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We address the problem of distinguishing among a finite collection of quantumstates, when the states are not entirely known. For completely specifiedstates, necessary and sufficient conditions on a quantum measurement minimizingthe probability of a detection error have been derived. In this work, we assumethat each of the states in our collection is a mixture of a known state and anunknown state. We investigate two criteria for optimality. The first isminimization of the worst-case probability of a detection error. For the secondwe assume a probability distribution on the unknown states, and minimize of theexpected probability of a detection error.  We find that under both criteria, the optimal detectors are equivalent to theoptimal detectors of an ``effective ensemble''. In the worst-case, theeffective ensemble is comprised of the known states with altered priorprobabilities, and in the average case it is made up of altered states with theoriginal prior probabilities.Comment: Refereed version. Improved numerical examples and figures. A few  typos fixe

Quantum detection with uncertain states