Separability and entanglement in classical eigenfunctions as a criterion for Hamiltonian chaos

TL;DR

This paper links classical chaos and integrability to the separability and entanglement of eigenfunctions of the Liouville operator, revealing that non-integrable systems exhibit entangled eigenfunctions.

Contribution

It establishes a novel criterion connecting classical chaos with quantum-like entanglement in eigenfunctions, based on Liouville's integrability theorem.

Findings

Eigenfunctions are separable if and only if the system is integrable.

Non-integrable systems have entangled eigenfunctions in all canonical coordinates.

The work bridges classical chaos theory with quantum mechanical concepts.

Abstract

We study the eigenfunctions of the classical Liouville operator and investigate the conditions they must obey to be separable as a product state. We point out that the conditions for separability are equivalent to requirements of Liouville's integrability theorem, this is, the eigenfunctions are separable if and only if the system is integrable. On the other hand, if the classical system is not integrable, then the eigenfunctions are entangled in all canonical coordinates. This results in a link between the classical notions of chaos and integrability with mathematical concepts that are usually restricted to quantum mechanics.