Relational bulk reconstruction from modular flow
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13130_2024_Article_23975.pdf
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Author(s)
Parrikar, Onkar
Rajgadia, Harshit
Singh, Vivek
Sorce, Jonathan
Date Issued
July 16, 2024
Journal
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Citation
Parrikar, O., Rajgadia, H., Singh, V. et al. Relational bulk reconstruction from modular flow. J. High Energ. Phys. 2024, 138 (2024).
Version
Final published version
Abstract
The entanglement wedge reconstruction paradigm in AdS/CFT states that for a bulk qudit within the entanglement wedge of a boundary subregion 𝐴⎯⎯⎯⎯
, operators acting on the bulk qudit can be reconstructed as CFT operators on 𝐴⎯⎯⎯⎯
. This naturally fits within the framework of quantum error correction, with the CFT states containing the bulk qudit forming a code protected against the erasure of the boundary subregion A. In this paper, we set up and study a framework for relational bulk reconstruction in holography: given two code subspaces both protected against erasure of the boundary region A, the goal is to relate the operator reconstructions between the two spaces. To accomplish this, we assume that the two code subspaces are smoothly connected by a one-parameter family of codes all protected against the erasure of A, and that the maximally-entangled states on these codes are all full-rank. We argue that such code subspaces can naturally be constructed in holography in a “measurement-based” setting. In this setting, we derive a flow equation for the operator reconstruction of a fixed code subspace operator using modular theory which can, in principle, be integrated to relate the reconstructed operators all along the flow. We observe a striking resemblance between our formulas for relational bulk reconstruction and the infinite-time limit of Connes cocycle flow, and take some steps towards making this connection more rigorous. We also provide alternative derivations of our reconstruction formulas in terms of a canonical reconstruction map we call the modular reflection operator.
MIT Department
Massachusetts Institute of Technology. Center for Theoretical Physics
Massachusetts Institute of Technology. Department of Physics
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DOI of Published Version
https://doi.org/10.1007/jhep07(2024)138
