Weak Markov Processes as Linear Systems

TL;DR

This paper explores the realization of noncommutative linear systems within weak Markov processes, providing a systematic framework for their classification, and linking quantum process structures to classical concepts like observability and stationary chains.

Contribution

It introduces a novel classification scheme for quantum processes using $mma$-extensions, connecting noncommutative systems with quantum Markov chains and scattering theory.

Findings

Complete classification of process constructions via $mma$-extensions.

Identification of cascade structures of noncommutative systems.

New insights into observability and asymptotic completeness in quantum Markov chains.

Abstract

A noncommutative Fornasini-Marchesini system (a multi-variable version of a linear system) can be realized within a weak Markov process (a model for quantum evolution). For a discrete time parameter the resulting structure is worked out systematically and some quantum mechanical interpretations are given. We introduce subprocesses and quotient processes and then the notion of a $\gamma$-extension for processes which leads to a complete classification of all the ways in which processes can be built from subprocesses and quotient processes. We show that within a $\gamma$-extension we have a cascade of noncommutative Fornasini-Marchesini systems. We study observability in this setting and as an application we gain new insights into stationary Markov chains where observability for the system is closely related to asymptotic completeness in a scattering theory for the chain.