  We consider a controlled quantum system whose finite dimensional state is
governed by a discrete-time nonlinear Markov process. In open-loop, the
measurements are assumed to be quantum non-demolition (QND). The eigenstates of
the measured observable are thus the open-loop stationary states: they are used
to construct a closed-loop supermartingale playing the role of a strict control
Lyapunov function. The parameters of this supermartingale are calculated by
inverting a Metzler matrix that characterizes the impact of the control input
on the Kraus operators defining the Markov process. The resulting state
feedback scheme, taking into account a known constant delay, provides the
almost sure convergence of the controlled system to the target state. This
convergence is ensured even in the case where the filter equation results from
imperfect measurements corrupted by random errors with conditional
probabilities given as a left stochastic matrix. Closed-loop simulations
corroborated by experimental data illustrate the interest of such nonlinear
feedback scheme for the photon box, a cavity quantum electrodynamics system.
