Quantum filtering using POVM measurements: Conditional Expectation

The objective of this work is to develop a recursive, discrete time quantum filtering equation for a system that interacts with a probe, on which measurements are performed according to the Positive Operator Valued Measures (POVMs) framework. POVMs are the most general measurements one can make on a quantum system and although in principle they can be reformulated as projective measurements on larger spaces, for which filtering results exist, a direct treatment of POVMs can be more natural and simplify the filter computations for some applications. Hence we formalize the notion of commuting (Davies) instruments which allows one to develop joint measurement statistics of two POVM type measurements as explicit functions of the POVMs. This allows us to prove the existence of a notion of conditional expectation POVM, which is essential for the development of a filtering equation. We demonstrate that under generally satisfied assumptions, the reduced model given by POVM elements is sufficient for the purpose of our quantum filtering task.