A Phase-space Formulation of the Belavkin-Kushner-Stratonovich Filtering Equation for Nonlinear Quantum Stochastic Systems
For researchers in quantum control and filtering, this work offers a new mathematical framework for nonlinear quantum filtering, but it is largely theoretical and incremental, extending existing phase-space methods to a specific class of systems.
This paper derives a phase-space formulation of the Belavkin-Kushner-Stratonovich filtering equation for nonlinear quantum stochastic systems with multichannel nondemolition measurements, obtaining a stochastic integro-differential equation for the posterior quasi-characteristic function. The approach provides a Fourier domain representation of the stochastic master equation, with a specific form for linear coupling and a Gaussian approximation outlined.
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.