Entanglement Entropy in Loop Quantum Gravity through Quantum Error Correction

TL;DR

This paper presents a new algebraic method using quantum error correction to compute entanglement entropy in Loop Quantum Gravity, successfully reproducing black hole entropy and deriving the Ryu-Takayanagi formula from first principles.

Contribution

It introduces an algebraic approach employing quantum error correction techniques to compute entanglement entropy in Loop Quantum Gravity, including a derivation of black hole entropy.

Findings

Reproduces black hole entropy using a canonical ensemble.

Derives the Ryu-Takayanagi formula within loop quantum gravity.

Provides a general algebraic framework for entanglement entropy across surfaces.

Abstract

We introduce a novel method for computing entanglement entropy across surfaces in Loop Quantum Gravity by employing techniques from quantum error correcting codes. In this construction, the redundancy encoded in the gauge invariant subspace is made manifest by embedding it in a larger Hilbert space. The enlarged Hilbert space of a surface does not factorize, which necessitates an algebraic formulation of the entanglement entropy using von Neumann algebras. Using this approach, we are able to reproduce the expected black hole entropy through the canonical ensemble. This includes a direct realization of the Ryu-Takayanagi formula, providing a first principles derivation of the black hole entropy within a kinematical framework of loop quantum gravity. The algebraic techniques developed in this work can be used to compute the entanglement entropy across arbitrary surfaces.