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

TL;DR

This paper constructs an explicit infinite-dimensional quantum error correcting code using von Neumann algebras, demonstrating entanglement wedge reconstruction and the equality of bulk and boundary relative entropies in AdS/CFT.

Contribution

It introduces a novel infinite-dimensional code model with von Neumann algebras, advancing understanding of bulk reconstruction in holography.

Findings

Explicit construction of infinite-dimensional quantum error correcting code.

Demonstration of entanglement wedge reconstruction in infinite-dimensional setting.

Equivalence of bulk and boundary relative entropies established.

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.