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arXiv:2311.07649 · Journal of High Energy Physics

Holographic entanglement distillation from the surface state correspondence

Ning Bao, Gün Süer

Published Jan 17, 2024• 1 citation
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Abstract

A bstract We study correlations between geometric subfactors living on the Ryu-Takayanagi surface that bounds the entanglement wedge. Using the surface-state correspondence and the bit threads program, we are able to calculate mutual information and conditional mutual information between subfactors. This enables us to count the shared Bell pairs between subfactors, and we propose an entanglement distillation procedure over these subsystems via a SWAP gate protocol. We comment on extending to multipartite entanglement.

DOI
10.1007/jhep01(2024)091

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A bstract We study correlations between geometric subfactors living on the Ryu-Takayanagi surface that bounds the entanglement wedge. Using the surface-state correspondence and the bit threads program, we are able to calculate mutual information and conditional mutual information between subfactors. This enables us to count the shared Bell pairs between subfactors, and we propose an entanglement distillation procedure over these subsystems via a SWAP gate protocol. We comment on extending to multipartite entanglement.
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Abstract We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured E P = E W relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entanglement wedge. Along the way, we comment on minimal purifications of holographic states, geometric purifications, and black hole geometries.
We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS3_33​/CFT2_{2}2​. With the guidance from the boundary and bulk modular flows we find a natural slicing of the entanglement wedge with the modular planes, which are co-dimension one bulk surfaces tangent to the modular flow everywhere. This gives an one-to-one correspondence between the points on the boundary interval A\mathcal{A}A and the points on the Ryu-Takayanagi (RT) surface EA\mathcal{E}_{\mathcal{A}}EA​. In the same sense an arbitrary subinterval A2\mathcal{A}_2A2​ of A\mathcal{A}A will correspond to a subinterval E2\mathcal{E}_2E2​ of EA\mathcal{E}_{\mathcal{A}}EA​. We propose that the length of E2\mathcal{E}_2E2​ captures the contribution from A2\mathcal{A}_2A2​ to the entanglement entropy SAS_{\mathcal{A}}SA​, and compare it with the mutual information between A2\mathcal{A}_2A2​ and the complement Ac\mathcal{A}^cAc.