Relational bulk reconstruction from modular flow

• URL: http://arxiv.org/abs/2403.02377v2
• Date: Thu, 15 Aug 2024 13:25:06 GMT
• Title: Relational bulk reconstruction from modular flow
• Authors: Onkar Parrikar, Harshit Rajgadia, Vivek Singh, Jonathan Sorce,
• Abstract summary: We study a framework for relational bulk reconstruction in holography. We derive a flow equation for the operator reconstruction of a fixed code subspace operator. We observe a striking resemblance between our formulas for relational bulk reconstruction and the infinite-time limit of Connes cocycle flow.
• Score: 0.206242362470764
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The entanglement wedge reconstruction paradigm in AdS/CFT states that for a bulk qudit within the entanglement wedge of a boundary subregion $\bar{A}$, operators acting on the bulk qudit can be reconstructed as CFT operators on $\bar{A}$. 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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