The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics

• URL: http://arxiv.org/abs/2005.05971v1
• Date: Tue, 12 May 2020 18:00:00 GMT
• Title: The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics
• Authors: Elliott Gesteau and Monica Jinwoo Kang
• Abstract summary: We construct an infinite-dimensional analog of the HaPPY code as a growing series of stabilizer codes defined respective to their Hilbert spaces. We construct a Hamiltonian that is compatible with the infinite-dimensional HaPPY code and further study the stabilizer of our code, which has an inherent fractal structure. This result shows the limits of the HaPPY code as a model of the AdS/CFT correspondence, but also hints that the relevance of quantum error correction in quantum gravity may not be limited to the CFT context.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We construct an infinite-dimensional analog of the HaPPY code as a growing series of stabilizer codes defined respective to their Hilbert spaces. The Hilbert spaces are related by isometric maps, which we define explicitly. We construct a Hamiltonian that is compatible with the infinite-dimensional HaPPY code and further study the stabilizer of our code, which has an inherent fractal structure. We use this result to study the dynamics of the code and map a nontrivial bulk Hamiltonian to the boundary. We find that the image of the mapping is scale invariant, but does not create any long-range entanglement in the boundary, therefore failing to reproduce the features of a CFT. This result shows the limits of the HaPPY code as a model of the AdS/CFT correspondence, but also hints that the relevance of quantum error correction in quantum gravity may not be limited to the CFT context.

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