Quantum detection with uncertain states

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 and an unknown state. 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 on the unknown states, and minimize the expected probability of a detection error. We find that under both criteria, the optimal detectors are equivalent to the optimal detectors of an effective ensemble. In the worst case, the effective ensemble is comprised of the known states with altered prior probabilities, and in the average case it is made up of altered states with the original prior probabilities.