Quantum mechanical linear filtering of vector signal processes

The problem of estimating a member of a discrete-time vector process from past and present quantum mechanical measurements is considered; specifically, the minimum-variance linear estimator based on optimal present measurements is studied. Necessary and sufficient conditions that characterize the optimal processing matrix coefficients and the optimal measurements are discussed and interpreted. The optimal linear filter is compared to the optimal quantum estimator without postprocessing of past data. When the signal sequence is pairwise Gaussian and the optimal quantum measurement without postproccssing has the properties that it is linear in a specific sense and that its outcome and the corresponding element of the signal sequence are jointly Gaussian, then the optimal linear filter separates. That is, the optimal measurement can be taken to be the same as thc optimal measurement without regard to past data, and the past and present data are processed classically. Thc results are illustrated by considering the filtering of the in-phase and quadrature amplitudes of a laser field received in a single-mode cavity along with thermal noise. In this case, when the random signal sequence satisfies a linear recursion, the estimate can be computed recursively in a very efficient manner.