The Ryu-Takayanagi Formula from Quantum Error Correction: An Algebraic Treatment of the Boundary CFT

It was recently shown by Harlow that any quantum error correcting code, satisfying the same complementary recovery properties as AdS/CFT, will obey a version of the Ryu-Takayanagi formula. In his most general result, Harlow allowed the bulk algebras to have nontrivial center, which was necessary for the "area operator" in this Ryu-Takayanagi formula to be nontrivial. However, the boundary Hilbert space was still assumed to factorise into Hilbert spaces associated with complementary boundary regions. We extend this work to include more general boundary theories, such as gauge theories, where the subalgebras associated with boundary regions may also have nontrivial center. We show the equivalence of a set of four conditions for a bulk algebra to be reconstructable from a boundary algebra, and then show that complementary recovery implies that the algebraic boundary entropy obeys a Ryu-Takayanagi formula. In contrast, we show that the distillable boundary entropy does not obey any such formula. If an additional "log dim R" term is added to the algebraic entropy, it will still obey a Ryu-Takayanagi formula, with a different area operator. However, since the "log dim R" term is a sum over local boundary contributions, we argue that it can only be related to the regularisation of the area at the bulk cut-off.