Quantum stochastic approach to the description of quantum measurements

In the present paper we consider the problem of description of an arbitrary generalized quantum measurement with outcomes in a measurable space. Analyzing the unitary invariants of a measuring process, we present the most general form of a possible integral representation of an instrument, which differs from the representations of an instrument available in the mathematical and physical literature. We introduce the notion of a quantum stochastic representation of an instrument, whose elements are wholly determined by the unitary invariants of a measuring process. We show that the description of a generalized direct quantum measurement can be considered in the frame of a new general approach, which we call the quantum stochastic approach (QSA), based on the notion of a family of quantum stochastic evolution operators, satisfying the orthonormality relation and describing the conditional evolution of a quantum system under a measurement. The QSA allows to give: a) the complete statistical description of any generalized direct quantum measurement (a POV measure and a family of posterior states); b) the complete description in a Hilbert space of the stochastic behaviour of a quantum system under a measurement in the sense of specification of the probabilistic transition law governing the change from the initial state of a quantum system to a final one under a single measurement; c) to formalize the consideration of all possible types of quantum measurements. For measurements continuous in time the QSA allows, in particular, to define in the most general case (without assuming any Markov property) the notion of posterior pure state trajectories (quantum trajectories) and to give their probabilistic treatment.