Abstract
The entanglement wedge reconstruction paradigm in AdS/CFT states that for a bulk qudit within the entanglement wedge of a boundary subregion \( \overline{A} \), operators acting on the bulk qudit can be reconstructed as CFT operators on \( \overline{A} \). 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.
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Acknowledgments
We thank Abhijit Gadde, Gautam Mandal, Shiraz Minwalla and Sandip Trivedi for helpful discussions and Tom Faulkner for helpful comments on an earlier version of this draft. We are grateful to the long term workshop YITP-T-23-01 held at YITP, Kyoto University, where part of this work was done. JS is supported by the AFOSR under award number FA9550-19-1-0360, by the DOE Early Career Award number DE-SC0021886, and by the Heising-Simons Foundation.
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Parrikar, O., Rajgadia, H., Singh, V. et al. Relational bulk reconstruction from modular flow. J. High Energ. Phys. 2024, 138 (2024). https://doi.org/10.1007/JHEP07(2024)138
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DOI: https://doi.org/10.1007/JHEP07(2024)138