The Ryu–Takayanagi Formula from Quantum Error Correction
By D. Harlow. Communications in Mathematical Physics, 2016
426 citations recorded by the source corpus.
I argue that a version of the quantum-corrected Ryu–Takayanagi formula holds in any quantum error-correcting code. I present this result as a series of theorems of increasing generality, with the final statement expressed in the language of operator-algebra quantum error correction. In AdS/CFT this gives a “purely boundary” interpretation of the formula. I also extend a recent theorem, which established entanglement-wedge reconstruction in AdS/CFT, when interpreted as a subsystem code, to the more general, and I argue more physical, case of subalgebra codes. For completeness, I include a self-contained presentation of the theory of von Neumann algebras on finite-dimensional Hilbert spaces, as well as the algebraic definition of entropy. The results confirm a close relationship between bulk gauge transformations, edge-modes/soft-hair on black holes, and the Ryu–Takayanagi formula. They also suggest a new perspective on the homology constraint, which basically is to get rid of it in a way that preserves the validity of the formula, but which removes any tension with the linearity of quantum mechanics. Moreover, they suggest a boundary interpretation of the “bit threads” recently introduced by Freedman and Headrick.
Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/5f57f7451bc03e07c38b0b32ccaa11a3bee06999
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Citations in the corpus
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- A general proof of the quantum null energy condition (2017, background): “ required to show that this bulk/boundary equivalence applies to modular ow; see footnote 5 for a justication [38]. For example this does not work for the quantum error correction codes discussed in [98] where the relation between bulk and boundary modular Hamiltonians requires a double sided projec”
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- Islands in cosmology (2020, methodology): “A more detailed understanding of this encoding can be found in the language of quantum error correction [14,36,37,96,97].”
- The Black Hole Information Problem (2016, background): “[60, 61, 62, 63, 64] suggest that quantum error correction allows one to reach a larger area, the entanglement wedge.”
- The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole (2019, background): “n into the bath, however, the bulk state is mixed: the right and left QESs are no longer required to coincide. Put dierently, complementary recovery is not guaranteed when the bulk state is not pure [46]. This discrepancy between left and right QESs, which in turn corresponds to the gap between the ”