  We study a class of Markov chains that model the evolution of a quantum
system subject to repeated measurements. Each Markov chain in this class is
defined by a measure on the space of matrices. It is then given by a random
product of correlated matrices taken from the support of the defining measure.
We give natural conditions on this support that imply that the Markov chain
admits a unique invariant probability measure. We moreover prove the geometric
convergence towards this invariant measure in the Wasserstein metric. Standard
techniques from the theory of products of random matrices cannot be applied
under our assumptions, and new techniques are developed, such as maximum
likelihood-type estimations.
