The problem of specifying the optimum quantum detector in multiple hypotheses testing is considered for application to optical communications. The quantum digital detection problem is formulated as a linear programming problem on an infinite-dimensional space. A necessary and sufficient condition is derived by the application of a general duality theorem specifying the optimum detector in terms of a set of linear operator equations and inequalities. Existence of the optimum quantum detector is also established. The optimality of commuting detection operators is discussed in some examples. The structure and performance of the optimal receiver are derived for the quantum detection of narrow-band coherent orthogonal and simplex signals. It is shown that modal photon counting is asymptotically optimum in the limit of a large signaling alphabet and that the capacity goes to infinity in the absence of a bandwidth limitation.

A quantum mechanical form of the Cramér-Rao inequality and a minimum-mcan-square-error quantum estimator for multiple parameters are derived, allowing all possible quantum measurements of the received field. The role of nonselfadjoint operators is emphasized in the formulation. Relations of our results to previous work on quantum estimation are discussed. For the estimation of complex mode amplitudes of coherent signals in Gaussian noise, it is shown that the optimal receiver measures the photon annihilation operator, which corresponds to optical heterodyning. This demonstrates the possible optimality of nonselfadjoint operators and clearly indicates the importance of considering more general quantum measurements in quantum signal detection.