Phase-space characterization of complexity in quantum many-body dynamics

TL;DR

This paper introduces a phase-space Wigner harmonics entropy measure for quantum many-body systems, capturing complexity, chaos, and entanglement effects, and demonstrating its effectiveness through numerical analysis of the Ising chain.

Contribution

It proposes a novel entropy measure for quantum complexity that accounts for chaos and entanglement, applicable to pure and mixed states in many-body systems.

Findings

Entropy grows linearly with time in both integrable and chaotic regimes.

Number of Wigner function harmonics grows exponentially over time.

Entropy growth rate can detect quantum phase transitions.

Abstract

We propose a phase-space Wigner harmonics entropy measure for many-body quantum dynamical complexity. This measure, which reduces to the well known measure of complexity in classical systems and which is valid for both pure and mixed states in single-particle and many-body systems, takes into account the combined role of chaos and entanglement in the realm of quantum mechanics. The effectiveness of the measure is illustrated in the example of the Ising chain in a homogeneous tilted magnetic field. We provide numerical evidence that the multipartite entanglement generation leads to a linear increase of entropy until saturation in both integrable and chaotic regimes, so that in both cases the number of harmonics of the Wigner function grows exponentially with time. The entropy growth rate can be used to detect quantum phase transitions. The proposed entropy measure can also distinguish between integrable and chaotic many-body dynamics by means of the size of long term fluctuations which become smaller when quantum chaos sets in.