The Ryu-Takayanagi Formula from Quantum Error Correction

TL;DR

This paper demonstrates that the quantum-corrected Ryu-Takayanagi formula applies broadly within quantum error correction codes, providing new insights into holography, entanglement, and boundary interpretations in AdS/CFT.

Contribution

It generalizes the quantum-corrected Ryu-Takayanagi formula to operator-algebra quantum error correction and extends entanglement-wedge reconstruction to subalgebra codes.

Findings

Validates the formula in general quantum error-correcting codes

Connects bulk gauge transformations with boundary entropy

Proposes a boundary interpretation of bit threads

Abstract

I argue that a version of the quantum-corrected Ryu-Takayanagi formula holds in any quantum error-correcting code. I present this result as a series of theorems of increasing generality, with the final statement expressed in the language of operator-algebra quantum error correction. In AdS/CFT this gives a "purely boundary" interpretation of the formula. I also extend a recent theorem, which established entanglement-wedge reconstruction in AdS/CFT, when interpreted as a subsystem code, to the more general, and I argue more physical, case of subalgebra codes. For completeness, I include a self-contained presentation of the theory of von Neumann algebras on finite-dimensional Hilbert spaces, as well as the algebraic definition of entropy. The results confirm a close relationship between bulk gauge transformations, edge-modes/soft-hair on black holes, and the Ryu-Takayanagi formula. They also suggest a new perspective on the homology constraint, which basically is to get rid of it in a way that preserves the validity of the formula, but which removes any tension with the linearity of quantum mechanics. Moreover they suggest a boundary interpretation of the "bit threads" recently introduced by Freedman and Headrick.