arXiv:1911.12384 · Physical Review Research

Diffusive scaling of Rényi entanglement entropy

Tianci Zhou, Andreas W. W. Ludwig

Published Jul 6, 2020• 28 citations

Abstract

The authors show a diffusive diffusive growth behavior in a random unitary circuit with a conservation law, and provide numerical evidence of its onset in a generic chaotic quantum spin chain possessing energy conservation.

DOI: 10.1103/physrevresearch.2.033020

Related papers

Entanglement growth in diffusive systems

Marko Žnidarič

arXiv:1912.03645Communications PhysicsJun 1, 202058 citations

Dynamics of Renyi entanglement entropy in local quantum circuits with charge conservation

Yichen Huang

arXiv:1902.00977Feb 3, 201961 citations

Sub-ballistic Growth of Rényi Entropies due to Diffusion

Tibor Rakovszky, Frank Pollmann, Curt W. von Keyserlingk

arXiv:1901.10502Physical Review LettersJun 25, 2019144 citations

Dynamics of Rényi entanglement entropy in diffusive qudit systems

Yichen Huang

arXiv:2008.00944IOP SciNotesDec 1, 202015 citations

Entanglement growth in diffusive systems with large spin

Tibor Rakovszky, Frank Pollmann, Curt W. von Keyserlingk

Recent works 1 , 2 argued that Rényi entropies S α with indices α > 1 exhibit a sub-ballistic, $\propto \sqrt{t}$ , growth in systems with diffusive transport. A subsequent work by Žnidarič 3 claims that such sub-ballistic growth occurs only in certain cases (in particular, d = 1 dimensional systems with q = 2 states per site) and is generically replaced by ballistic growth in the absence of further fine-tuning. Below, we try to make precise the conditions needed for diffusive (rather than ballistic) Rényi growth and argue that they apply to a much wider class of systems than what is suggested by Žnidarič 3 . In particular, the examples presented by Žnidarič that avoid diffusive growth do so because of the presence of additional non-conserved degrees of freedom, not merely their larger local Hilbert space.

Communications PhysicsMay 10, 20218 citations

Comment on "Entanglement growth in diffusive systems"

Tibor Rakovszky, Frank Pollmann, Curt W. von Keyserlingk

In a recent paper (Commun. Phys. 3, 100) Znidaric studies the growth of higher Renyi entropies in diffusive systems and claims that they generically grow ballistically in time, except for spin-1/2 models in d=1 dimension. Here, we point out that the necessary conditions for sub-ballistic growth of Renyi entropies are in fact much more general, and apply to a large class of systems, including experimentally relevant ones in arbitrary dimension and with larger local Hilbert spaces.

arXiv:2010.07969Oct 15, 20205 citations

arXiv:1911.12384 · Physical Review Research

Diffusive scaling of Rényi entanglement entropy

Tianci Zhou, Andreas W. W. Ludwig

Published Jul 6, 2020• 28 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevresearch.2.033020

Related papers

Entanglement growth in diffusive systems

Marko Žnidarič

Abstract Entanglement helps in understanding diverse phenomena, going from quantifying complexity to classifying phases of matter. Here we study the influence of conservation laws on entanglement growth. Focusing on systems with U(1) symmetry, i.e., conservation of charge or magnetization, that exhibits diffusive dynamics, we theoretically predict the growth of entanglement, as quantified by the Rényi entropy, in lattice systems in any spatial dimension d and for any local Hilbert space dimension q (qudits). We find that the growth depends both on d and q, and is in generic case first linear in time, similarly as for systems without any conservation laws. Exception to this rule are chains of 2-level systems where the dependence is a square-root of time at all times. Predictions are numerically verified by simulations of diffusive Clifford circuits with upto ~ 105 qubits. Such efficiently simulable circuits should be a useful tool for other many-body problems.Abstract Entanglement helps in understanding diverse phenomena, going from quantifying complexity to classifying phases of matter. Here we study the influence of conservation laws on entanglement growth. Focusing on systems with U(1) symmetry, i.e., conservation of charge or magnetization, that exhibits diffusive dynamics, we theoretically predict the growth of entanglement, as quantified by the Rényi entropy, in lattice systems in any spatial dimension d and for any local Hilbert space dimension q (qudits). We find that the growth depends both on d and q, and is in generic case first linear in time, similarly as for systems without any conservation laws. Exception to this rule are chains of 2-level systems where the dependence is a square-root of time at all times. Predictions are numerically verified by simulations of diffusive Clifford circuits with upto ~ 105 qubits. Such efficiently simulable circuits should be a useful tool for other many-body problems.

arXiv:1912.03645Communications PhysicsJun 1, 202058 citations

Dynamics of Renyi entanglement entropy in local quantum circuits with charge conservation

Yichen Huang

In local quantum circuits with charge conservation, we initialize the system in random product states and study the dynamics of the Renyi entanglement entropy $R_α$ . We rigorously prove that $R_α$ with Renyi index $α>1$ at time $t$ is $\le O(\sqrt{t\ln t})$ if the transport of charges is diffusive. Very recent numerical results of Rakovszky et al. show that this upper bound is saturated (up to the sub-logarithmic correction) in random local quantum circuits with charge conservation.In local quantum circuits with charge conservation, we initialize the system in random product states and study the dynamics of the Renyi entanglement entropy $R_α$. We rigorously prove that $R_α$ with Renyi index $α>1$ at time $t$ is $\le O(\sqrt{t\ln t})$ if the transport of charges is diffusive. Very recent numerical results of Rakovszky et al. show that this upper bound is saturated (up to the sub-logarithmic correction) in random local quantum circuits with charge conservation.

