Symmetries and ergodic properties in quantum probability

TL;DR

This paper explores the structure of noncommutative stochastic processes using operator algebras, analyzing symmetries and ergodic properties across various quantum probability frameworks.

Contribution

It provides a unified operator algebra approach to classical and quantum probability processes, detailing symmetries and ergodic properties in diverse quantum contexts.

Findings

Unified framework for classical and quantum stochastic processes

Analysis of symmetries like stationarity and exchangeability

Detailed ergodic properties in quantum probability models

Abstract

We deal with the general structure of (noncommutative) stochastic processes by using the standard techniques of Operator Algebras. Any stochastic process is associated to a state on a universal object, i.e. the free product $C^*$-algebra in a natural way. In this setting one recovers the classical (i.e. commutative) probability scheme and many others, like those associated to the Monotone, Boolean and the $q$-deformed canonical commutation relations including the Bose/Fermi and Boltzmann cases. Natural symmetries like stationarity and exchangeability, as well as the ergodic properties of the stochastic processes are reviewed in detail for many interesting cases arising from Quantum Physics and Probability.