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The minimum-error probability of ambiguous discrimination for two quantumstates is the well-known {\it Helstrom limit} presented in 1976. Since then, ithas been thought of as an intractable problem to obtain the minimum-errorprobability for ambiguously discriminating arbitrary $m$ quantum states. Inthis paper, we obtain a new lower bound on the minimum-error probability forambiguous discrimination and compare this bound with six other bounds in theliterature. Moreover, we show that the bound between ambiguous and unambiguousdiscrimination does not extend to ensembles of more than two states.Specifically, the main technical contributions are described as follows: (1) Wederive a new lower bound on the minimum-error probability for ambiguousdiscrimination among arbitrary $m$ mixed quantum states with given priorprobabilities, and we present a necessary and sufficient condition to show thatthis lower bound is attainable. (2) We compare this new lower bound with sixother bounds in the literature in detail, and, in some cases, this bound isoptimal. (3) It is known that if $m=2$, the optimal inconclusive probability ofunambiguous discrimination $Q_{U}$ and the minimum-error probability ofambiguous discrimination $Q_{E}$ between arbitrary given $m$ mixed quantumstates 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 ingeneral, and there may be no supremum of $Q_{U}/Q_{E}$ for more than twostates, which may also reflect an essential difference between discriminationfor two-states and multi-states. (4) A number of examples are constructed.

Minimum-error discrimination of quantum states: New bounds and comparison