Recursive Filtering of Operator Valued Processes in Quantum Estimation

The unified approach to estimation of a stochastic signal satisfying a dynamical equation through the use of martingale and multiplicity theory is applied, in its infinite dimensional version, to the problem of estimation of a signal via quantum mechanical measurements in an optical communication system. The "state" is an operator valued stochastic process representing a density operator in the quantum mechanical sense acting in an appropriate Hilbert space. The aim is the derivation of recursion relations for the optimum measurement operators that minimize a quadratic cost function. The Markovian nature of the density operator indexed by the signal process allows for optimum or suboptimum recursive filtering relations that are analogous to the ones in the classical case.