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Thermal states are vital: Entanglement Wedge Reconstruction from Operator-Pushing

Elliott Gesteau, Monica Jinwoo Kang

TL;DR

The paper develops an operator-pushing framework for bulk reconstruction in infinite-dimensional holographic settings by starting from a bulk $C^*$-algebra with a bulk-to-boundary isometry and a boundary $ ext{KMS}$ state $oldsymbol{\omega}$. Using GNS representations of the algebras, it constructs physically relevant Hilbert spaces and a bulk/boundary isometry $u$ that implement the bulk reconstruction, while proving conservation of Araki relative entropy and equality of modular flows between bulk and boundary. It thereby provides a state-dependent, algebraic realization of a wormhole’s other side, connecting ideas like the thermofield double and Papadodimas–Raju mirror operators, and applies the construction to an infinite-dimensional HaPPY code to realize a holographic wormhole with entanglement wedge reconstruction. The work blends algebraic quantum field theory tools (Tomita–Takesaki theory, Araki entropy) with holographic quantum error correction, highlighting the central role of thermal equilibrium via the $ ext{KMS}$ condition in enabling operator-level bulk reconstruction in the infinite-dimensional regime.

Abstract

We give a general construction of a setup that verifies bulk reconstruction, conservation of relative entropies, and equality of modular flows between the bulk and the boundary, for infinite-dimensional systems with operator-pushing. In our setup, a bulk-to-boundary map is defined at the level of the $C^*$-algebras of state-independent observables. We then show that if the boundary dynamics allow for the existence of a KMS state, physically relevant Hilbert spaces and von Neumann algebras can be constructed directly from our framework. Our construction should be seen as a state-dependent construction of the other side of a wormhole and clarifies the meaning of black hole reconstruction claims such as the Papadodimas-Raju proposal. As an illustration, we apply our result to construct a wormhole based on the HaPPY code, which satisfies all properties of entanglement wedge reconstruction.

Thermal states are vital: Entanglement Wedge Reconstruction from Operator-Pushing

TL;DR

The paper develops an operator-pushing framework for bulk reconstruction in infinite-dimensional holographic settings by starting from a bulk -algebra with a bulk-to-boundary isometry and a boundary state . Using GNS representations of the algebras, it constructs physically relevant Hilbert spaces and a bulk/boundary isometry that implement the bulk reconstruction, while proving conservation of Araki relative entropy and equality of modular flows between bulk and boundary. It thereby provides a state-dependent, algebraic realization of a wormhole’s other side, connecting ideas like the thermofield double and Papadodimas–Raju mirror operators, and applies the construction to an infinite-dimensional HaPPY code to realize a holographic wormhole with entanglement wedge reconstruction. The work blends algebraic quantum field theory tools (Tomita–Takesaki theory, Araki entropy) with holographic quantum error correction, highlighting the central role of thermal equilibrium via the condition in enabling operator-level bulk reconstruction in the infinite-dimensional regime.

Abstract

We give a general construction of a setup that verifies bulk reconstruction, conservation of relative entropies, and equality of modular flows between the bulk and the boundary, for infinite-dimensional systems with operator-pushing. In our setup, a bulk-to-boundary map is defined at the level of the -algebras of state-independent observables. We then show that if the boundary dynamics allow for the existence of a KMS state, physically relevant Hilbert spaces and von Neumann algebras can be constructed directly from our framework. Our construction should be seen as a state-dependent construction of the other side of a wormhole and clarifies the meaning of black hole reconstruction claims such as the Papadodimas-Raju proposal. As an illustration, we apply our result to construct a wormhole based on the HaPPY code, which satisfies all properties of entanglement wedge reconstruction.

Paper Structure

This paper contains 28 sections, 12 theorems, 59 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Analysis in infinite dimensions
  3. Analysis in infinite dimensions
  4. Topolgies on $\mathcal{B}(\mathcal{H})$
  5. $C^*$-algebras
  6. Von Neumann algebras
  7. Tools from algebraic quantum field theory
  8. $C^*$-algebras, von Neumann algebras, and state dependence
  9. The GNS representation
  10. KMS states
  11. Tomita--Takesaki theory
  12. Araki's relative entropy
  13. Infinite-dimensional entanglement wedge reconstruction
  14. Main theorem and proof
  15. The main theorem and its physical implications
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\mathcal{A}_{code}$ and $\mathcal{A}_{phys}$ be two $C^*$-algebras, and let $\iota:\mathcal{A}_{code}\longrightarrow \mathcal{A}_{phys}$ be an isometric $C^*$-homomorphism. Let $\sigma_t$ be a strongly continuous one-parameter group of isometries of $\mathcal{A}_{phys}$ such that $\sigma_t(\iot

Figures (4)

  • Figure 8.1: This is the HaPPY code at level 2 with bulk and boundary nodes. The black qubits are the bulk nodes and the white dangling qubits are the boundary nodes. The blue region corresponds to level 1 bulk, which is represented with a single central node. The yellow regions are the new additional tiles for the bulk of level 2. (Level $n$ bulk always include level $n-1$ bulk. For example, the blue tile from level 1 is also a part of bulk for level $2$ and beyond.) The green regions are the addendums for the bulk of level 3. The red regions are the addendums of the bulk of level 4.
  • Figure 8.2: The straight-forward HaPPY code from the center and its reverse engineering from the boundary for the infinite-dimensional analog of the HaPPY code.
  • Figure 8.3: Pushing a trapeze operator to the first boundary layer.
  • Figure 8.4: The holographic HaPPY wormhole with its two infinite boundaries.

Theorems & Definitions (27)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Banach-Alaoglu BanachAlaoglu
  • Theorem 2.5: Takesaki Takesaki
  • Definition 2.6
  • Theorem 2.7: Kaplansky Kaplansky
  • Definition 2.8
  • Theorem 2.9: Jones-vNalg, page 12
  • ...and 17 more