Islands for Reflected Entropy

TL;DR

This work extends the island paradigm to the reflected entropy by proposing a covariant islands formula that accounts for island contributions to S_R(A:B). The authors derive the formula from gravitational path integrals using replica-trick saddles and double holography, and verify that it satisfies fundamental S_R inequalities. Through JT gravity with a bath and black hole setups, they uncover phase transitions in S_R tied to entanglement-entropy transitions, revealing rich multipartite entanglement structure beyond von Neumann entropy. The framework suggests natural generalizations to other correlation measures and provides a foundation for understanding entanglement in Hawking radiation beyond traditional entropy measures.

Abstract

Recent work has demonstrated the need to include contributions from entanglement islands when computing the entanglement entropy in QFT states coupled to regions of semiclassical gravity. We propose a new formula for the reflected entropy that includes additional contributions from such islands. We derive this formula from the gravitational path integral by finding additional saddles that include generalized replica wormholes. We also demonstrate that our covariant formula satisfies all the inequalities required of the reflected entropy. We use this formula in various examples that demonstrate its relevance in illustrating the structure of multipartite entanglement that are invisible to the entropies.

Figures (11)

• Figure 1: A $d$ dimensional BCFT has a $d$ dimensional effective description in terms of a gravitating brane coupled to flat space. In the presence of holographic matter, this effective theory itself has a $d+1$ dimensional bulk dual. The reflected entropy of the regions $A$ and $B$ in the BCFT can be computed using the entanglement wedge cross section $\text{EW}(A:B)$ in the $d+1$ dimensional bulk dual. From the perspective of the effective $d$ dimensional theory, this leads to the islands formula of Eqn. (\ref{['eq:islandSR']}).
• Figure 2: The gravitational region $\mathcal{M}^{\text{bulk}}_{m}$ (shaded yellow) of the manifold $\mathcal{M}_{m}$ that computes $Z_m$ for $m=4$ is depicted here. In addition to a cyclic $\mathbb{Z}_m$ symmetry, we have a $\mathbb{Z}_2$ reflection symmetry which allows us to consider the bulk dual to the state $\ket{\rho_{AB}^{m/2}}$ by cutting open the path integral in half about the horizontal axis $\Sigma_m$. The Cauchy slice $\Sigma_m$ is made up of two pieces, that are denoted $\text{Is}(AB)_m$ and $\text{Is}(A'B')_m$, which become the entanglement islands of the respective regions in the limit $m\rightarrow1$. The red dot denotes the fixed point of $\mathbb{Z}_m$ symmetry that becomes the quantum extremal surface as $m\rightarrow1$. The dashed lines represent the complementary region to the island which has been traced out.
• Figure 3: The manifold $\mathcal{M}_{m,n}$ involves gluing the subregions B cyclically in the vertical $\mu$ direction, whereas the subregions A are glued together cyclically in the vertical direction upto a cyclic twist, in the horizontal $\nu$ direction, at $\mu=0,\frac{m}{2}$.
• Figure 4: The time slice $\Sigma_m$ consists of a gravitating region (denoted red) where two copies of the island region $\text{Is}(AB)$ are glued together at $\partial\text{Is}(AB)$ (denoted purple). The non-gravitating region involves twist operators inserted at $\partial A$ and $\partial B$ (denoted yellow). The effect of these twist operators can be thought of as inducing two kinds of cosmic branes in the gravitating region, which we call Type-$m$ and Type-$n$ branes.
• Figure 5: The Penrose diagram for the vacuum $\text{AdS}_2$ setup consisting of a finite subregion $A$ and a semi-infinite subregion $B$ in a half-Minkowski space (bath) eternally coupled to a gravitating region with the correspond island $a$ and cross-section $a'$.