Entanglement spreading and emergent locality in Brownian SYK chains

TL;DR

This paper investigates how quantum information spreads and localizes in a Brownian SYK chain, revealing a sharp light-cone structure governed by a non-linear diffusion equation, with implications for holographic duality.

Contribution

It provides an analytical study of information spreading and emergent locality in a chaotic quantum system using QEC tools, highlighting the role of FKPP equations.

Findings

Information of injected qudit shows a sharp transition at the butterfly velocity.

The sharp light-cone emerges from FKPP domain wall solutions.

Operator growth properties explain the physical basis of the domain walls.

Abstract

The Ryu-Takayanagi (RT) formula and its interpretation in terms of quantum error correction (QEC) implies an emergent locality for the spread of quantum information in holographic CFTs, where information injected at a point in the boundary theory spreads within a sharp light-cone corresponding to the butterfly velocity. This emergent locality is a necessary condition for the existence of a geometric bulk dual with an RT-like formula for entanglement entropy. In this paper, we use tools from QEC to study the spread of quantum information and the emergence of a sharp light-cone in an analytically tractable model of chaotic dynamics, namely a one-dimensional Brownian SYK chain. We start with an infinite temperature state in this model and inject a qudit at time $t=0$ at some point $p$ on the chain. We then explicitly calculate the amount of information of the qudit contained in an interval of length $2\ell$ (centered around $p$) at some later time $t=T$. We find that at strong coupling, this quantity shows a sharp transition as a function of $\ell$ from near zero to near maximal correlation. The transition occurs at $\ell \sim v_B T$, with $v_B$ being the butterfly velocity. Underlying the emergence of this sharp light-cone is a non-linear generalization of the diffusion equation called the FKPP equation, which admits sharp domain wall solutions at late times and strong coupling. These domain wall solutions can be understood on physical grounds from properties of operator growth in chaotic systems.