  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.
