  We derive a general lower bound on the minimum-error probability for {\it
ambiguous discrimination} between arbitrary $m$ mixed quantum states with given
prior probabilities. When $m=2$, this bound is precisely the well-known
Helstrom limit. Also, we give a general lower bound on the minimum-error
probability for discriminating quantum operations. Then we further analyze how
this lower bound is attainable for ambiguous discrimination of mixed quantum
states by presenting necessary and sufficient conditions related to it.
Furthermore, with a restricted condition, we work out a upper bound on the
minimum-error probability for ambiguous discrimination of mixed quantum states.
Therefore, some sufficient conditions are obtained for the minimum-error
probability attaining this bound. Finally, under the condition of the
minimum-error probability attaining this bound, we compare the minimum-error
probability for {\it ambiguously} discriminating arbitrary $m$ mixed quantum
states with the optimal failure probability for {\it unambiguously}
discriminating the same states.
