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

In this paper we translate the two higher levels of the Ergodic Hierarchy  [11], the Kolmogorov level and the Bernoulli level, to quantum language. Moreover, this paper can be considered as the second part of  [3]. As in  [3], we consider the formalism where the states are positive functionals on the algebra of observables and we use the properties of the Wigner transform  [12]. We illustrate the physical relevance of the Quantum Ergodic Hierarchy with two emblematic examples of the literature: the Casati-Prosen model [13], [14] and the kicked rotator  [6], [7], [8].