Interscale entanglement production in a quantum system simulating classical chaos

TL;DR

This paper introduces a quantum lattice model that simulates classical chaos through an interscale entanglement entropy, revealing how chaos emerges from quantum mechanics and linking quantum entanglement growth to classical entropy.

Contribution

It develops a quantum framework that exactly reproduces classical chaos in the continuum limit and introduces a novel interscale entanglement entropy to quantify chaos-related entanglement.

Findings

Interscale entanglement entropy becomes positive only when chaos emerges.

Entropy growth rate correlates with classical Gibbs entropy.

Long-time average of entropy indicates chaos in the quantum system.

Abstract

It is a fundamental problem how the universal concept of classical chaos emerges from the microscopic description of quantum mechanics. We here study standard classical chaos in a framework of quantum mechanics. In particular, we design a quantum lattice system that exactly simulates classical chaos after an appropriate continuum limit, which is called the "Hamiltonian equation limit". The key concept of our analysis is an entanglement entropy defined by dividing the lattice into many blocks of equal size and tracing out the degrees of freedom within each block. We refer to this entropy as the "interscale entanglement entropy" because it measures the amount of entanglement between the microscopic degrees of freedom within each block and the macroscopic degrees of freedom that define the large-scale structure of the wavefunction. By numerically simulating a quantum lattice system corresponding to the Hamiltonian of the kicked rotor, we find that the long-time average of the interscale entanglement entropy becomes positive only when chaos emerges in the Hamiltonian equation limit, and the growth rate of the entropy in the initial stage is proportional to that of the coarse-grained Gibbs entropy of the corresponding classical system.