Chaos and relative entropy

TL;DR

This paper investigates how the relative entropy between quantum states in conformal field theories and spin chains decays over time, revealing universal early-time behavior and model-dependent late-time dynamics related to chaos and integrability.

Contribution

It demonstrates universal exponential decay of relative entropy at early times in CFTs with gravity duals and numerically compares this behavior in integrable and non-integrable spin chain models.

Findings

Exponential decay of relative entropy until scrambling time in CFTs.

Universal early-time decay exponent independent of model details.

Large revivals of relative entropy in integrable models, absent in non-integrable models.

Abstract

One characterization of a chaotic system is the quick delocalization of quantum information (fast scrambling). One therefore expects that in such a system a state quickly becomes locally indistinguishable from its perturbations. In this paper we study the time dependence of the relative entropy between the reduced density matrices of the thermofield double state and its perturbations in two dimensional conformal field theories. We show that in a CFT with a gravity dual, this relative entropy exponentially decays until the scrambling time. This decay is not uniform. We argue that the early time exponent is universal while the late time exponent is sensitive to the butterfly effect. This large $c$ answer breaks down at the scrambling time, therefore we also study the relative entropy in a class of spin chain models numerically. We find a similar universal exponential decay at early times, while at later times we observe that the relative entropy has large revivals in integrable models, whereas there are no revivals in non-integrable models.