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Entanglement Wedge Reconstruction of Infinite-dimensional von Neumann Algebras using Tensor Networks

Monica Jinwoo Kang, David K. Kolchmeyer

TL;DR

This work constructs an explicit quantum error correcting code with infinite-dimensional code and physical spaces, hosting a Type $II_1$ von Neumann algebra on each side. It employs an infinite tensor-network isometry to reconstruct code operations as boundary operators, demonstrating entanglement wedge reconstruction and exact equality between bulk and boundary relative entropies (JLMS) in this infinite-dimensional setting. Relative entropy is shown to arise as the limit of relative entropies for finite-dimensional subalgebras of a hyperfinite algebra, providing a practical computational route. The framework points toward deeper connections between holography, modular theory, and infinite-dimensional tensor networks, with potential extensions to Type III algebras and HaPPY-like constructions.

Abstract

Quantum error correcting codes with finite-dimensional Hilbert spaces have yielded new insights on bulk reconstruction in AdS/CFT. In this paper, we give an explicit construction of a quantum error correcting code where the code and physical Hilbert spaces are infinite-dimensional. We define a von Neumann algebra of type II$_1$ acting on the code Hilbert space and show how it is mapped to a von Neumann algebra of type II$_1$ acting on the physical Hilbert space. This toy model demonstrates the equivalence of entanglement wedge reconstruction and the exact equality of bulk and boundary relative entropies in infinite-dimensional Hilbert spaces.

Entanglement Wedge Reconstruction of Infinite-dimensional von Neumann Algebras using Tensor Networks

TL;DR

This work constructs an explicit quantum error correcting code with infinite-dimensional code and physical spaces, hosting a Type von Neumann algebra on each side. It employs an infinite tensor-network isometry to reconstruct code operations as boundary operators, demonstrating entanglement wedge reconstruction and exact equality between bulk and boundary relative entropies (JLMS) in this infinite-dimensional setting. Relative entropy is shown to arise as the limit of relative entropies for finite-dimensional subalgebras of a hyperfinite algebra, providing a practical computational route. The framework points toward deeper connections between holography, modular theory, and infinite-dimensional tensor networks, with potential extensions to Type III algebras and HaPPY-like constructions.

Abstract

Quantum error correcting codes with finite-dimensional Hilbert spaces have yielded new insights on bulk reconstruction in AdS/CFT. In this paper, we give an explicit construction of a quantum error correcting code where the code and physical Hilbert spaces are infinite-dimensional. We define a von Neumann algebra of type II acting on the code Hilbert space and show how it is mapped to a von Neumann algebra of type II acting on the physical Hilbert space. This toy model demonstrates the equivalence of entanglement wedge reconstruction and the exact equality of bulk and boundary relative entropies in infinite-dimensional Hilbert spaces.

Paper Structure

This paper contains 44 sections, 19 theorems, 184 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Infinite-dimensional von Neumann algebras
  3. Hilbert Space and Bounded Operators
  4. Topologies on $\mathcal{B}(\mathcal{H})$
  5. Definition of von Neumann algebras
  6. Classification of von Neumann algebras
  7. Type I factors
  8. Type II factors
  9. Type III factors
  10. Relative Entropy from Tomita-Takesaki theory
  11. The isometry between two infinite-dimensional Hilbert spaces
  12. The three-qutrit code and a finite tensor network
  13. The code and physical Hilbert spaces
  14. The tensor network of isometries
  15. Defining von Neumann algebras
  16. ...and 29 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. $u$ 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 spaces.

Figures (3)

  • Figure 1.1: Our setup consists of infinitely many collections of code (black) qutrit pairs which are related to physical (white) qutrits via a tensor network that consists of infinitely many disconnected graphs. The von Neumann algebra $M_{phys}$ acts on the first row of qutrits. The algebras $M_{code}$, $M_{code}^\prime$, and $M_{phys}^\prime$ act on the second, third, and fourth rows of qutrits respectively.
  • Figure 2.1: As shown in Takesaki, the norm operator topology is stronger than the strong operator topology and the ultraweak operator topology, which are both stronger than the weak operator topology.
  • Figure 4.1: The black code subspace qutrits are mapped to the white physical qutrits via an isometry defined by this tensor network.

Theorems & Definitions (63)

  • Theorem 1.1: Kang-Kolchmeyer HolographicEntropy
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Uniform Boundedness Principle Sokal
  • Theorem 2.5
  • proof
  • Theorem 2.6: Bounded Linear Transformation (BLT) Theorem ReedSimon
  • Definition 2.7
  • Definition 2.8
  • ...and 53 more