Relative Entropy of Random States and Black Holes

TL;DR

This paper develops a diagrammatic approach to compute the relative entropy of highly excited quantum states, applies it to black hole microstates in AdS/CFT, and finds that such states are distinguishable even with minimal access, supporting the subsystem Eigenstate Thermalization Hypothesis.

Contribution

It introduces an exact large-N diagrammatic method for relative entropy in random states and applies it to holographic black holes, revealing their distinguishability properties.

Findings

Random states' relative entropy matches numerical simulations.

Black hole microstates are distinguishable with arbitrarily small access.

Holographic systems obey the subsystem Eigenstate Thermalization Hypothesis.

Abstract

We study the relative entropy of highly excited quantum states. First, we sample states from the Wishart ensemble and develop a large-N diagrammatic technique for the relative entropy. The solution is exactly expressed in terms of elementary functions. We compare the analytic results to small-N numerics, finding precise agreement. Furthermore, the random matrix theory results accurately match the behavior of chaotic many-body eigenstates, a manifestation of eigenstate thermalization. We apply this formalism to the AdS/CFT correspondence where the relative entropy measures the distinguishability between different black hole microstates. We find that black hole microstates are distinguishable even when the observer has arbitrarily small access to the quantum state, though the distinguishability is nonperturbatively small in Newton's constant. Finally, we interpret these results in the context of the subsystem Eigenstate Thermalization Hypothesis (sETH), concluding that holographic systems obey sETH up to subsystems half the size of the total system.