  In this paper we present the solution to the problem of optimally
discriminating among quantum states, i.e., identifying the states with maximum
probability of success when a certain fixed rate of inconclusive answers is
allowed. By varying the inconclusive rate, the scheme optimally interpolates
between Unambiguous and Minimum Error discrimination, the two standard
approaches to quantum state discrimination. We introduce a very general method
that enables us to obtain the solution in a wide range of cases and give a
complete characterization of the minimum discrimination error as a function of
the rate of inconclusive answers. A critical value of this rate is identified
that coincides with the minimum failure probability in the cases where
unambiguous discrimination is possible and provides a natural generalization of
it when states cannot be unambiguously discriminated. The method is illustrated
on two explicit examples: discrimination of two pure states with arbitrary
prior probabilities and discrimination of trine states.
