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Open Systems & Information Dynamics

Viacheslav Pavlovich Belavkin, 1946–2012: In Memory of Professor V. P. Belavkin

P. Staszewski

Published Jun 1, 2013• 1 citation

Abstract

Viacheslav P. Belavkin was born in Lvov on 30 May 1946. A 1970 graduate of Physics Department of Moscow State University, he received his PhD in 1973. His thesis, entitled Optimal Estimation and Measurements in Quantum Systems, written under the supervision of Professor Ruslan Stratonovich, was a pioneering work that would lay the foundations for the new field of quantum optimal filtering and control. As a student and then a collaborator of Stratonovich, Slava learnt very well stochastic calculus along with the theory of nonlinear filtering. His scientific programme included an extension of the ideas of filtering and control from classical to the quantum domain. In this area of research Slava worked on the development of quantum Markov models and quantum stochastic processes, dynamical nondemolition principle for quantum continuous measurements and, finally, on the formulation of quantum filtering theory. During his life Slava obtained a number of remarkable and highly original results in mathematical physics and quantum probability, and there are theorems and equations named after him. One of such results is the Belavkin quantum filtering equation — the evolution equation (an analog of the Schrodinger equation) for observed quantum systems. Overall Professor Belavkin published more than 200 papers. His doctoral dissertation (habilitation) was presented and defended at the Steklov Institute of Mathematical Science in Moscow in 1991. In 1996, Belavkin shared the Main State Prize of the Russian Federation with R. Stratonovich. In 1978/79 Slava spent one year visiting quantum probability group of Professor Roman S. Ingarden in Torun. It was the origin of his long standing

DOI: 10.1142/s1230161213770015

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Viacheslav P. Belavkin was born in Lvov on 30 May 1946. A 1970 graduate of Physics Department of Moscow State University, he received his PhD in 1973. His thesis, entitled Optimal Estimation and Measurements in Quantum Systems, written under the supervision of Professor Ruslan Stratonovich, was a pioneering work that would lay the foundations for the new field of quantum optimal filtering and control. As a student and then a collaborator of Stratonovich, Slava learnt very well stochastic calculus along with the theory of nonlinear filtering. His scientific programme included an extension of the ideas of filtering and control from classical to the quantum domain. In this area of research Slava worked on the development of quantum Markov models and quantum stochastic processes, dynamical nondemolition principle for quantum continuous measurements and, finally, on the formulation of quantum filtering theory. During his life Slava obtained a number of remarkable and highly original results in mathematical physics and quantum probability, and there are theorems and equations named after him. One of such results is the Belavkin quantum filtering equation — the evolution equation (an analog of the Schrodinger equation) for observed quantum systems. Overall Professor Belavkin published more than 200 papers. His doctoral dissertation (habilitation) was presented and defended at the Steklov Institute of Mathematical Science in Moscow in 1991. In 1996, Belavkin shared the Main State Prize of the Russian Federation with R. Stratonovich. In 1978/79 Slava spent one year visiting quantum probability group of Professor Roman S. Ingarden in Torun. It was the origin of his long standing

In this thesis we study continuously observed open quantum systems from the point of view of quantum filtering theory. The conditioned state of the open system obeys a non-commutative or quantum analogue of the Kushner-Stratonovich equation in classical filtering theory. The thesis consists out of four chapters. The first chapter is a brief introduction to quantum probability theory and quantum stochastic calculus. The second chapter is a description of a photon counting experiment in continuous time within the framework of Davies and Srinivas. The third chapter focusses on the derivation of the quantum filtering or Belavkin equation from the quantum stochastic differential equation governing the interaction between the open system and its environment, the electromagnetic field. The fourth chapter shows how the quantum filtering equation can be used to control quantum systems.

These notes are intended as an introduction to noncommutative (quantum) filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as the least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. Next we describe the Hudson-Parthasarathy quantum Ito calculus and its use in the modelling of physical systems. Finally, we use a reference probability method to obtain quantum filtering equations, in the Belavkin-Zakai (unnormalized) form, for several system-observation models from quantum optics. The normalized (Belavkin-Kushner-Stratonovich) form is obtained through a noncommutative analogue of the Kallianpur-Striebel formula.

We derive the filtering equation for Markovian systems undergoing ho-modyne measurement in the situation where the output processes being monitored are squeezed. The filtering theory applies to case where the system is driven by Fock noise (that, quantum input processes in a co-herent state) and where the output is mixed with a squeezed signal. It also applies to the case of a system driven by squeezed noise, but here there is a physical restriction to emission/absorption coupling only. For the special case of a cavity mode where the dynamics is linear, we are able to derive explicitly the filtered estimate pit(a) for the mode annihilator a based on the homodyne quadrature observations up to time t. In memory of Slava Belavkin. 1

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

The theory of quantum filtering (of quantum continuous measurements) was developed by V.P. Belavkin about 40 years ago. Since then it attracted attention of numerous investigators including mathematicians, theoretical and experimental physicists. However, the rigorous mathematical theory of the filtering equations for mixed states in basic infinite-dimensional quantum systems remained an open problem, which was resolved by the author recently. This survey paper presents in full the mathematical theory of quantum filtering equations, their rigorous derivation from basic principles, the corresponding law of large number limits (propagation of chaos) and related topics. Applications to feedback control, quantum dynamic and mean-field games are discussed.

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.