The Page Curve for Reflected Entropy

TL;DR

This work analyzes the reflected entropy in the West Coast Model—a JT gravity setup with ETW branes—to test holographic duality to the entanglement wedge cross section and to understand phase-transition behavior. Using gravitational path integrals and an extended resolvent method, the authors obtain the full reflected entanglement spectrum, revealing two superselection sectors corresponding to disconnected and connected purifications; area fluctuations broaden the Page-like transition in the canonical ensemble and give Renyi-index dependent transition locations. They demonstrate that S_R(R1:R2) scales with the connected-sector probability p_c and the minimal radiation entropy, while the spectrum and Renyi variants are controlled by p_d and p_c through m-dependent Catalan-like structures. The results illuminate how geometry and non-perturbative effects shape multipartite entanglement in holographic evaporation scenarios and suggest deeper ties to reconstruction maps such as the Petz map. Overall, the paper provides a concrete, exact handle on Page-curve-like behavior for reflected entropy and clarifies the role of fluctuations and Renyi generalizations in holographic entanglement measures.

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.

Figures (11)

• Figure 1: The Lorentzian description of the state we consider in the West Coast Model, a JT gravity black hole with an ETW brane. The ETW brane carries two sub-flavours, denoted black and green, that are entangled (dashed, coloured lines) with radiation systems $R_1$ and $R_2$ respectively. The extremal surface is denoted in purple and the island that dominates after the Page time is coloured gray.
• Figure 2: A spatial slice of AdS with $A$ and $B$ chosen to be two intervals. The figure depicts the entanglement wedge of $AB$ (gray), the entanglement wedge of $C$ (green), the RT surface $\gamma_{AB}$ and the entanglement wedge cross section $\Gamma_{A:B}$, which divides the entanglement wedge into regions $a$ and $b$ homologous to $A$ and $B$ respectively.
• Figure 3: The main result: (a) The reflected entanglement spectrum of $\rho_{AA^*}$ in the canonically purified state $\ket{\sqrt{\rho_{AB}}}$ is a mixture of two superselection sectors: a single pole of weight $p_d$ corresponding to the disconnected purification, and a mound of approximately $k_1^2$ eigenvalues (assuming $k_1<k_2$) with weight $p_c$ corresponding to a connected purification involving a closed universe with the entanglement wedge cross section denoted in orange. (b) The probability of the connected purification $p_c$ as we vary $k$ across the Page transition. We show analytic plots for the microcanonical and canonical ensemble. The latter shows a spread in the phase transition of $O(\sqrt{G_N})$. We also show plots for $p_c$ when considering the $(m,1)$-Renyi reflected entropy; these undergo sharp transitions at $m$-dependent locations in the canonical ensemble.
• Figure 4: (a) The thermal spectrum of the JT black hole. (b) The approximate spectrum of $\rho_{R_1 R_2}$ is a cutoff thermal spectrum which is obtained by truncating the spectrum to the $k$ largest eigenvalues and shifting it by $\lambda_0$ to make it normalized.
• Figure 5: (a) For $m>1$, the Renyi entropy (denoted by the green line) has a sharp transition at $s_k=s_k^{(m)}$ where the two replica symmetric saddle contributions (denoted by blue and red dotted lines) exchange dominance. Phase transitions in the two terms in Eq. \ref{['eq:max_m']} happen at $s_k=s_1,s_m$ (depicted with curved, gray dashed lines) and thus don't affect the Renyi entropies. (b) For $m<1$, the Renyi entropy (denoted by the green line) has a transition in the range $s_k \in [s_1,s_m]$ whereas the two naive replica symmetric saddles (denoted by blue and red dotted lines) exchange dominance at $s_k=s_k^{(m)}$. Thus, the phase transition is over a large window of size $O(\frac{1}{\beta})$ and the Renyi entropy has $O(\frac{1}{\beta})$ corrections compared to a naive analytic continuation of the replica symmetric saddles.