Holographic Reflected Entropy: Islands and Defect Phases

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.

💡 Research Summary

This research addresses one of the most profound challenges in modern physics: the Black Hole Information Paradox. While recent advancements involving the “island prescription” have provided a potential pathway to resolving how information is preserved during black hole evaporation, previous studies have largely focused on single radiation subsystems. This paper significantly expands the scope of inquiry by investigating the entanglement structure of multiple radiation subsystems that are either disjoint or adjacent, presenting a much more complex and realistic physical scenario.

The technical cornerstone of this study is the implementation of “Reflected Entropy” as a primary diagnostic tool. In the context of a black hole coupled with radiation, the system exists in a mixed state. Traditional measures like Von Neumann entropy, while effective for pure states, often fail to capture the true entanglement present in mixed states, potentially overestimating the degree of correlation. By utilizing reflected entropy, the researchers were able to quantify the intricate entanglement structure within these mixed states with much higher precision.

Methodologically, the paper employs a dual-approach to verify the robustness of holographic duality. On the boundary side, the “island prescription” is utilized to track information localization, while on the bulk side, the “Defect Extremal Surface (DES) formula” is applied to calculate geometric properties. The study demonstrates a remarkable consistency between these two independent frameworks, proving that the reflected entropy in the boundary theory is dual to the Entanglement Wedge Cross Section (EWCS) in the bulk. This verification reinforces the fundamental principles of the AdS/BCFT correspondence.

Furthermore, the research provides a detailed derivation of the “Page curves” for both entanglement entropy and reflected entropy. The Page curve is a critical indicator of the information recovery process during black hole evaporation, showing the transition from increasing entropy to decreasing entropy after the Page time. The authors reveal that the entanglement structure undergoes distinct “phase transitions” depending on the size and spatial configuration of the radiation subsystems. This discovery suggests that the entanglement within radiation is not a simple, monotonic process but a complex evolution of topological phases. Ultimately, this work highlights that the correlations between different subsystems are vital to understanding the underlying mechanism of information preservation in quantum gravity.