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Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality

Xi Dong, Daniel Harlow, Aron C. Wall

TL;DR

The paper proves that all bulk operators within a boundary entanglement wedge can be reconstructed as operators supported on a boundary subregion, using a quantum-error-correction framework and the JLMS relative-entropy relation. It formulates a constructive reconstruction theorem that guarantees the existence of a boundary operator O_A with identical action to a bulk operator O within a designated code subspace, under assumptions about quantum extremal surfaces. The approach unifies causal-wedge reconstruction and entanglement-wedge ideas, strengthening subregion duality and clarifying the error-correcting structure of AdS/CFT, with explicit connections to the HKLL method. The results pave the way for explicit bulk reconstructions at all orders in 1/N within the entanglement wedge, subject to the generalized entropy prescription.

Abstract

In this Letter we prove a simple theorem in quantum information theory, which implies that bulk operators in the Anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence can be reconstructed as CFT operators in a spatial subregion $A$, provided that they lie in its entanglement wedge. This is an improvement on existing reconstruction methods, which have at most succeeded in the smaller causal wedge. The proof is a combination of the recent work of Jafferis, Lewkowycz, Maldacena, and Suh on the quantum relative entropy of a CFT subregion with earlier ideas interpreting the correspondence as a quantum error correcting code.

Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality

TL;DR

The paper proves that all bulk operators within a boundary entanglement wedge can be reconstructed as operators supported on a boundary subregion, using a quantum-error-correction framework and the JLMS relative-entropy relation. It formulates a constructive reconstruction theorem that guarantees the existence of a boundary operator O_A with identical action to a bulk operator O within a designated code subspace, under assumptions about quantum extremal surfaces. The approach unifies causal-wedge reconstruction and entanglement-wedge ideas, strengthening subregion duality and clarifying the error-correcting structure of AdS/CFT, with explicit connections to the HKLL method. The results pave the way for explicit bulk reconstructions at all orders in 1/N within the entanglement wedge, subject to the generalized entropy prescription.

Abstract

In this Letter we prove a simple theorem in quantum information theory, which implies that bulk operators in the Anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence can be reconstructed as CFT operators in a spatial subregion , provided that they lie in its entanglement wedge. This is an improvement on existing reconstruction methods, which have at most succeeded in the smaller causal wedge. The proof is a combination of the recent work of Jafferis, Lewkowycz, Maldacena, and Suh on the quantum relative entropy of a CFT subregion with earlier ideas interpreting the correspondence as a quantum error correcting code.

Paper Structure

This paper contains 6 sections, 1 theorem, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Entanglement Wedge Reconstruction as Quantum Error Correction
  3. Review of the JLMS Argument
  4. A Reconstruction Theorem
  5. Discussion
  6. Appendix: Theorem of Operator Algebra Quantum Error Correction

Key Result

Theorem 1

Let $\mathcal{H}$ be a finite-dimensional Hilbert space, $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_{\overline{A}}$ be a tensor factorization, and $\mathcal{H}_{code}$ be a subspace of $\mathcal{H}$. Let $O$ be an operator that, together with its Hermitian conjugate, acts within $\mathcal{H}_{ then both of the following statements are true:

Figures (1)

  • Figure 1: Factorizing the bulk and boundary on a time slice. The entanglement wedge $\mathcal{E}_A$ is shaded. For simplicity we have shown a connected boundary region $A$, although this might not be the case.

Theorems & Definitions (2)

  • Theorem
  • proof