Centre for Mathematical Physics and Stochastics

Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

MPS-RR 2000-39

September 2000

In the present paper we consider the problem of description of a generalized quantum measurement with outcomes in a measurable space. Analyzing the concepts of operational approach in quantum measurement theory, we introduce the notion of a quantum stochastic representation of an instrument. We show that the description of a generalized quantum measurement can be considered in the frame of a new general approach based on the notion of a family of quantum stochastic evolution operators. Such approach gives not only the complete statistical description of any quantum measurement but the complete description in a Hilbert space of the stochastic behaviour of a quantum system under a measurement. In the frame of the proposed approach, which we call quantum stochastic, all possible schemes of measurements upon a quantum system can be considered. In the case of repeated or continuous in time measurements the quantum stochastic approach allows to define in the most general case the notion of a family of posterior pure state trajectories (quantum trajectories in discrete or continuous time) in a Hilbert space of a quantum system and to give their probabilistic treatment.

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