An Optimization Problem from Linear Filtering with Quantum Measurements

We consider the problem of optimal (in the sense of minimum error variance) linear filtering a vector discrete-time signal process, which influences a quantum mechanical field, utilizing quantum mechanical measurements. The nonclassical characteristic of the problem is the joint optimization over the measurement process and the linear signal processing scheme. The problem is formulated as an optimization problem of a functional over a set of operator-valued measures and matrices. We prove the existence of optimal linear filters and provide necessary and sufficient conditions for optimality.