On the Sufficient Optimality Condition for Quantum Information   Processing
  The necessary and sufficient conditions of optimality of the decoding of quantum signals minimizing the Bayesian risk are generalized for the Shannon mutual information criteria. It is shown that for a linear channel with Gaussian boson noise these conditions are satisfied by coherent quasi-measurement of the canonical annihilation amplitudes in the received superposition. 
V. P. Belavkin and A. G. Vantsyan