Quantum extensions of dynamical systems and of Markov semigroups

TL;DR

This paper develops methods to extend classical Markov operators to quantum channels, linking dilation theory with quantum extensions, and applies these to Lévy processes and finite-dimensional systems.

Contribution

It introduces a new recipe for quantum extensions of Markov operators, connecting dilation theory with quantum dynamical systems, and classifies CP maps stabilizing finite or infinite classical algebras.

Findings

Quantum extensions exist for certain Markov operators via dilation techniques.

The method applies to continuous-time Lévy processes, ensuring quantum extensions.

Classified all CP maps stabilizing finite or infinite classical algebras with physical examples.

Abstract

We investigate some particular completely positive maps which admit a stable commutative Von Neumann subalgebra. The restriction of such maps to the stable algebra is then a Markov operator. In the first part of this article, we propose a recipe in order to find a quantum extension of a given Markov operator in the above sense. We show that the existence of such an extension is linked with the existence of a special form of dilation for the Markov operator studied by Attal in \cite{Att1}, reducing the problem to the extension of dynamical system. We then apply our method to the same problem in continuous time, proving the existence of a quantum extension for L\'evy processes. In the second part of this article, we focus on the case where the commutative algebra is isomorphic to $\Acal=l^\infty(1,...,N)$ with $N$ either finite or infinite. We propose a classification of the CP maps leaving $\Acal$ stable, producing physical examples of each classes.