  We reformulate entanglement wedge reconstruction in the language of
operator-algebra quantum error correction with infinite-dimensional physical
and code Hilbert spaces. Von Neumann algebras are used to characterize
observables in a boundary subregion and its entanglement wedge. Assuming that
the infinite-dimensional von Neumann algebras associated with an entanglement
wedge and its complement may both be reconstructed in their corresponding
boundary subregions, we prove that the relative entropies measured with respect
to the bulk and boundary observables are equal. We also prove the converse:
when the relative entropies measured in an entanglement wedge and its
complement equal the relative entropies measured in their respective boundary
subregions, entanglement wedge reconstruction is possible. Along the way, we
show that the bulk and boundary modular operators act on the code subspace in
the same way. For holographic theories with a well-defined entanglement wedge,
this result provides a well-defined notion of holographic relative entropy.