arXiv:1902.00977Feb 3, 201961 citations

Sub-ballistic Growth of Rényi Entropies due to Diffusion

Tibor Rakovszky, Frank Pollmann, Curt W. von Keyserlingk

We investigate the dynamics of quantum entanglement after a global quench and uncover a qualitative difference between the behavior of the von Neumann entropy and higher Rényi entropies. We argue that the latter generically grow sub-ballistically, as ∝sqrt[t], in systems with diffusive transport. We provide strong evidence for this in both a U(1) symmetric random circuit model and in a paradigmatic nonintegrable spin chain, where energy is the sole conserved quantity. We interpret our results as a consequence of local quantum fluctuations in conserved densities, whose behavior is controlled by diffusion, and use the random circuit model to derive an effective description. We also discuss the late-time behavior of the second Rényi entropy and show that it exhibits hydrodynamic tails with three distinct power laws occurring for different classes of initial states.We investigate the dynamics of quantum entanglement after a global quench and uncover a qualitative difference between the behavior of the von Neumann entropy and higher Rényi entropies. We argue that the latter generically grow sub-ballistically, as ∝sqrt[t], in systems with diffusive transport. We provide strong evidence for this in both a U(1) symmetric random circuit model and in a paradigmatic nonintegrable spin chain, where energy is the sole conserved quantity. We interpret our results as a consequence of local quantum fluctuations in conserved densities, whose behavior is controlled by diffusion, and use the random circuit model to derive an effective description. We also discuss the late-time behavior of the second Rényi entropy and show that it exhibits hydrodynamic tails with three distinct power laws occurring for different classes of initial states.

arXiv:1901.10502Physical Review LettersJun 25, 2019144 citations

Dynamics of Rényi entanglement entropy in diffusive qudit systems

Yichen Huang

Abstract My previous work [arXiv: 1902.00977 ] studied the dynamics of Rényi entanglement entropy R α in local quantum circuits with charge conservation. Initializing the system in a random product state, it was proved that R α with Rényi index α > 1 grows no faster than ‘diffusively’ (up to a sublogarithmic correction) if charge transport is not faster than diffusive. The proof was given only for qubit or spin-1/2 systems. In this note, I extend the proof to qudit systems, i.e., spin systems with local dimension d ≥ 2 .Abstract My previous work [arXiv: 1902.00977 ] studied the dynamics of Rényi entanglement entropy R α in local quantum circuits with charge conservation. Initializing the system in a random product state, it was proved that R α with Rényi index α > 1 grows no faster than ‘diffusively’ (up to a sublogarithmic correction) if charge transport is not faster than diffusive. The proof was given only for qubit or spin-1/2 systems. In this note, I extend the proof to qudit systems, i.e., spin systems with local dimension d ≥ 2 .

arXiv:2008.00944IOP SciNotesDec 1, 202015 citations

Entanglement growth in diffusive systems with large spin

Tibor Rakovszky, Frank Pollmann, Curt W. von Keyserlingk

Recent works 1 , 2 argued that Rényi entropies S α with indices α > 1 exhibit a sub-ballistic, $\propto \sqrt{t}$ , growth in systems with diffusive transport. A subsequent work by Žnidarič 3 claims that such sub-ballistic growth occurs only in certain cases (in particular, d = 1 dimensional systems with q = 2 states per site) and is generically replaced by ballistic growth in the absence of further fine-tuning. Below, we try to make precise the conditions needed for diffusive (rather than ballistic) Rényi growth and argue that they apply to a much wider class of systems than what is suggested by Žnidarič 3 . In particular, the examples presented by Žnidarič that avoid diffusive growth do so because of the presence of additional non-conserved degrees of freedom, not merely their larger local Hilbert space.Recent works 1 , 2 argued that Rényi entropies S α with indices α > 1 exhibit a sub-ballistic, $\propto \sqrt{t}$ , growth in systems with diffusive transport. A subsequent work by Žnidarič 3 claims that such sub-ballistic growth occurs only in certain cases (in particular, d = 1 dimensional systems with q = 2 states per site) and is generically replaced by ballistic growth in the absence of further fine-tuning. Below, we try to make precise the conditions needed for diffusive (rather than ballistic) Rényi growth and argue that they apply to a much wider class of systems than what is suggested by Žnidarič 3 . In particular, the examples presented by Žnidarič that avoid diffusive growth do so because of the presence of additional non-conserved degrees of freedom, not merely their larger local Hilbert space.

Communications PhysicsMay 10, 20218 citations

Comment on "Entanglement growth in diffusive systems"

Tibor Rakovszky, Frank Pollmann, Curt W. von Keyserlingk

arXiv:2010.07969Oct 15, 20205 citations

In local quantum circuits with charge conservation, we initialize the system in random product states and study the dynamics of the Renyi entanglement entropy

R_α

. We rigorously prove that

R_α

with Renyi index

α>1

at time

t

\le O(\sqrt{t\ln t})

if the transport of charges is diffusive. Very recent numerical results of Rakovszky et al. show that this upper bound is saturated (up to the sub-logarithmic correction) in random local quantum circuits with charge conservation.

Recent works 1 , 2 argued that Rényi entropies S α with indices α > 1 exhibit a sub-ballistic,

\propto \sqrt{t}

, growth in systems with diffusive transport. A subsequent work by Žnidarič 3 claims that such sub-ballistic growth occurs only in certain cases (in particular, d = 1 dimensional systems with q = 2 states per site) and is generically replaced by ballistic growth in the absence of further fine-tuning. Below, we try to make precise the conditions needed for diffusive (rather than ballistic) Rényi growth and argue that they apply to a much wider class of systems than what is suggested by Žnidarič 3 . In particular, the examples presented by Žnidarič that avoid diffusive growth do so because of the presence of additional non-conserved degrees of freedom, not merely their larger local Hilbert space.