Holographic Relative Entropy in Infinite-Dimensional Hilbert Spaces

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.

Automated summary

We reframe entanglement wedge reconstruction using the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are employed to characterize observables in a boundary subregion and its entanglement wedge. By assuming the reconstruction of infinite-dimensional von Neumann algebras for both the entanglement wedge and its complement in their respective boundary subregions, we demonstrate the equality of relative entropies measured with respect to the bulk and boundary observables. Moreover, we establish the converse by showing that if the relative entropies measured in an entanglement wedge and its complement are equal to those measured in their respective boundary subregions, entanglement wedge reconstruction is possible. Along the way, we find that the bulk and boundary modular operators act similarly on the code subspace, providing a well-defined notion of holographic relative entropy for holographic theories with well-defined entanglement wedges.