  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 is more natural and can simplify the filter computations for
some applications. Hence we formalize the notion of strongly commuting (Davies)
instruments which allows one to develop joint measurement statistics for POVM
type measurements. This allows us to prove the existence of conditional POVMs,
which is essential for the development of a filtering equation. We demonstrate
that under generally satisfied assumptions, knowing the observed probe POVM
operator is sufficient to uniquely specify the quantum filtering evolution for
the system.
