Quantum Filtering of Markov Signals with White Quantum Noise

Time-continuous non-anticipating quantum processes of nondemolition measurements are introduced as the dynamical realizations of the causal quasi-measurements, which are described in this paper by the adapted operator-valued probability measures on the trajectory spaces of the generalized temporal observations in quantum open systems. In particular, the notion of physically realizable quantum filter is defined and the problem of its optimization to obtain the best a posteriori quantum state is considered. It is proved that the optimal filtering of a quantum Markovian Gaussian signal with the Gaussian white quantum noise is described as a coherent Markovian linear filter generalizing the classical Kalman filter. As an example, the problem of optimal measurement of complex amplitude for a quantum Markovian open oscillator, loaded to a quantum wave communication line, is considered and solved.