Towards a definition of the Quantum Ergodic Hierarchy: Kolmogorov and Bernoulli systems

TL;DR

This paper extends the classical ergodic hierarchy to quantum systems, translating the Kolmogorov and Bernoulli levels into quantum language and illustrating their relevance with key models.

Contribution

It formalizes the quantum ergodic hierarchy, building on previous work, and applies it to well-known quantum models to demonstrate physical significance.

Findings

Quantum ergodic hierarchy formalized

Application to Casati-Prosen model

Application to kicked rotator

Abstract

In this paper we translate the two higher levels of the Ergodic Hierarchy [1], the Kolmogorov level and the Bernoulli level, to quantum language. Moreover, this paper can be considered as the second part of [2]. As in paper [2], we consider the formalism where the states are positive functionals on the algebra of observables and we use the properties of the Wigner transform [3]. We illustrate the physical relevance of the Quantum Ergodic Hierarchy with two emblematic examples of the literature: the Casati-Prosen model [4], [5] and the kicked rotator [6], [7], [8].