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A phase-space formulation and Gaussian approximation of the filtering equations for nonlinear quantum stochastic systems

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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.

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The author thanks the anonymous reviewers for useful comments.

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  1. College of Engineering and Computer Science, Australian National University, Canberra, ACT 2601, Australia

    Igor G. Vladimirov

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Correspondence to Igor G. Vladimirov.

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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.

Igor G. VLADIMIROV received M.Sc. degree in Control Systems in 1989 and Ph.D. degree in Physics and Mathematics (with specialization in Mathematical Cybernetics) in 1992 from the Department of Control and Applied Mathematics of the Moscow Institute (State University) of Physics and Technology, Russia. He worked as a Senior Research Associate at the State Research Institute of Aviation Systems in 1993–1997 and the Institute for Information Transmission Problems, the Russian Academy of Sciences, Moscow, in 1994–2007, in the areas of applied Kalman filtering, stochastic robust control, hysteresis systems and spatially discretized dynamical systems. From 1997 to 2008, Dr. Vladimirov held research academic positions at the Mathematics Department and School of Engineering at the University of Queensland, Brisbane, Australia, working in the above areas and in stochastic modelling of econometric time series, lattice models of statistical mechanics and transport phenomena in random media. In 2000, he also had a visiting research fellowship at the School of Mathematical Sciences, Queen Mary and Westfield College, University of London, working on p-adic analysis of Hamiltonian roundoff. From 2009 to 2016, Dr. Vladimirov was a Senior Research Fellow at the University of New South Wales Canberra, doing research on quantum stochastic filtering and control, which he continues after moving to the Australian National University in 2017. In 2013, Dr. Vladimirov was awarded B.N.Petrov prize of the Russian Academy of Sciences for his works on the anisotropy-based theory of stochastic robust filtering and control.

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Vladimirov, I.G. A phase-space formulation and Gaussian approximation of the filtering equations for nonlinear quantum stochastic systems. Control Theory Technol. 15, 177–192 (2017). https://doi.org/10.1007/s11768-017-7012-2

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