The Page Curve for Reflected Entropy

TL;DR

This paper investigates the reflected entropy in a toy black hole evaporation model, confirming its holographic duality, analyzing non-perturbative effects, and exploring phase transitions and their dependence on parameters.

Contribution

It provides an analytical study of reflected entropy in the West Coast Model, revealing its spectrum, phase transition smoothing, and Renyi generalization effects.

Findings

Reflected entropy matches entanglement wedge cross section away from phase transitions.

Non-perturbative effects smooth out phase transition discontinuities.

Phase transition location varies with Renyi parameter.

Abstract

We study the reflected entropy $S_R$ in the West Coast Model, a toy model of black hole evaporation consisting of JT gravity coupled to end-of-the-world branes. We demonstrate the validity of the holographic duality relating it to the entanglement wedge cross section away from phase transitions. Further, we analyze the important non-perturbative effects that smooth out the discontinuity in the $S_R$ phase transition. By performing the gravitational path integral, we obtain the reflected entanglement spectrum analytically. The spectrum takes a simple form consisting of superselection sectors, which we interpret as a direct sum of geometries, a disconnected one and a connected one involving a closed universe. We find that area fluctuations of $O(\sqrt{G_N})$ spread out the $S_R$ phase transition in the canonical ensemble, analogous to the entanglement entropy phase transition. We also consider a Renyi generalization of the reflected entropy and show that the location of the phase transition varies as a function of the Renyi parameter.