Centre for Mathematical Physics and Stochastics

Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

MPS-RR 1999-45

November 1999

We propose a new approach to the description in the general case of continuous in time indirect measurement of an open system. Our approach is based not on the concept of a generating map of an instrument as the way for the description of an indirect measurement process and not on quantum stochastic calculus as a tool of consideration but on the methods of quantum theory and the Schrödinger equation.

Our approach is valid for a broad class of quantum measurement models and quantum input processes but not only in the case of the Markovian approximation.

In the general case we introduce the operator describing the evolution of an open system under the condition that the output process was continuously observed until the moment t and found to have the definite trajectory. We derive the integral equation describing the quantum stochastic evolution of an open system in the general case of nondemolition observation.

As an example of application of our results to concrete measurement models we consider the special measurement model which is the extended variant (including the gauge term) usually considered in the frame of so called quantum stochastic mechanics. We get the new equation describing the quantum stochastic evolution of an open system under continuous in time diffusion observation.

This equation can be rewritten in the stochastic form which in case of the vacuum initial state of a reservoir and the absence of the gauge term coincides with the well known quantum filtering equation in quantum stochastic mechanics introduced by V.P.Belavkin.

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