Tripartite mutual information, entanglement, and scrambling in permutation symmetric systems with an application to quantum chaos

TL;DR

This paper investigates permutation symmetric quantum states, deriving their entanglement properties, analyzing tripartite mutual information, and exploring their behavior in quantum chaos and scrambling phenomena.

Contribution

It introduces a new random matrix ensemble for permutation symmetric states and studies their entanglement and information sharing in chaotic regimes.

Findings

Permutation symmetric states are marginally entangled with typically positive TMI.

Entanglement and mutual information increase and saturate in chaotic regimes.

OTOCs evolve exponentially, indicating phase space scrambling.

Abstract

Many-body states that are invariant under particle relabelling, the permutation symmetric states, occur naturally when the system dynamics is described by symmetric processes or collective spin operators. We derive expressions for the reduced density matrix for arbitrary subsystem decomposition for these states and study properties of permutation symmetric states and their subsystems when the joint system is picked randomly and uniformly. Thus defining a new random matrix ensemble, we find the average linear entropy and von Neumann entropy which implies that random permutation symmetric states are marginally entangled and as a consequence the tripartite mutual information (TMI) is typically positive, preventing information from being shared globally. Applying these results to the quantum kicked top viewed as a multi-qubit system we find that entanglement, mutual information and TMI all increase for large subsystems across the Ehrenfest or log-time and saturate at the random state values if there is global chaos. During this time the out-of-time order correlators (OTOC) evolve exponentially implying scrambling in phase space. We discuss how positive TMI may coexist with such scrambling.