Dynamics of Pseudoentanglement

TL;DR

This paper introduces models of quantum dynamics that achieve thermal equilibrium with minimal entanglement, using pseudoentanglement concepts and classical Markov chains to analyze the emergence of equilibrium states.

Contribution

It presents a novel framework of ensemble pseudothermalization via random circuits that replicate thermal predictions with low entanglement, and analyzes their dynamics through Markov chain models.

Findings

Pseudoentangled ensembles mimic thermal equilibrium predictions.

Relaxation times differ from classical Markov chain behavior.

Conjecture of sharp dynamical transition to equilibrium (cutoff phenomenon).

Abstract

The dynamics of quantum entanglement plays a central role in explaining the emergence of thermal equilibrium in isolated many-body systems. However, entanglement is notoriously hard to measure. Recent works have introduced a notion of pseudoentanglement describing ensembles of many-body states that, while only weakly entangled, cannot be efficiently distinguished from states with much higher entanglement, such as random states in the Hilbert space. This prompts the question: how much entanglement is truly necessary to achieve thermal equilibrium in quantum systems? In this work we address this question by introducing random circuit models of quantum dynamics that, at late times, equilibrate to pseudoentangled ensembles -- a phenomenon we name ensemble pseudothermalization. These models replicate all the efficiently observable predictions of thermal equilibrium, while generating only a small (and tunable) amount of entanglement. We examine (i) how a pseudoentangled ensemble on a small subsystem spreads to the whole system as a function of time, and (ii) how a pseudoentangled ensemble can be generated from an initial product state. We map the above problems onto a family of classical Markov chains on subsets of the computational basis. The mixing times of such Markov chains are related to the time scales at which the states produced from the dynamics become indistinguishable from Haar-random states at the level of each statistical moment, or number of copies. Based on a combination of rigorous bounds and conjectures supported by numerics, we argue that each Markov chain's relaxation time and mixing time have different asymptotic behavior in the limit of large system size. This is a necessary condition for a cutoff phenomenon: an abrupt dynamical transition to equilibrium. We thus conjecture that our random circuits give rise to asymptotically sharp distinguishability transitions.