Abstract

Unitary circuits subject to repeated projective measurements can undergo an entanglement phase transition (EPT) as a function of the measurement rate. This transition is generally understood in terms of a competition between the scrambling effects of unitary dynamics and the disentangling effects of measurements. We find that, surprisingly, EPTs are possible even in the absence of scrambling unitary dynamics, where they are best understood as arising from measurements alone. This motivates us to introduce \emph{measurement-only models}, in which the "scrambling" and "un-scrambling" effects driving the EPT are fundamentally intertwined and cannot be attributed to physically distinct processes. This represents a novel form of an EPT, conceptually distinct from that in hybrid unitary-projective circuits. We explore the entanglement phase diagrams, critical points, and quantum code properties of some of these measurement-only models. We find that the principle driving the EPTs in these models is \emph{frustration}, or mutual incompatibility, of the measurements. Suprisingly, an entangling (volume-law) phase is the generic outcome when measuring sufficiently long but still local ($\gtrsim 3$-body) operators. We identify a class of exceptions to this behavior ("bipartite ensembles") which cannot sustain an entangling phase, but display dual area-law phases, possibly with different kinds of quantum order, separated by self-dual critical points. Finally, we introduce a measure of information spreading in dynamics with measurements and use it to demonstrate the emergence of a statistical light-cone, despite the non-locality inherent to quantum measurements.

Highlights

  • The study of out-of-equilibrium quantum dynamics is an exciting research frontier that has seen many important developments in recent years, especially in relation to the dynamics of isolated many-body quantum systems [1,2,3,4,5,6,7,8]

  • Ideas from quantum information theory are playing a pivotal role in the development of this new toolkit: In particular, (i) random circuits have emerged as a versatile tool for the study of many-body dynamics in various contexts [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] and (ii) universality in the dynamics of quantum entanglement has emerged as a novel and incisive paradigm for characterizing many-body systems ranging from electrons in solids to cold atomic gases to black holes [10,25,26,27,28,29,30,31,32,33,34,35,36]

  • The present work adds a third novel member to the set of known transitions in entanglement dynamics; by studying measurement-only nonunitary circuits, we find entanglement phase transition (EPT) that are qualitatively distinct from both the many-body localization (MBL) transition and the hybrid-unitary projective measurement transition

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Summary

INTRODUCTION

The study of out-of-equilibrium quantum dynamics is an exciting research frontier that has seen many important developments in recent years, especially in relation to the dynamics of isolated many-body quantum systems [1,2,3,4,5,6,7,8]. We show how these nonscrambling models can equivalently be described as measurement-only dynamics, where the unitary gates are discarded altogether, at the expense of introducing multisite (but still local) measurements With this motivation, the present work adds a third novel member to the set of known transitions in entanglement dynamics; by studying measurement-only nonunitary circuits, we find EPTs that are qualitatively distinct from both the MBL transition and the hybrid-unitary projective measurement transition. (b) Introducing a novel EPT separating volume- and area-law entangled phases in MOMs.—This transition is distinct from both the MBL EPT and the hybrid unitary-projective EPT and adds a conceptually new member to the set of known EPTs. In MOMs, the scrambling and unscrambling effects are fundamentally intertwined, being produced by the same physical phenomenon (measurement); unlike the unitaryprojective case, where the balance between scrambling and unscrambling is naturally controlled by the measurement rate p, here we find that the operative property is the “frustration” of the measurement ensemble. This new probe is a nontrivial advance, because “standard” measures for information spreading, such as out-of-time ordered commutators, do not readily generalize to the nonunitary setting

Relation to previous work
Structure of the paper
MOTIVATION: “MEASUREMENT-ENABLED ENTANGLEMENT” IN l-BIT CIRCUITS
Removing scrambling: l-bit models
Removing unitary gates
MEASUREMENT-ONLY DYNAMICS
Measurements in the stabilizer formalism
Simple limits
Measurement frustration
PHENOMENOLOGY
Models
Entanglement phases
Critical properties
BIPARTITE ENSEMBLES AND QUANTUM ORDER
Absence of volume-law phase
Ordered area-law phases
QUANTUM CODE PROPERTIES
LOCALITY AND INFORMATION SPREADING
VIII. DISCUSSION
Ensemble
Connection with l-bit circuit model
Entanglement transition
Simulation of the dynamics
Equivalence between ensembles
Frustration of factorizable ensembles

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