On entanglement spreading from holography

A global quench is an interesting setting where we can study thermalization of subsystems in a pure state. We investigate entanglement entropy (EE) growth in global quenches in holographic field theories and relate some of its aspects to quantities characterizing chaos. More specifically we obtain four key results: 1. We prove holographic bounds on the entanglement velocity v E and the butterfly effect speed v B that arises in the study of chaos. 2. We obtain the EE as a function of time for large spherical entangling surfaces analytically. We show that the EE is insensitive to the details of the initial state or quench protocol. 3. In a thermofield double state we determine analytically the two-sided mutual information between two large concentric spheres separated in time. 4. We derive a bound on the rate of growth of EE for arbitrary shapes, and develop an expansion for EE at early times. In a companion paper [1], these results are put in the broader context of EE growth in chaotic systems: we relate EE growth to the chaotic spreading of operators, derive bounds on EE at a given time, and compare the holographic results to spin chain numerics and toy models. In this paper, we perform holographic calculations that provide the basis of arguments presented in that paper.A global quench is an interesting setting where we can study thermalization of subsystems in a pure state. We investigate entanglement entropy (EE) growth in global quenches in holographic field theories and relate some of its aspects to quantities characterizing chaos. More specifically we obtain four key results: 1. We prove holographic bounds on the entanglement velocity v E and the butterfly effect speed v B that arises in the study of chaos. 2. We obtain the EE as a function of time for large spherical entangling surfaces analytically. We show that the EE is insensitive to the details of the initial state or quench protocol. 3. In a thermofield double state we determine analytically the two-sided mutual information between two large concentric spheres separated in time. 4. We derive a bound on the rate of growth of EE for arbitrary shapes, and develop an expansion for EE at early times. In a companion paper [1], these results are put in the broader context of EE growth in chaotic systems: we relate EE growth to the chaotic spreading of operators, derive bounds on EE at a given time, and compare the holographic results to spin chain numerics and toy models. In this paper, we perform holographic calculations that provide the basis of arguments presented in that paper.

arXiv:1612.00082Journal of High Energy PhysicsMay 1, 201764 citations

Spread of entanglement for small subsystems in holographic CFTs

Sandipan Kundu, Juan F. Pedraza

We develop an analytic perturbative expansion to study the propagation of entanglement entropy for small subsystems after a global quench, in the context of the $\mathrm{AdS}/\mathrm{CFT}$ correspondence. Opposite to the large interval limit, in this case the evolution of the system takes place at time scales that are shorter in comparison to the local equilibration scale and, thus, different physical mechanisms govern the dynamics and subsequent thermalization. In particular, we show that the heuristic picture in terms of an ``entanglement tsunami'' does not apply in this regime. We find two crucial differences: First, that the instantaneous rate of growth of the entanglement is not constrained by causality, but rather its time average and, second, that the approach to saturation is always continuous, regardless of the shape of the entangling surface. Our analytic expansion also enables us to verify some previous numerical results, namely, that the saturation time is nonmonotonic with respect to the chemical potential. All of our results are pertinent to CFTs with a classical gravity dual formulation.We develop an analytic perturbative expansion to study the propagation of entanglement entropy for small subsystems after a global quench, in the context of the $\mathrm{AdS}/\mathrm{CFT}$ correspondence. Opposite to the large interval limit, in this case the evolution of the system takes place at time scales that are shorter in comparison to the local equilibration scale and, thus, different physical mechanisms govern the dynamics and subsequent thermalization. In particular, we show that the heuristic picture in terms of an ``entanglement tsunami'' does not apply in this regime. We find two crucial differences: First, that the instantaneous rate of growth of the entanglement is not constrained by causality, but rather its time average and, second, that the approach to saturation is always continuous, regardless of the shape of the entangling surface. Our analytic expansion also enables us to verify some previous numerical results, namely, that the saturation time is nonmonotonic with respect to the chemical potential. All of our results are pertinent to CFTs with a classical gravity dual formulation.

arXiv:1602.05934Physical review. D/Physical review. D.Apr 10, 201740 citations

Spread of entanglement and causality

Horacio Casini, Hong Liu, Márk Mezei

We investigate causality constraints on the time evolution of entanglement entropy after a global quench in relativistic theories. We first provide a general proof that the so-called tsunami velocity is bounded by the speed of light. We then generalize the free particle streaming model of [1] to general dimensions and to an arbitrary entanglement pattern of the initial state. In more than two spacetime dimensions the spread of entanglement in these models is highly sensitive to the initial entanglement pattern, but we are able to prove an upper bound on the normalized rate of growth of entanglement entropy, and hence the tsunami velocity. The bound is smaller than what one gets for quenches in holographic theories, which highlights the importance of interactions in the spread of entanglement in many-body systems. We propose an interacting model which we believe provides an upper bound on the spread of entanglement for interacting relativistic theories. In two spacetime dimensions with multiple intervals, this model and its variations are able to reproduce intricate results exhibited by holographic theories for a significant part of the parameter space. For higher dimensions, the model bounds the tsunami velocity at the speed of light. Finally, we construct a geometric model for entanglement propagation based on a tensor network construction for global quenches.We investigate causality constraints on the time evolution of entanglement entropy after a global quench in relativistic theories. We first provide a general proof that the so-called tsunami velocity is bounded by the speed of light. We then generalize the free particle streaming model of [1] to general dimensions and to an arbitrary entanglement pattern of the initial state. In more than two spacetime dimensions the spread of entanglement in these models is highly sensitive to the initial entanglement pattern, but we are able to prove an upper bound on the normalized rate of growth of entanglement entropy, and hence the tsunami velocity. The bound is smaller than what one gets for quenches in holographic theories, which highlights the importance of interactions in the spread of entanglement in many-body systems. We propose an interacting model which we believe provides an upper bound on the spread of entanglement for interacting relativistic theories. In two spacetime dimensions with multiple intervals, this model and its variations are able to reproduce intricate results exhibited by holographic theories for a significant part of the parameter space. For higher dimensions, the model bounds the tsunami velocity at the speed of light. Finally, we construct a geometric model for entanglement propagation based on a tensor network construction for global quenches.

