  We study the discrimination of N mixed quantum states in an optimal
measurement that maximizes the probability of correct results while the
probability of inconclusive results is fixed at a given value. After
considering the discrimination of N states in a d-dimensional Hilbert space, we
focus on the discrimination of qubit states. We develop a method to determine
an optimal measurement for discriminating arbitrary qubit states, taking into
account that often the optimal measurement is not unique and the maximum
probability of correct results can be achieved by several different
measurements. Analytical results are derived for a number of examples, mostly
for the discrimination between qubit states which possess a partial symmetry,
but also for discriminating N equiprobable qubit states and for the
dicrimination between a pure and a uniformly mixed state in d dimensions. In
the special case where the fixed rate of inconclusive results is equal to zero,
our method provides a treatment for the minimum-error discrimination of
arbitrary qubit states which differs from previous approaches.
