Symmetry-Resolved Relative Entropy of Random States

TL;DR

This paper calculates the symmetry-resolved relative entropy of random states with $U(1)$ symmetry using large-$N$ techniques, revealing universal behavior and deriving the symmetry-resolved Page curve to understand their distinguishability.

Contribution

It introduces a diagrammatic method for computing symmetry-resolved relative entropy in symmetric random states, providing new insights into their statistical properties.

Findings

Symmetry-resolved relative entropy exhibits universal statistical behavior.

Derived the symmetry-resolved Page curve for random states.

Method applicable to states with $U(1)$ symmetry.

Abstract

We use large-$N$ diagrammatic techniques to calculate the relative entropy of symmetric random states drawn from the Wishart ensemble. These methods are specifically designed for symmetric sectors, allowing us to determine the relative entropy for random states exhibiting $U(1)$ symmetry. This calculation serves as a measure of distinguishability within the symmetry sectors of random states. Our findings reveal that the symmetry-resolved relative entropy of random pure states displays universal statistical behavior. Furthermore, we derive the symmetry-resolved Page curve. These results deepen our understanding of the properties of these random states.