On the quantum theory of measurements continuous in time

Abstract

A quantum theory of measurements continuous in time has been developed at various levels of generality and by means of various mathematical tools. Here, only some points of this theory are presented. By using the notion of convolution semigroup of instruments one can give a prescription for extracting from quantum mechanics a consistent set of finite-dimensional probabilities (multi-time joint probabilities for some family of observables) describing the continuous measurement. Some results on the structure of such semigroups are given. Moreover, one can go beyond finite- dimensional distributions by using more elaborate families of instruments which can be constructed via certain stochastic differential equations for processes with values in the trace-class operators (related to the notion of a posteriori states). This kind of equations gives a detailed description of the behaviour of the quantum system and allows for generalizations of the theory (such generalizations, which allow to handle certain memory effects, are not described here); moreover, independently from continuous measurement theory, these stochastic equations are now used also for numerical simulations of master equations in quantum optics. A higher level of description, involving quantum stochastic calculus, dilations of quantum dynamical semigroups, quantum stochastic processes, …, is left out from this paper.