  We consider two different optimized measurement strategies for the
discrimination of nonorthogonal quantum states. The first is conclusive
discrimination with a minimum probability of inferring an erroneous result, and
the second is unambiguous, i. e. error-free, discrimination with a minimum
probability of getting an inconclusive outcome, where the measurement fails to
give a definite answer. For distinguishing between two mixed quantum states, we
investigate the relation between the minimum error probability achievable in
conclusive discrimination, and the minimum failure probability that can be
reached in unambiguous discrimination of the same two states. The latter turns
out to be at least twice as large as the former for any two given states. As an
example, we treat the case that the state of the quantum system is known to be,
with arbitrary prior probability, either a given pure state, or a uniform
statistical mixture of any number of mutually orthogonal states. For this case
we derive an analytical result for the minimum probability of error and perform
a quantitative comparison to the minimum failure probability.
