Relative entropy in higher spin holography

TL;DR

This paper investigates the behavior of relative entropy in higher-spin holography, confirming theoretical expectations and computing specific state differences in a 3D higher-spin gravity/CFT duality context.

Contribution

It provides a detailed holographic calculation of relative entropy in higher-spin gravity, including the vacuum and high-temperature states with chemical potentials, using Wilson line functionals.

Findings

Relative entropy vanishes for small entangling intervals.

The modular Hamiltonian expectation value difference matches entanglement entropy difference.

Computed relative entropy for smooth solutions in the $SL(2, ext{Z})$ family.

Abstract

We examine relative entropy in the context of the higher-spin/CFT duality. We consider 3$d$ bulk configurations in higher spin gravity which are dual to the vacuum and a high temperature state of a CFT with $\mathcal{W}$-algebra symmetries in presence of a chemical potential for a higher spin current. The relative entropy between these states is then evaluated using the Wilson line functional for holographic entanglement entropy. In the limit of small entangling intervals, the relative entropy should vanish for a generic quantum system. We confirm this behaviour by showing that the difference in the expectation values of the modular Hamiltonian between the states matches with the difference in the entanglement entropy in the short-distance regime. Additionally, we compute the relative entropy of states corresponding to smooth solutions in the $SL(2,\mathbb{Z})$ family with respect to the vacuum.