Quantum filtering for multiple diffusive and Poissonian measurements

TL;DR

This paper rigorously derives a quantum filter for multiple measurements, including diffusive and Poissonian types, applicable to quantum optics scenarios like homodyne detection and photon counting, extending previous single-measurement filters.

Contribution

It provides a comprehensive derivation of a quantum filter for multiple measurement processes, including necessary conditions for measurement commutation and an explicit example involving beam splitter outputs.

Findings

Derived a necessary and sufficient condition for measurement commutation.

Extended single-measurement quantum filters to multiple measurements.

Corrected and explicitly derived quantum filter for homodyne and photon counting measurements.

Abstract

We provide a rigorous derivation of a quantum filter for the case of multiple measurements being made on a quantum system. We consider a class of measurement processes which are functions of bosonic field operators, including combinations of diffusive and Poissonian processes. This covers the standard cases from quantum optics, where homodyne detection may be described as a diffusive process and photon counting may be described as a Poissonian process. We obtain a necessary and sufficient condition for any pair of such measurements taken at different output channels to satisfy a commutation relationship. Then, we derive a general, multiple measurement quantum filter as an extension of a single-measurement quantum filter. As an application we explicitly obtain the quantum filter corresponding to homodyne detection and photon counting at the output ports of a beam splitter, correcting an earlier result.