Relation between minimum-error discrimination and optimum unambiguous discrimination
Daowen Qiu, Lvjun Li · Physical Review A · 2010
In this paper, we investigate the relationship between the minimum-error probability ${Q}_{E}$ of ambiguous discrimination and the optimal inconclusive probability ${Q}_{U}$ of unambiguous discrimination. It is known that for discriminating two states, the inequality ${Q}_{U}\ensuremath{\geqslant}2{Q}_{E}$ has been proved in the literature. The main technical results are as follows: (1) We show that, for discriminating more than two states, ${Q}_{U}\ensuremath{\geqslant}2{Q}_{E}$ may not hold again, but the infimum of ${Q}_{U}/{Q}_{E}$ is $1,$ and there is no supremum of ${Q}_{U}/{Q}_{E}$, which implies that the failure probabilities of the two schemes for discriminating some states may be narrowly or widely gapped. (2) We derive two concrete formulas of the minimum-error probability ${Q}_{E}$ and the optimal inconclusive probability ${Q}_{U}$, respectively, for ambiguous discrimination and unambiguous discrimination among arbitrary $m$ simultaneously diagonalizable mixed quantum states with given prior probabilities. In addition, we show that ${Q}_{E}$ and ${Q}_{U}$ satisfy the relationship that ${Q}_{U}\ensuremath{\geqslant}\frac{m}{m\ensuremath{-}1}{Q}_{E}$.