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Holographic Relative Entropy in Infinite-dimensional Hilbert Spaces

Monica Jinwoo Kang, David K. Kolchmeyer

TL;DR

This work extends holographic relative entropy to infinite-dimensional Hilbert spaces by casting bulk and boundary observables as von Neumann algebras $M_{code}$ and $M_{phys}$ within an operator-algebra quantum error-correction framework. Using Tomita–Takesaki theory, it proves a necessary-and-sufficient condition: the equality of relative entropies computed on bulk vs boundary algebras (and their commutants) is equivalent to entanglement wedge reconstruction, with bulk and boundary modular operators acting identically on the code subspace. The results provide a rigorous, horizon-independent notion of holographic relative entropy and connect modular flow in the bulk and boundary, generalizing previous finite-dimensional QEC insights to the physically relevant infinite-dimensional setting. The discussion situates the findings in the context of QFT, AdS/CFT, and the Reeh–Schlieder theorem, and outlines approximate reconstructions and type III$_1$ factor considerations that support applicability to realistic holographic theories. Overall, the paper clarifies how equal bulk-boundary relative entropies encode reconstructibility and offers an explicit operator mapping that realizes bulk operators on the boundary in the infinite-dimensional regime.

Abstract

We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a boundary subregion and its entanglement wedge. Assuming that the infinite-dimensional von Neumann algebras associated with an entanglement wedge and its complement may both be reconstructed in their corresponding boundary subregions, we prove that the relative entropies measured with respect to the bulk and boundary observables are equal. We also prove the converse: when the relative entropies measured in an entanglement wedge and its complement equal the relative entropies measured in their respective boundary subregions, entanglement wedge reconstruction is possible. Along the way, we show that the bulk and boundary modular operators act on the code subspace in the same way. For holographic theories with a well-defined entanglement wedge, this result provides a well-defined notion of holographic relative entropy.

Holographic Relative Entropy in Infinite-dimensional Hilbert Spaces

TL;DR

This work extends holographic relative entropy to infinite-dimensional Hilbert spaces by casting bulk and boundary observables as von Neumann algebras and within an operator-algebra quantum error-correction framework. Using Tomita–Takesaki theory, it proves a necessary-and-sufficient condition: the equality of relative entropies computed on bulk vs boundary algebras (and their commutants) is equivalent to entanglement wedge reconstruction, with bulk and boundary modular operators acting identically on the code subspace. The results provide a rigorous, horizon-independent notion of holographic relative entropy and connect modular flow in the bulk and boundary, generalizing previous finite-dimensional QEC insights to the physically relevant infinite-dimensional setting. The discussion situates the findings in the context of QFT, AdS/CFT, and the Reeh–Schlieder theorem, and outlines approximate reconstructions and type III factor considerations that support applicability to realistic holographic theories. Overall, the paper clarifies how equal bulk-boundary relative entropies encode reconstructibility and offers an explicit operator mapping that realizes bulk operators on the boundary in the infinite-dimensional regime.

Abstract

We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a boundary subregion and its entanglement wedge. Assuming that the infinite-dimensional von Neumann algebras associated with an entanglement wedge and its complement may both be reconstructed in their corresponding boundary subregions, we prove that the relative entropies measured with respect to the bulk and boundary observables are equal. We also prove the converse: when the relative entropies measured in an entanglement wedge and its complement equal the relative entropies measured in their respective boundary subregions, entanglement wedge reconstruction is possible. Along the way, we show that the bulk and boundary modular operators act on the code subspace in the same way. For holographic theories with a well-defined entanglement wedge, this result provides a well-defined notion of holographic relative entropy.

Paper Structure

This paper contains 18 sections, 19 theorems, 106 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bounded and Unbounded Operators
  3. Functions of bounded operators
  4. Unbounded operators
  5. The spectral theorem for unbounded operators
  6. Review of Tomita-Takesaki theory
  7. Proof of Theorem \ref{['thm:otherdirection']}
  8. Special case of bounded relative Tomita operator
  9. General proof of Theorem \ref{['thm:otherdirection']}
  10. Proof of Theorem \ref{['thm:maintheorem']}
  11. Discussion
  12. Von Neumann algebras in quantum field theory
  13. Approximate entanglement wedge reconstruction
  14. The Reeh--Schlieder theorem
  15. Physical motivation for the assumptions of Theorem \ref{['thm:maintheorem']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $u : \mathcal{H}_{code}\rightarrow \mathcal{H}_{phys}$ be an isometryThis means that $u$ is a norm-preserving map. The map $u$ need not be a bijection. In general, $u^\dagger u$ is the identity on $\mathcal{H}_{code}$ and $uu^\dagger$ is a projection on $\mathcal{H}_{phys}$. between two Hilbert

Figures (1)

  • Figure 3.1: Two Rindler wedges in Minkowski space. The generators $K_ r$ and $K_\ell$ correspond to boosts, as shown.

Theorems & Definitions (52)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 42 more