Holographic Reflected Entropy: Islands and Defect Phases

TL;DR

The paper investigates mixed-state entanglement in a time-dependent defect AdS3/BCFT2 setup by computing the reflected entropy for disjoint and adjacent radiation subsystems. Using both the island prescription and the defect extremal surface (DES) formula, it reveals a rich set of entanglement-entropy phases and shows exact holographic duality between island and DES descriptions via the entanglement wedge cross section (EWCS). It presents Page curves for entanglement entropy and reflected entropy across these phases, and demonstrates consistency between the two holographic approaches across time evolution and subsystem configurations. The work extends previous analyses to more complex mixed-state configurations, providing cross-checks and laying groundwork for future explorations of multipartite entanglement and higher-dimensional brane-world models.

Abstract

We investigate the mixed state entanglement structure through the reflected entropy for disjoint radiation subsystems coupled to a 2d eternal brane world black hole in a time dependent defect AdS$_3$/BCFT$_2$ scenario. Utilizing the island prescription and the defect extremal surface (DES) formula, we demonstrate a complex mixed state entanglement structure through the reflected entropy corresponding to distinct entanglement entropy phases. In each case we verify the holographic duality of the reflected entropy with the bulk entanglement wedge cross section (EWCS) and also obtain the Page curves for both the entanglement entropy and the associated reflected entropy phases. Furthermore, we extend our analysis to adjacent radiation subsystems and obtain consistent results using both the island and the DES prescription.

Figures (19)

• Figure 1: Holographic dual of a BCFT$_2$ defined on a half plane $(x>0)$. The EOW brane is considered to be at a constant hyperbolic angle $\sigma_0$. Figure modified from Deng:2020ent.
• Figure 2: A set of global conformal transformations takes us from a Euclidean AdS$_3$/BCFT$_2$ setup to a thermofield double state. Subsequently, by applying the analytic continuation $\tau' \to it'$ we obtain the Lorentzian solution. Figures modified from Chu:2021gdb.
• Figure 3: (a) Connected and (b) disconnected phases of the extremal surfaces. Figures modified from Chu:2021gdb.
• Figure 4: EE Phase-1 for two disjoint intervals $A$ and $B$. The RT surfaces corresponding to $A \cup B$ are represented by the green lines, while the dashed lines depict the different configurations of the EWCS.
• Figure 5: EE Phase-2 for two disjoint intervals $A$ and $B$. The RT surfaces corresponding to $A \cup B$ are represented by the green lines, while the dashed lines depict the different configurations of the EWCS.