On entanglement spreading in chaotic systems

TL;DR

The paper proposes two speed-based bounds on the time evolution of subsystem entropies in chaotic quantum systems: an emergent light-cone bound with speed $v_{LC}$ and a bound on the rate of entropy growth $rac{d}{dt}S[A(t)] \le v_E\, s_{\text{th}}\, \text{area}(A)$, where $v_E$ is the entanglement velocity. It analyzes how these bounds interplay across holographic theories and chaotic spin chains, introducing an operator-growth model in which $v_B=v_{LC}$ and $v_E$ arises from a rare non-growth tail, and provides a boundary-dynamics perspective linking $v_B$ to entanglement-wedge reconstruction in AdS/CFT. The authors find that holographic systems often saturate the combined bound for large regions, while spin-chain data show near-saturation with some deviations, and they demonstrate a boundary-level derivation of $v_B$ via the near-horizon entanglement wedge. Altogether, the work offers a coarse-grained, speed-based framework for entanglement spreading in chaotic many-body systems and connects information sprawl to geometric notions in holography.

Abstract

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.

Figures (10)

• Figure 1: Left: The "$v_{LC}$ cone" for a disk $A'(t')$ passes through $A(t)$. Any information in $A'(t')$ is contained in $A(t)$, hence $A'(t')$ is a subsystem of $A(t)$ and we can use the monotonicity of relative entropy \ref{['relent']}. Right: The tsunami wavefront for different times corresponding to the case where $\partial A$ is an ellipse (purple). Lighter color means later time.
• Figure 2: The bound \ref{['boundS']} for $v_E<\frac{v_{LC}}{d-1}$ (left) where only the $v_E$ constraint plays a role, $\frac{v_{LC}}{d-1}<v_E<v_{LC}$ (middle) where both are important, and $v_E = v_{LC}$ (right) where only the $v_{LC}$ constraint is relevant. In the middle plot the change of color indicates the position of $t_0$ defined in \ref{['boundS']}.
• Figure 3: We plot $S[A(t)]$ until saturation for a large sphere. Black/dashed is our upper bound using the speeds \ref{['holspeeds']}, blue/dotted is the bound from relativistic causality Hartman:2015apr and red/solid is the holographic computation Mezei:2016zxg. The dashed lines should be understood as one-parameter ($v_E$) fits, because $v_{LC}$ is determined by an independent calculation based on commutators (or equivalently, out-of-time order correlation functions).
• Figure 4: We plot $I[B_R(t),A_L]$ as a function of $r_B/r_A$. The different curves correspond to $t/r_A = 0,0.5,1,2,3$. Dashed/black is the lower bound, solid/red is the holographic result.
• Figure 5: If $B$ is smaller than shown in (a), the lower bound and the holographic answer indicate that $B$ contains none of the information in $A$. If $B$ is larger than shown in (b), it contains all of the information in $A$. The dashed lines are $v_{LC}$ cones, not light cones.