Non-commutative Markov Chains and Multi-analytic Operators

We study a model of repeated interaction between quantum systems which can be thought of as a non-commutative Markov chain. It is shown that there exists an outgoing Cuntz scattering system associated to this model which induces an input-output formalism with a transfer function corresponding to a multi-analytic operator, in the sense of multivariate operator theory. Finally we show that observability for this system is closely related to the scattering theory of non-commutative Markov chains.