Reversible part of a quantum dynamical system

TL;DR

This paper investigates the reversible component of quantum dynamical systems, exploring its structure, properties, and relationship with ergodicity and canonical decompositions, providing insights into the system's long-term behavior.

Contribution

It introduces a detailed analysis of the reversible part of quantum dynamical systems and its connection to ergodic properties and linear contraction decompositions.

Findings

Reversible part characterized by a von Neumann sub-algebra and automorphism.

Ergodicity linked to the triviality of the reversible algebra.

Relationships established with Nagy-Fojas canonical decomposition.

Abstract

In this work a quantum dynamical system $(\mathfrak M,\Phi, \varphi)$ is constituted by a von Neumann algebra $\mathfrak M$, by a unital Schwartz map $\Phi:\mathfrak{M\rightarrow M}$ and by a $\Phi$-invariant normal faithful state $\varphi$ on $\mathfrak M$. The ergodic properties of a quantum dynamical system, depends on its reversible part $(\mathfrak{D}_\infty,\Phi_\infty, \varphi_\infty)$. It is constituted by a von Neumann sub-algebra $\mathfrak{D}_\infty$ of $\mathfrak M$ by an automorphism $\Phi_\infty$ and a normal state $\varphi_\infty$, the restrictions of $\Phi$ and $\varphi$ on $\mathfrak{D}_\infty$ respectively. Moreover, if $\mathfrak{D}_\infty$ is a trivial algebra the quantum dynamical system is ergodic. Furthermore we will give some properties of the reversible part of quantum dynamical system, in particular, we will study its relationships with the canonical decomposition of Nagy-Fojas of linear contraction related to the quantum dynamical system.