Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1808.05949 · Physical review. B./Physical review. B

Unitary-projective entanglement dynamics

Amos Chan, Rahul Nandkishore, Michael Pretko, Graeme Smith

Published Jun 24, 2019• 474 citations
arXivHTMLPDF

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 noninteracting 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 a Floquet random unitary circuit, where the answers depend on the order of limits---one order of limits yields area-law entanglement for any nonzero measurement rate, whereas a different order of limits allows for an arealaw--volumelaw 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.

DOI
10.1103/physrevb.99.224307

Related papers

Entanglement Transitions in Unitary Circuit Games

Raúl Morral-Yepes, Adam Smith, Shivaji L. Sondhi +1 more

Repeated projective measurements in unitary circuits can lead to an entanglement phase transition as the measurement rate is tuned. In this work, we consider a different setting in which the projective measurements are replaced by dynamically chosen unitary gates that minimize the entanglement. This can be seen as a one-dimensional unitary circuit game in which two players get to place unitary gates on randomly assigned bonds at different rates: the “entangler” applies a random local unitary gate with the aim of generating extensive (volume-law) entanglement. The “disentangler,” based on limited knowledge about the state, chooses a unitary gate to reduce the entanglement entropy on the assigned bond with the goal of limiting to only finite (area-law) entanglement. In order to elucidate the resulting entanglement dynamics, we consider three different scenarios: (i) a classical discrete height model, (ii) a Clifford circuit, and (iii) a general U(4) unitary circuit. We find that both the classical and Clifford circuit models exhibit phase transitions as a function of the rate that the disentangler places a gate, which have similar properties that can be understood through a connection to the stochastic Fredkin chain. In contrast, the entangler always wins when using Haar random unitary gates and we observe extensive, volume-law entanglement for all nonzero rates of entangling. Published by the American Physical Society 2024

arXiv:2304.12965•PRX Quantum•Jan 24, 2024•13 citations

Measurement-Induced Phase Transitions in the Dynamics of Entanglement

Brian J. Skinner, Jonathan Ruhman, Adam Nahum

© 2019 authors. Published by the American Physical Society. We define dynamical universality classes for many-body systems whose unitary evolution is punctuated by projective measurements. In cases where such measurements occur randomly at a finite rate p for each degree of freedom, we show that the system has two dynamical phases: "entangling" and "disentangling." The former occurs for p smaller than a critical rate pc and is characterized by volume-law entanglement in the steady state and "ballistic" entanglement growth after a quench. By contrast, for p>pc the system can sustain only area-law entanglement. At p=pc the steady state is scale invariant, and in 1+1D, the entanglement grows logarithmically after a quench. To obtain a simple heuristic picture for the entangling-disentangling transition, we first construct a toy model that describes the zeroth Rényi entropy in discrete time. We solve this model exactly by mapping it to an optimization problem in classical percolation. The generic entangling-disentangling transition can be diagnosed using the von Neumann entropy and higher Rényi entropies, and it shares many qualitative features with the toy problem. We study the generic transition numerically in quantum spin chains and show that the phenomenology of the two phases is similar to that of the toy model but with distinct "quantum" critical exponents, which we calculate numerically in 1+1D. We examine two different cases for the unitary dynamics: Floquet dynamics for a nonintegrable Ising model, and random circuit dynamics. We obtain compatible universal properties in each case, indicating that the entangling-disentangling phase transition is generic for projectively measured many-body systems. We discuss the significance of this transition for numerical calculations of quantum observables in many-body systems.

•Jan 1, 2019•619 citations

Quantum Zeno effect and the many-body entanglement transition

Yaodong Li, Xiao Chen, Matthew P. A. Fisher

We introduce and explore a one-dimensional ``hybrid'' quantum circuit model consisting of both unitary gates and projective measurements. While the unitary gates are drawn from a random distribution and act uniformly in the circuit, the measurements are made at random positions and times throughout the system. By varying the measurement rate we can tune between the volume law entangled phase for the random unitary circuit model (no measurements) and a ``quantum Zeno phase'' where strong measurements suppress the entanglement growth to saturate in an area law. Extensive numerical simulations of the quantum trajectories of the many-particle wave functions (exploiting Clifford circuitry to access systems up to 512 qubits) provide evidence for a stable ``weak measurement phase'' that exhibits volume-law entanglement entropy, with a coefficient decreasing with increasing measurement rate. We also present evidence for a continuous quantum dynamical phase transition between the ``weak measurement phase'' and the ``quantum Zeno phase,'' driven by a competition between the entangling tendencies of unitary evolution and the disentangling tendencies of projective measurements. Detailed steady-state and dynamic critical properties of this quantum entanglement transition are accessed.

