Infinite Dimensional Analysis Quantum Probability and Related Topics

On the Asymptotic Behaviour of Some Stochastic Differential Equations for Quantum States

Alberto Barchielli, Anna Maria Paganoni

Published Jun 1, 2003• 23 citations

Abstract

In this article we study the long time behaviour of a class of stochastic differential equations introduced in the theory of measurements continuous in time for quantum open systems. Such equations give the time evolution of the a posteriori states for a system underlying a continual measurement. First of all we give conditions for the equation to preserve pure states and then, in the case of a finite-dimensional Hilbert space, we obtain sufficient conditions from which the stochastic equation for a posteriori states is ensured to map, for t → +∞, mixed states into pure ones. Finally we study existence and uniqueness of an invariant measure for the equations which preserve pure states. We give a general theorem for the purely diffusive case, again in the finite-dimensional case; we then apply it to some physical examples. For the purely jump case, an example is discussed in which the invariant measure exists and is unique.

DOI: 10.1142/s0219025703001146

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