Optimum quantum decision between a pure and a mixed state with application to detection of TCS signals

A method is given for calculating the minimum average error probability attainable in deciding between a mixed state and a pure state of a quantum system. It is applied, first, to the detection of a noisy coherent pulse represented by a state with a Gaussian P-representation and, second, to the detection of a signal represented at the transmitter by a two-photon coherent state (TCS). In both problems the field of the receiver is free of background noise and, when no signal was transmitted, is in the ground state.