Detection of an ensemble mixed with unknown states

Full Text:We address the problem of distinguishing among a finite collection of quantum states, when the states are not entirely known. For completely specified states, necessary and sufficient conditions on a quantum measurement minimizing the probability of a detection error have been derived. In this work, we assume that each of the states in our collection is a mixture of a known state pi and an unknown state ρi1, so that ?i qipi + (1- qi)pi where 0 <=≤ qi<=≤ 1 are known. We investigate two criteria for optimality. The first is minimization of the worst-case probability of a detection error. For the second we assume a probability distribution for the unknown states, and seek minimization of the expected probability of a detection error. We derive necessary and sufficient conditions for optimality under both criteria, and interpret them