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Multipartite Purification, Multiboundary Wormholes and Islands in AdS$_{3}$/CFT$_{2}$

Aranya Bhattacharya

TL;DR

The paper investigates how multipartite purification (EoP) and island physics in AdS3/CFT2 can be reconciled using multiboundary wormholes as a geometric toy model. It argues that the boundary of nontrivial islands corresponds to the multipartite entanglement wedge cross-section, offering a purification-based lens on islands and the Page curve. A central subtlety is potential overcounting of bulk entropy when computing island contributions with multipartite EWCS, which the author addresses by considering bipartite limits and time-dependent quantum extremal surfaces. The study highlights connections to quantum error correction and entanglement shadows, and suggests extensions to higher dimensions and tensor-network frameworks to deepen the island-purification dictionary.

Abstract

The holographic duals of Entanglement of Purification through the Entanglement Wedge Cross Section has been a well-discussed topic in the literature recently. More general entanglement measures involving multipartite information and their holographic duals have also been proposed. On the other hand, the recent intriguing program reproducing the Page Curve in Black hole entropy using the notion of islands has also been an obvious issue of attraction. A toy model involving Multiboundary wormholes in AdS$_{3}$ was able to capture many interesting facts about such calculations. In such a toy model, the notion of islands was intuitively connected to quantum error correction. We try to bridge the ideas of the two programs especially in AdS$_{3}$/CFT$_{2}$ and give a description of the islands in terms of multipartite entanglement of purification. This clarifies a few simplified assumptions made while describing the toy model and also enables us to understand the familiar information paradox within the framework of the same model.

Multipartite Purification, Multiboundary Wormholes and Islands in AdS$_{3}$/CFT$_{2}$

TL;DR

The paper investigates how multipartite purification (EoP) and island physics in AdS3/CFT2 can be reconciled using multiboundary wormholes as a geometric toy model. It argues that the boundary of nontrivial islands corresponds to the multipartite entanglement wedge cross-section, offering a purification-based lens on islands and the Page curve. A central subtlety is potential overcounting of bulk entropy when computing island contributions with multipartite EWCS, which the author addresses by considering bipartite limits and time-dependent quantum extremal surfaces. The study highlights connections to quantum error correction and entanglement shadows, and suggests extensions to higher dimensions and tensor-network frameworks to deepen the island-purification dictionary.

Abstract

The holographic duals of Entanglement of Purification through the Entanglement Wedge Cross Section has been a well-discussed topic in the literature recently. More general entanglement measures involving multipartite information and their holographic duals have also been proposed. On the other hand, the recent intriguing program reproducing the Page Curve in Black hole entropy using the notion of islands has also been an obvious issue of attraction. A toy model involving Multiboundary wormholes in AdS was able to capture many interesting facts about such calculations. In such a toy model, the notion of islands was intuitively connected to quantum error correction. We try to bridge the ideas of the two programs especially in AdS/CFT and give a description of the islands in terms of multipartite entanglement of purification. This clarifies a few simplified assumptions made while describing the toy model and also enables us to understand the familiar information paradox within the framework of the same model.

Paper Structure

This paper contains 13 sections, 31 equations, 14 figures.

Table of Contents

  1. Introduction:
  2. A Note on Entanglement of Purification:
  3. Definitions and Properties:
  4. Holographic Duals:
  5. A Note on Page Curve Study and Toy Model:
  6. Information Paradox and Resolutions (Islands) :
  7. Multiboundary Wormholes and a simple toy model:
  8. Connections between Purification and MBW Toy Model:
  9. Connections to be drawn:
  10. Realization of Overcounting through Shared Interior:
  11. Discussion and Outlook:
  12. More on Multiboundary Wormhole Construction:
  13. Limitations and Clarifications:

Figures (14)

  • Figure 1: A schematic diagram of bipartite purification ($A A^{\prime} B B^{\prime}$ forms a pure state and EoP is entanglement between $A A^{\prime}$ and $B B^{\prime}$)
  • Figure 2: Entanglement Wedge Cross Section (holographic dual of Bipartite Entanglement of Purification)
  • Figure 3: EWCS for tripartite EoP: The combination of A, B, C and the the HRT geodesics are combined as $\Tilde{A}\Tilde{B}\Tilde{C}$ and considered to form a tripartite geometric pure state.
  • Figure 4: Penrose diagram of two sided Black Hole with nontrivial island included (red and blue lines represent Hawking partner modes outside and inside the black hole horizon respectively.)
  • Figure 5: Choice of islands before and after Page time and Page Curve
  • ...and 9 more figures