  We study an optimized measurement which discriminates N mixed quantum states
occurring with given prior robabilities. 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 \leq N) and the discrimination of N
symmetric mixed qubit states.
