SciPost Submission Page
Partial entanglement network and bulk geometry reconstruction in AdS/CFT
by Jiong Lin, Yizhou Lu and Qiang Wen
Submission summary
| Authors (as registered SciPost users): | Qiang Wen |
| Submission information | |
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| Preprint Link: | scipost_202405_00012v1 (pdf) |
| Date submitted: | 2024-05-08 10:38 |
| Submitted by: | Wen, Qiang |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In the context of Anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence, we present a general scheme to reconstruct bulk geometric quantities in static AdS background with the partial entanglement entropy (PEE), which is a measure of the entanglement structure on the boundary CFT. The PEE between any two points $\mathcal{I}(\vec x, \vec y)$ is the fundamental building block of the PEE structure. Following \cite{Lin:2023rbd}, we geometrize any two-point PEE $\mathcal{I}(\vec x, \vec y)$ into the bulk geodesic connecting the two boundary points $\vec x$ and $\vec y$, which we refer to as the PEE thread. Thus, in the AdS bulk we get a continues ``network'' of the PEE threads, with the density of the threads determined by the boundary PEE structure. In this paper, we demonstrate that the strength of the PEE flux at any bulk point along any direction is $1/4G$. This observation give us a reformulation for the RT formula. More explicitly, for any static boundary region $A$ the homologous surface $\Sigma_{A}$ that has the minimal flux of the PEE threads passing through it is exactly the Ryu-Takayanagi (RT) surface of $A$, and the minimal flux coincides with the holographic entanglement entropy of $A$. Furthermore, we demonstrate that any geometric quantities can be reconstructed by the PEE threads passing through it, which can further be interpreted as an integration of the boundary two-point PEEs
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Weaknesses
1. Contains essentially the same results as ref. [1]. Nothing substantial is added to merit a separate publication.
2. The proposed thread prescription depends on the bulk geometry and state. It only seems to apply to spheres in vacuum AdS.
3. The 'reformulation' of the RT prescription is wrong in general.
4. The boundedness condition is not checked or used, which is essential to the max flow program.
Report
The manuscript 'Partial Entanglement Network and Bulk Geometry Reconstruction in AdS/CFT' seeks to offer a novel characterization of bulk geometry through the concept of PPE bit threads. However, the paper predominantly comprises an extensive review of [1], with the original contributions confined to section 4, which spans only one page. Consequently, the manuscript does not present enough new material to justify publication. Additionally, the paper is deficient in detail and contains several erroneous claims, some of which are inherited from [1]. I will elaborate on these issues below.
Firstly, the two-point PPE I(x,y) highly depends on the state. The one used in this paper works for the AdS vacuum, but does not easily generalize to other geometries or excited states. Since, I(x,y) is linked to a geodesic, it effectively encodes a two-point function of the boundary theory in the large conformal dimension limit. However, in general geometries and states, boundary-anchored geodesics span a region in the bulk that is parametrically smaller than the region spanned by minimal surfaces. Therefore, I do not believe the construction presented in this paper is universally applicable. A simple case illustrating this limitation is the presence of a black hole. In such a scenario, some of the threads must necessarily reach the horizon, and thus cannot be encoded in the boundary two-point function.
Secondly, even in the AdS vacuum, the analysis for multiple regions appears to have several flaws. First, the 'reformulation' is not entirely independent of the RT prescription. The RT surfaces are required to determine the weights w_i, which must be manually inserted into the threads program. In a true reformulation, these weights should be dynamic variables. Second, the assigned weights in different phases (connected/disconnected) imply that entanglement sometimes accounts for threads connecting two points within region A (or its complement), which is contrary to what the max flow program actually does. According to the original bit threads paper (ref. [23] in the article), S_A should only count the number of threads connecting region A and its complement, representing the EPR pairs that can be distilled from the state. Threads connecting two points within A (or its complement) should not affect the entropy.
What does this imply? While Equation (17) in the paper is correct in terms of entropies (or areas), the proposed thread version of this equation will NOT solve the max flow program. It will result in a divergenceless vector field that yields the correct entropy, but its norm will not necessarily be bounded. Furthermore, the authors do not attempt to prove the boundedness of their proposed thread constructions in the general case. They also fail to provide an explicit expression for V^mu in the case of multiple regions (not even provided in ref. [1]), which could be used to explicitly check whether the norm bound is violated or satisfied in that particular case.
Based on the issues outlined above, I believe the fundamental premise of the paper is flawed and does not meet the standards for publication in SciPost.
Recommendation
Reject