Unitary-projective entanglement dynamics

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. CONTENTS

Entanglement transition from variable-strength weak measurements

“…The fate of this transition in open quantum systems is the subject of intense current investigations [13][14][15][16][17] . A promising path to address the entanglement transition in this setting is to consider a randomly driven system on which the external environment acts as a quantum detector [18][19][20][21] . In this scenario, the free unitary evolution of the unobserved system leads to a ballistic temporal growth of entanglement [22][23][24] , until it settles into a highly entangled quasi-stationary state that follows the volume law.…”

“…This tendency is counteracted by local measurements, which induce a stochastic nonunitary backaction. Very recently it has been shown that when local projective measurements are performed frequently enough one encounters an entanglement transition to a quasi-stationary state characterized by an area law [18][19][20] .…”

Quantum Zeno effect appears in stages

“…Tracking quantum trajectories has been exploited as a tool to engineering quantum states via continuous feedback control [8][9][10] and entanglement distillation [11,12]. It has been used to observe fundamental properties of quantum measurements [13][14][15][16][17] and, recently, to predict topological transitions in measurement-induced geometric phases [18][19][20][21] and many-body entanglement phase transitions in random unitary circuits, invisible to the average dynamics [22][23][24][25].…”

Entanglement Membrane in Chaotic Many-Body Systems

“…The effective spin model may also be extended to dynamics with measurement, or other types of interaction with an environment, for example, to clarify the relationship between different universality classes of measurementinduced criticality [88,89,[93][94][95][96][97][98][99][100][101][102][103].…”