Time-continuous quantum state estimation

Controlling single quantum systems is an important issue in quantum information processing technology. Recent experimental advancements have made feasible time-continuous weakly disturbing quantum measurements on a single system. New perspectives of quantum feed-back control have raised an immediate related task: the time-continuous estimation and real-time determination of a quantum state. We discuss the time-continuous quantum state estimation problem and present a new non-linear stochastic master equation that governs the time-evolution of the estimated quantum state. Its differential evolution corresponds to the infinitesimal updates that depend on the time-continuous measurement of the true quantum state. The new stochastic master equation couples to the two standard stochastic differential equations of time-continuous quantum measurement. We prove that the calculated estimate almost always converges to the true state. We demonstrate this convergence by a numerically simulated evolution of the true and the estimated wave function of a particle in a double-well potential