Multiparameter filtering with Quantum Measurements

We analyze here the problem of estimating a member of a vector discrete time process utilizing past and present quantum mechanical measurements. The minimum variance linear estimator based on optimal present measure­ment selection combined with an optimal linear processing of past measurements is studied. A necessary condition for the optimal extended measurement and the optimal coefficient matri­ces is given which leads to the generation of several more specialized necessary conditions. Certain operator equations related to these necessary conditions are studied and it is found that the conditions are necessary and sufficient in the commutative case. Finally when the average quantum measurement is linear in the random signal and the signal process is pairwise Gaussian, the filter separates: the optimal measurement can be taken the same as the optimal measurement with no regard to past data and the past and present data are processed classically. The results are illustrated by considering the estimator of the amplitude and phase of a laser received in a single-mode ca­vity along with thermal noise; when the random signal sequence satisfies a linear recursion, the estimate can be computed recursively.