Quantum binary channels with mixed states

We address binary communication channels with symbols encoded in two states of a finite dimensional Hilbert spaces. For pure states we confirm that the optimal decoding stage, maximizing the mutual information,coincides with the projective measure that minimizes the error probability (Bayes criterion). On the other hand, we prove that for communication schemes based on mixed states the optimal decoding, still being a projective measurement, is generally different from the Bayes’ one, unless the two density operators commute.