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 in...

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Bibliographic Details
Published in:Communications in mathematical physics 2023-06, Vol.400 (3), p.1665-1695
Main Authors: Kang, Monica Jinwoo, Kolchmeyer, David K.
Format: Article
Language:English
Subjects:
Algebra
Classical and Quantum Gravitation
Complex Systems
Entropy
Error correction
Hilbert space
Holography
Mathematical and Computational Physics
Mathematical Physics
Operators (mathematics)
Physics
Physics and Astronomy
Quantum Physics
Reconstruction
Relativity Theory
Theoretical
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Summary: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.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-022-04627-z