A Phase-space Formulation of the Belavkin-Kushner-Stratonovich Filtering Equation for Nonlinear Quantum Stochastic Systems

TL;DR

This paper develops a phase-space formulation for nonlinear quantum stochastic systems' filtering problem, deriving a stochastic integro-differential equation for the posterior quasi-characteristic function using the Wigner-Moyal framework.

Contribution

It introduces a phase-space approach to quantum filtering, providing a new integro-differential equation for the posterior QCF in nonlinear quantum systems.

Findings

Derivation of a stochastic integro-differential equation for the posterior QCF

Representation of the filtering equation in the Fourier domain

Discussion of Gaussian approximation for the posterior quantum state

Abstract

This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic systems with multichannel nondemolition measurements. The system-observation dynamics are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation driven by quantum Wiener processes of bosonic fields in vacuum state. The Hamiltonian and system-field coupling operators, as functions of the system variables, are represented in a Weyl quantization form. Using the Wigner-Moyal phase-space framework, we obtain a stochastic integro-differential equation for the posterior quasi-characteristic function (QCF) of the system conditioned on the measurements. This equation is a spatial Fourier domain representation of the Belavkin-Kushner-Stratonovich stochastic master equation driven by the innovation process associated with the measurements. We also discuss a more specific form of the posterior QCF dynamics in the case of linear system-field coupling and outline a Gaussian approximation of the posterior quantum state.