Unitary-projective entanglement dynamics

TL;DR

This paper investigates how the interplay of unitary evolution and projective measurements affects entanglement in quantum many-body systems, revealing conditions for entanglement transitions and universal signatures of measurement effects.

Contribution

It introduces a toy Bell pair model and analyzes Clifford and Floquet circuits to demonstrate entanglement phase transitions driven by measurement rate, providing analytical and rigorous insights.

Findings

Measurements can maintain low entanglement states contrasting unitary growth.

An entanglement transition occurs at a critical measurement rate in generic models.

A universal subleading correction term indicates volume law entanglement in the presence of measurements.

Abstract

Starting from a state of low quantum entanglement, local unitary time evolution increases the entanglement of a quantum many-body system. In contrast, local projective measurements disentangle degrees of freedom and decrease entanglement. We study the interplay of these competing tendencies by considering time evolution combining both unitary and projective dynamics. We begin by constructing a toy model of Bell pair dynamics which demonstrates that measurements can keep a system in a state of low (i.e. area law) entanglement, in contrast with the volume law entanglement produced by generic pure unitary time evolution. While the simplest Bell pair model has area law entanglement for any measurement rate, as seen in certain non-interacting systems, we show that more generic models of entanglement can feature an area-to-volume law transition at a critical value of the measurement rate, in agreement with recent numerical investigations. As a concrete example of these ideas, we analytically investigate Clifford evolution in qubit systems which can exhibit an entanglement transition. We are able to identify stabilizer size distributions characterizing the area law, volume law and critical 'fixed points.' We also discuss Floquet random circuits, where the answers depend on the order of limits - one order of limits yields area law entanglement for any non-zero measurement rate, whereas a different order of limits allows for an area law - volume law transition. Finally, we provide a rigorous argument that a system subjected to projective measurements can only exhibit a volume law entanglement entropy if it also features a subleading correction term, which provides a universal signature of projective dynamics in the high-entanglement phase. Note: The results presented here supersede those of all previous versions of this manuscript, which contained some erroneous claims.