measurement to other discrimination schemes is also
discussed.Comment: 7 pages, 1 figure, accepted for publication in Phys. Rev.