  The minimum-error probability of ambiguous discrimination for two quantum
states is the well-known {\it Helstrom limit} presented in 1976. Since then, it
has been thought of as an intractable problem to obtain the minimum-error
probability for ambiguously discriminating arbitrary $m$ quantum states. In
this paper, we obtain a new lower bound on the minimum-error probability for
ambiguous discrimination and compare this bound with six other bounds in the
literature. Moreover, we show that the bound between ambiguous and unambiguous
discrimination does not extend to ensembles of more than two states.
Specifically, the main technical contributions are described as follows: (1) We
derive a new lower bound on the minimum-error probability for ambiguous
discrimination among arbitrary $m$ mixed quantum states with given prior
probabilities, and we present a necessary and sufficient condition to show that
this lower bound is attainable. (2) We compare this new lower bound with six
other bounds in the literature in detail, and, in some cases, this bound is
optimal. (3) It is known that if $m=2$, the optimal inconclusive probability of
unambiguous discrimination $Q_{U}$ and the minimum-error probability of
ambiguous discrimination $Q_{E}$ between arbitrary given $m$ mixed quantum
states have the relationship $Q_{U}\geq 2Q_{E}$. In this paper, we show that,
however, if $m>2$, the relationship $Q_{U}\geq 2Q_{E}$ may not hold again in
general, and there may be no supremum of $Q_{U}/Q_{E}$ for more than two
states, which may also reflect an essential difference between discrimination
for two-states and multi-states. (4) A number of examples are constructed.
