Entanglement entropy converges to classical entropy around periodic orbits

TL;DR

This paper demonstrates that in chaotic Hamiltonian systems, the entanglement entropy of oscillators driven periodically converges to a linear growth rate determined by classical Lyapunov exponents, linking quantum entanglement to classical chaos.

Contribution

It establishes a connection between quantum entanglement entropy growth and classical Lyapunov exponents in chaotic systems, including dependence on coarse-graining choices.

Findings

Entanglement entropy asymptotes to linear growth rate given by positive Lyapunov exponents.

The asymptotic entropy growth rate is robust against different coarse-graining methods.

The results bridge quantum entanglement dynamics with classical chaos theory.

Abstract

We consider oscillators evolving subject to a periodic driving force that dynamically entangles them, and argue that this gives the linearized evolution around periodic orbits in a general chaotic Hamiltonian dynamical system. We show that the entanglement entropy, after tracing over half of the oscillators, generically asymptotes to linear growth at a rate given by the sum of the positive Lyapunov exponents of the system. These exponents give a classical entropy growth rate, in the sense of Kolmogorov, Sinai and Pesin. We also calculate the dependence of this entropy on linear mixtures of the oscillator Hilbert space factors, to investigate the dependence of the entanglement entropy on the choice of coarse-graining. We find that for almost all choices the asymptotic growth rate is the same.