Non-Commutative Markov Chains and Multi-Analytic Operators

TL;DR

This paper explores non-commutative Markov chains in quantum systems, establishing a connection with multi-analytic operators and scattering theory, and introduces an input-output formalism based on Cuntz scattering systems.

Contribution

It introduces a novel framework linking non-commutative Markov chains with multi-analytic operators and scattering theory, expanding the mathematical tools for quantum system analysis.

Findings

Existence of an outgoing Cuntz scattering system for the model

Input-output formalism with transfer functions as multi-analytic operators

Relationship between system observability and scattering theory

Abstract

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.