Multiple-parameter quantum estimation and measurement of nonselfadjoint observables

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.