Long Time Behavior of the Solutions of the Belavkin Quantum Filtering Equation

Vassili Nikitich Kolokoltsov

Vassili Nikitich Kolokoltsov

In the article, the complete investigation is given of the large times behavior of the solutions of the Belavkin quantum filtering equation describing a quantum particle with continuously observed coordinate. It turns out that these solutions have an extraordinary property, namely: as t → ∞, any solution will tend to a Gaussian function (for almost all realizations of innovating Wiener process). As a consequence, it follows that the dispersion of coordinate will always tend to the same limit, as t → ∞, not depending on initial data.

Naman Garg, Ankur Kumar, Harish Parthasarathy

The Belavkin filter is a real-time quantum filter for dynamically estimating atomic observables and states based on non-demolition measurements, i.e. measurements that commute at different times and with future values of the observables. This filter is a generalisation of the classical Kushner filter based on quantum mechanics, and we propose to simulate it by using the H-P stochastic equation and the Belavkin equations without randomness. Randomness enters only when we consider the distribution of observables in specific states of the system and bath. We apply the Belavkin filter to estimate the spin of the electron.

Luigi Accardi

P. Staszewski

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

Hendra I. Nurdin, Naoki Yamamoto