arXiv:1808.06134•Physical review. B./Physical review. B•Nov 19, 2018•709 citations

Entanglement Phase Transitions in Measurement-Only Dynamics

Matteo Ippoliti, Michael J. Gullans, Sarang Gopalakrishnan +2 more

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 finding motivates us to introduce measurement-only models, in which the "scrambling" and "unscrambling" effects driving the EPT are fundamentally intertwined and cannot be attributed to physically distinct processes.These models represent 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 frustration, or mutual incompatibility, of the measurements.Surprisingly, an entangling (volume-law) phase is the generic outcome when measuring sufficiently long but still local (≳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 nonlocality inherent to quantum measurements.

arXiv:2004.09560•Physical Review X•Feb 15, 2021•277 citations

Entanglement transitions via space-time rotation of quantum circuits

Tsung-Cheng Lu, Tarun Grover

Time evolution of quantum many-body systems typically leads to a state with maximal entanglement allowed by symmetries. Two distinct routes to impede entanglement growth are inducing localization via spatial disorder, or subjecting the system to non-unitary evolution, e.g., via projective measurements. Here we employ the idea of space-time rotation of a circuit to explore the relation between systems that fall into these two classes. In particular, by space-time rotating unitary Floquet circuits that display a localization transition, we construct non-unitary circuits that display a rich variety of entanglement scaling and phase transitions. One outcome of our approach is a non-unitary circuit for free fermions in 1d that exhibits an entanglement transition from logarithmic scaling to volume-law scaling. This transition is accompanied by a 'purification transition' analogous to that seen in hybrid projective-unitary circuits. We follow a similar strategy to construct a non-unitary 2d Clifford circuit that shows a transition from area to volume-law entanglement scaling. Similarly, we space-time rotate a 1d spin chain that hosts many-body localization to obtain a non-unitary circuit that exhibits an entanglement transition. Finally, we introduce an unconventional correlator and argue that if a unitary circuit hosts a many-body localization transition, then the correlator is expected to be singular in its non-unitary counterpart as well.

•Mar 12, 2021•24 citations

Measurement-induced criticality in random quantum circuits

Chao‐Ming Jian, Yi‐Zhuang You, Romain Vasseur +1 more

A new class of quantum entanglement transitions separating phases with different entanglement entropy scaling has been observed in recent numerical studies. Despite the numerical efforts, an analytical understanding of such transitions has remained elusive. Here, the authors propose a theory for the area-law to volume-law entanglement transition in many-body systems that undergo both random unitary evolutions and projective measurements. Using the replica method, the authors map analytically this entanglement transition to an ordering transition in a classical statistical mechanics model. They derive the general entanglement scaling properties at the transition and show a solvable limit where this transition can be mapped onto two-dimensional percolation.