arXiv:1509.05044Journal of High Energy PhysicsJul 1, 2016110 citations

Black Holes Often Saturate Entanglement Entropy the Fastest

Márk Mezei, Wilke van der Schee

There is a simple bound on how fast the entanglement entropy of a subregion of a many-body quantum system can saturate in a quench: t_{sat}≥R/v_{B}, where t_{sat} is the saturation time, R the radius of the largest inscribed sphere, and v_{B} the butterfly velocity characterizing operator growth. By combining analytic and numerical approaches, we show that in systems with a holographic dual, the saturation time is equal to this lower bound for a variety of differently shaped entangling surfaces, implying that the dual black holes saturate the entanglement entropy as fast as possible. This finding adds to the growing list of tasks that black holes are the fastest at. We furthermore analyze the complete time evolution of entanglement entropy for large regions with a variety of shapes, yielding more detailed information about the process of thermalization in these systems.There is a simple bound on how fast the entanglement entropy of a subregion of a many-body quantum system can saturate in a quench: t_{sat}≥R/v_{B}, where t_{sat} is the saturation time, R the radius of the largest inscribed sphere, and v_{B} the butterfly velocity characterizing operator growth. By combining analytic and numerical approaches, we show that in systems with a holographic dual, the saturation time is equal to this lower bound for a variety of differently shaped entangling surfaces, implying that the dual black holes saturate the entanglement entropy as fast as possible. This finding adds to the growing list of tasks that black holes are the fastest at. We furthermore analyze the complete time evolution of entanglement entropy for large regions with a variety of shapes, yielding more detailed information about the process of thermalization in these systems.

arXiv:2001.03172Physical Review LettersMay 20, 202012 citations

Comments on entanglement propagation

Moshe Rozali, Alexandre Vincart-Emard

We extend our work on entanglement propagation following a local quench in 2+1 dimensional holographic conformal field theories. We find that entanglement propagates along an emergent lightcone, whose speed of propagation v E seems distinct from other measures of quantum information spreading. We compare the relations we find to information and hydrodynamic velocities in strongly coupled 2+1 dimensional theories. While early-time entanglement velocities corresponding to small entangling regions are numerically close to the butterfly velocity, late-time entanglement velocities for large regions show less regularity. We also generalize and extend our previous results regarding the late-time decay of the entanglement entropy back to its equilibrium value.We extend our work on entanglement propagation following a local quench in 2+1 dimensional holographic conformal field theories. We find that entanglement propagates along an emergent lightcone, whose speed of propagation v E seems distinct from other measures of quantum information spreading. We compare the relations we find to information and hydrodynamic velocities in strongly coupled 2+1 dimensional theories. While early-time entanglement velocities corresponding to small entangling regions are numerically close to the butterfly velocity, late-time entanglement velocities for large regions show less regularity. We also generalize and extend our previous results regarding the late-time decay of the entanglement entropy back to its equilibrium value.

arXiv:1702.05869Journal of High Energy PhysicsJun 1, 20175 citations

On entanglement spreading in chaotic systems

Márk Mezei, Douglas Stanford

We discuss the time dependence of subsystem entropies in interacting quantum systems. As a model for the time dependence, we suggest that the entropy is as large as possible given two constraints: one follows from the existence of an emergent light cone, and the other is a conjecture associated to the “entanglement velocity” v E . We compare this model to new holographic and spin chain computations, and to an operator growth picture. Finally, we introduce a second way of computing the emergent light cone speed in holographic theories that provides a boundary dynamics explanation for a special case of entanglement wedge subregion duality in AdS/CFT.We discuss the time dependence of subsystem entropies in interacting quantum systems. As a model for the time dependence, we suggest that the entropy is as large as possible given two constraints: one follows from the existence of an emergent light cone, and the other is a conjecture associated to the “entanglement velocity” v E . We compare this model to new holographic and spin chain computations, and to an operator growth picture. Finally, we introduce a second way of computing the emergent light cone speed in holographic theories that provides a boundary dynamics explanation for a special case of entanglement wedge subregion duality in AdS/CFT.

arXiv:1608.05101Journal of High Energy PhysicsMay 1, 2017210 citations

We develop an analytic perturbative expansion to study the propagation of entanglement entropy for small subsystems after a global quench, in the context of the

\mathrm{AdS}/\mathrm{CFT}

correspondence. Opposite to the large interval limit, in this case the evolution of the system takes place at time scales that are shorter in comparison to the local equilibration scale and, thus, different physical mechanisms govern the dynamics and subsequent thermalization. In particular, we show that the heuristic picture in terms of an ``entanglement tsunami'' does not apply in this regime. We find two crucial differences: First, that the instantaneous rate of growth of the entanglement is not constrained by causality, but rather its time average and, second, that the approach to saturation is always continuous, regardless of the shape of the entangling surface. Our analytic expansion also enables us to verify some previous numerical results, namely, that the saturation time is nonmonotonic with respect to the chemical potential. All of our results are pertinent to CFTs with a classical gravity dual formulation.