A Survey on the Classical Limit of Quantum Dynamical Entropies

TL;DR

This paper reviews how quantum dynamical entropies relate to classical chaos measures, focusing on hyperbolic automorphisms of the 2-torus and showing their correspondence over specific time scales.

Contribution

It provides a detailed semi-classical analysis of quantum dynamical entropies approaching the classical limit using coherent states.

Findings

Quantum entropies match Kolmogorov-Sinai entropy over logarithmic time scales.

The hyperbolic automorphisms model illustrates the quantum-classical transition.

Dynamical localization plays a key role in the analysis.

Abstract

We analyze the behavior of quantum dynamical entropies production from sequences of quantum approximants approaching their (chaotic) classical limit. The model of the quantized hyperbolic automorphisms of the 2-torus is examined in detail and a semi-classical analysis is performed on it using coherent states, fulfilling an appropriate dynamical localization property. Correspondence between quantum dynamical entropies and the Kolmogorov-Sinai invariant is found only over time scales that are logarithmic in the quantization parameter.