A phase-space formulation of the Belavkin-Kushner-Stratonovich filtering equation for nonlinear quantum stochastic systems

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