Quantum filtering using POVM measurements

TL;DR

This paper develops a recursive quantum filtering equation for systems interacting with probes measured via POVMs, enabling more natural and potentially simplified filtering computations in quantum systems.

Contribution

It formalizes strongly commuting instruments and proves the existence of conditional POVMs, facilitating direct quantum filtering with POVM measurements.

Findings

Established a recursive quantum filtering equation for POVM measurements.

Proved the uniqueness of the filtering evolution given the observed POVM operator.

Formalized the notion of strongly commuting instruments for joint measurement statistics.

Abstract

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.