Control Theory Tech, Vol. 15, No. 3, pp. 177–192, August 2017 Control Theory and Technology http://link.springer.com/journal/11768 A phase-space formulation and Gaussian approximation of the filtering equations for nonlinear quantum stochastic systems Igor G. VLADIMIROV College of Engineering and Computer Science, Australian National University, Canberra, ACT 2601, Australia Received 31 January 2017; revised 5 May 2017; accepted 5 May 2017 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 assumed to be 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 discuss a 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. Keywords: Quantum stochastic system, quantum filtering equation, Gaussian approximation DOI 10.1007/s11768-017-7012-2 1 Introduction Estimation of the unknown current state of a stochas- tic system, based on the past history of a statistically dependent random process, is the central problem in the stochastic filtering theory which dates back to the works of Kolmogorov and Wiener of the 1940s [1,2]. The performance of state estimators is usually quantified by mean square values of the estimation errors which have to be minimized. In the framework of quadratic cost functionals, optimal estimators are delivered by condi- tional expectations of the state of the system, condi- E-mail: igor.g.vladimirov@gmail.com. This paper is dedicated to Professor Ian R. Petersen on the occasion of his 60th birthday. This work was initiated while the author was with the UNSW Canberra, Australia, where it was supported by the Australian Research Council, and was completed at the Australian National University under support of the Air Force Office of Scientific Research (AFOSR) under agreement number FA2386-16-1-4065. A brief version [80] of this paper was presented at the IEEE 2016 Conference on Norbert Wiener in the 21st Century, 13-15 July 2016, Melbourne, Australia. © 2017 South China University of Technology, Academy of Mathematics and Systems Science, CAS, and Springer-Verlag Berlin Heidelberg