Entanglement Entropy in Loop Quantum Gravity through Quantum Error Correction

TL;DR

The paper develops a quantum-error-correction-inspired, algebraic framework for calculating entanglement entropy across surfaces in Loop Quantum Gravity by embedding the gauge-invariant code subspace into a larger ambient Hilbert space and using von Neumann algebras to define reduced states. This approach yields a surface entropy with a bulk (center-fixed) contribution plus an edge-mode (RT-like) term, and, in a canonical ensemble for black holes, reproduces the Bekenstein-Hawking area law with a renormalized Newton constant; it also provides a first-principles derivation of a Ryu-Takayanagi–type boundary term within a kinematical LQG setting. The method unifies edge-mode counting, center fixing, and cylindrical consistency to give a general procedure for entanglement across arbitrary surfaces, linking quantum information concepts to geometric entropy in quantum gravity. While dynamical aspects are left for future work, the framework offers a principled route to connect LQG with holographic ideas and to explore entropy in complex quantum geometries.

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.