  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.