arXiv:1908.08051•Physical review. B./Physical review. B•Mar 3, 2020•452 citations
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 noninteracting 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 a Floquet random unitary circuit, where the answers depend on the order of limits---one order of limits yields area-law entanglement for any nonzero measurement rate, whereas a different order of limits allows for an arealaw--volumelaw 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.
Repeated projective measurements in unitary circuits can lead to an entanglement phase transition as the measurement rate is tuned. In this work, we consider a different setting in which the projective measurements are replaced by dynamically chosen unitary gates that minimize the entanglement. This can be seen as a one-dimensional unitary circuit game in which two players get to place unitary gates on randomly assigned bonds at different rates: the “entangler” applies a random local unitary gate with the aim of generating extensive (volume-law) entanglement. The “disentangler,” based on limited knowledge about the state, chooses a unitary gate to reduce the entanglement entropy on the assigned bond with the goal of limiting to only finite (area-law) entanglement. In order to elucidate the resulting entanglement dynamics, we consider three different scenarios: (i) a classical discrete height model, (ii) a Clifford circuit, and (iii) a general U(4) unitary circuit. We find that both the classical and Clifford circuit models exhibit phase transitions as a function of the rate that the disentangler places a gate, which have similar properties that can be understood through a connection to the stochastic Fredkin chain. In contrast, the entangler always wins when using Haar random unitary gates and we observe extensive, volume-law entanglement for all nonzero rates of entangling. Published by the American Physical Society 2024
© 2019 authors. Published by the American Physical Society. We define dynamical universality classes for many-body systems whose unitary evolution is punctuated by projective measurements. In cases where such measurements occur randomly at a finite rate p for each degree of freedom, we show that the system has two dynamical phases: "entangling" and "disentangling." The former occurs for p smaller than a critical rate pc and is characterized by volume-law entanglement in the steady state and "ballistic" entanglement growth after a quench. By contrast, for p>pc the system can sustain only area-law entanglement. At p=pc the steady state is scale invariant, and in 1+1D, the entanglement grows logarithmically after a quench. To obtain a simple heuristic picture for the entangling-disentangling transition, we first construct a toy model that describes the zeroth Rényi entropy in discrete time. We solve this model exactly by mapping it to an optimization problem in classical percolation. The generic entangling-disentangling transition can be diagnosed using the von Neumann entropy and higher Rényi entropies, and it shares many qualitative features with the toy problem. We study the generic transition numerically in quantum spin chains and show that the phenomenology of the two phases is similar to that of the toy model but with distinct "quantum" critical exponents, which we calculate numerically in 1+1D. We examine two different cases for the unitary dynamics: Floquet dynamics for a nonintegrable Ising model, and random circuit dynamics. We obtain compatible universal properties in each case, indicating that the entangling-disentangling phase transition is generic for projectively measured many-body systems. We discuss the significance of this transition for numerical calculations of quantum observables in many-body systems.
We introduce and explore a one-dimensional ``hybrid'' quantum circuit model consisting of both unitary gates and projective measurements. While the unitary gates are drawn from a random distribution and act uniformly in the circuit, the measurements are made at random positions and times throughout the system. By varying the measurement rate we can tune between the volume law entangled phase for the random unitary circuit model (no measurements) and a ``quantum Zeno phase'' where strong measurements suppress the entanglement growth to saturate in an area law. Extensive numerical simulations of the quantum trajectories of the many-particle wave functions (exploiting Clifford circuitry to access systems up to 512 qubits) provide evidence for a stable ``weak measurement phase'' that exhibits volume-law entanglement entropy, with a coefficient decreasing with increasing measurement rate. We also present evidence for a continuous quantum dynamical phase transition between the ``weak measurement phase'' and the ``quantum Zeno phase,'' driven by a competition between the entangling tendencies of unitary evolution and the disentangling tendencies of projective measurements. Detailed steady-state and dynamic critical properties of this quantum entanglement transition are accessed.
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 finding motivates us to introduce measurement-only models, in which the "scrambling" and "unscrambling" effects driving the EPT are fundamentally intertwined and cannot be attributed to physically distinct processes.These models represent 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 frustration, or mutual incompatibility, of the measurements.Surprisingly, an entangling (volume-law) phase is the generic outcome when measuring sufficiently long but still local (≳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 nonlocality inherent to quantum measurements.
Time evolution of quantum many-body systems typically leads to a state with maximal entanglement allowed by symmetries. Two distinct routes to impede entanglement growth are inducing localization via spatial disorder, or subjecting the system to non-unitary evolution, e.g., via projective measurements. Here we employ the idea of space-time rotation of a circuit to explore the relation between systems that fall into these two classes. In particular, by space-time rotating unitary Floquet circuits that display a localization transition, we construct non-unitary circuits that display a rich variety of entanglement scaling and phase transitions. One outcome of our approach is a non-unitary circuit for free fermions in 1d that exhibits an entanglement transition from logarithmic scaling to volume-law scaling. This transition is accompanied by a 'purification transition' analogous to that seen in hybrid projective-unitary circuits. We follow a similar strategy to construct a non-unitary 2d Clifford circuit that shows a transition from area to volume-law entanglement scaling. Similarly, we space-time rotate a 1d spin chain that hosts many-body localization to obtain a non-unitary circuit that exhibits an entanglement transition. Finally, we introduce an unconventional correlator and argue that if a unitary circuit hosts a many-body localization transition, then the correlator is expected to be singular in its non-unitary counterpart as well.
A new class of quantum entanglement transitions separating phases with different entanglement entropy scaling has been observed in recent numerical studies. Despite the numerical efforts, an analytical understanding of such transitions has remained elusive. Here, the authors propose a theory for the area-law to volume-law entanglement transition in many-body systems that undergo both random unitary evolutions and projective measurements. Using the replica method, the authors map analytically this entanglement transition to an ordering transition in a classical statistical mechanics model. They derive the general entanglement scaling properties at the transition and show a solvable limit where this transition can be mapped onto two-dimensional percolation.