  A stochastic filter uses a series of measurements over time to produce
estimates of unknown variables based on a dynamic model. For a quantum system,
such an algorithm is provided by a quantum filter, which is also known as a
stochastic master equation (SME). For a linear quantum system subject to linear
measurements and Gaussian noise, the quantum filter reduces to a quantum Kalman
filter. In this article, we introduce a quantum extended Kalman filter (quantum
EKF), which applies a commutative approximation and a time-varying
linearization to non-commutative quantum stochastic differential equations
(QSDEs). We will show that there are conditions under which a filter similar to
the classical EKF can be implemented for quantum systems. The boundedness of
estimation errors and the filtering problems with `state-dependent' covariances
for process and measurement noises are also discussed. We demonstrate the
effectiveness of the quantum EKF by applying it to systems which involve
multiple modes, nonlinear Hamiltonians and simultaneous jump-diffusive
measurements.
