Complexity and non-separability of classical Liouvillian dynamics

TL;DR

This paper introduces a new complexity measure called separability entropy for classical Liouvillian dynamics, which better captures the system's complexity than traditional entropy measures by reflecting exponential instability and non-Markovian behavior.

Contribution

The paper proposes the separability entropy as a simple, effective indicator of classical Liouvillian complexity, providing a stricter criterion than Kolmogorov-Sinai entropy.

Findings

Linear growth of separability entropy indicates exponential instability.

Separability entropy captures non-linear and non-Markovian dynamics.

It offers a more stringent complexity criterion than Kolmogorov-Sinai entropy.

Abstract

We propose a simple complexity indicator of classical Liouvillian dynamics, namely the separability entropy, which determines the logarithm of an effective number of terms in a Schmidt decomposition of phase space density with respect to an arbitrary fixed product basis. We show that linear growth of separability entropy provides stricter criterion of complexity than Kolmogorov-Sinai entropy, namely it requires that dynamics is exponentially unstable, non-linear and non-markovian.