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In the present paper we consider the problem of description of an arbitrarygeneralized quantum measurement with outcomes in a measurable space. Analyzingthe unitary invariants of a measuring process, we present the most general formof a possible integral representation of an instrument, which differs from therepresentations of an instrument available in the mathematical and physicalliterature. We introduce the notion of a quantum stochastic representation ofan instrument, whose elements are wholly determined by the unitary invariantsof a measuring process. We show that the description of a generalized directquantum 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 afamily of quantum stochastic evolution operators, satisfying the orthonormalityrelation and describing the conditional evolution of a quantum system under ameasurement. The QSA allows to give: a) the complete statistical description ofany generalized direct quantum measurement (a POV measure and a family ofposterior states); b) the complete description in a Hilbert space of thestochastic behaviour of a quantum system under a measurement in the sense ofspecification of the probabilistic transition law governing the change from theinitial 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 inthe most general case (without assuming any Markov property) the notion ofposterior pure state trajectories (quantum trajectories) and to give theirprobabilistic treatment.Comment: 37 pages, no figure

Quantum stochastic approach to the description of quantum measurements