The Ryu Takayanagi Formula from Quantum Error Correction
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.