Temporal entanglement transition in chaotic quantum many-body dynamics

TL;DR

This paper investigates the scaling behavior of temporal entanglement in chaotic quantum many-body systems, revealing a transition from volume-law to area-law scaling that simplifies the simulation of local observables.

Contribution

It introduces a coarse-graining method that induces a transition in temporal entanglement scaling and demonstrates that local dynamics can be efficiently captured by an area-law influence matrix.

Findings

TE is extensive for low bath growth rates and indicates non-Markovianity.

A transition from volume- to area-law TE scaling occurs with measurement coarse-graining.

Standard compression algorithms accurately describe local evolution via an area-law influence matrix.

Abstract

Temporal entanglement (TE) of an influence matrix (IM) has been proposed as a measure of complexity of simulating dynamics of local observables in a many-body system. Foligno et al. [Phys. Rev. X 13, 041008 (2023)] recently argued that the TE in chaotic 1d quantum circuits obeys linear (volume-law) scaling with evolution time. To reconcile this apparent high complexity of IM with the rapid thermalization of local observables, here we study the relation between TE, non-Markovianity, and local temporal correlations for chaotic quantum baths. By exactly solving a random-unitary bath model, and bounding distillable entanglement between future and past degrees of freedom, we argue that TE is extensive for low enough bath growth rate, and it reflects genuine non-Markovianity. This memory, however, is entirely contained in highly complex temporal correlations, and its effect on few-point temporal correlators is negligible. An IM coarse-graining procedure, reducing the allowed frequency of measurements of the probe system, results in a transition from volume- to area-law TE scaling. We demonstrate the generality of this TE transition in 1d circuits by analyzing the kicked Ising model analytically at dual-unitary points, as well as numerically away from them. This finding indicates that dynamics of local observables are fully captured by an area-law IM. We provide evidence that the compact IM MPS obtained via standard compression algorithms accurately describes local evolution.