Quantum Filtering for a Qubit System Subject to Classical Disturbances

In this paper, we consider the filtering problem for a hybrid system where a quantum qubit system is disturbed by a classical signal. The quantum filtering theory, which is based on quantum probability theory, can not be directly applied to a hybrid system where a classical stochastic process is also needed in describing the system dynamics. An optical cavity system is employed to model the classical disturbance. By designing the parameters of the auxiliary cavity system, the expectation of the quadrature operator of the cavity shares the same dynamics with the classical signal. With this correspondence guaranteed, one can obtain the real time expectation of the classical signal. The quantum concatenation product is adopted to describe the quantum system which contains both the qubit subsystem and the cavity subsystem. A stochastic master equation, which provides estimates for the quantum state and the classical signal, is given. To reduce the computational complexity, the quantum extended Kalman filter is also applied to this system.