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Ownerless island and partial entanglement entropy in island phases
by Debarshi Basu, Jiong Lin, Yizhou Lu, Qiang Wen
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Submission summary
Authors (as registered SciPost users):  Debarshi Basu · Yizhou Lu · Qiang Wen 
Submission information  

Preprint Link:  scipost_202308_00014v1 (pdf) 
Date submitted:  20230809 09:00 
Submitted by:  Wen, Qiang 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
In the context of partial entanglement entropy (PEE), we study the entanglement structure of the island phases realized in several 2dimensional holographic setups. The selfencoding property of the island phase changes the way we evaluate the PEE. With the contributions from islands taken into account, we give a generalized prescription to construct PEE and balanced partial entanglement entropy (BPE). Here the ownerless island region, which lies inside the island $\text{Is}(AB)$ of $A\cup B$ but outside $\text{Is}(A)\cup \text{Is}(B)$, plays a crucial role. Remarkably, we find that under different assignments for the ownerless island, we get different BPEs, which exactly correspond to different saddles of the entanglement wedge crosssection (EWCS) in the entanglement wedge of $A\cup B$. The assignments can be settled by choosing the one that minimizes the BPE. Furthermore, under this assignment we study the PEE and give a geometric picture for the PEE in holography, which is consistent with the geometric picture in the noisland phases.
Author comments upon resubmission
List of changes
The manuscript has been significally revised. See the blue colored part of the revised version and the newly added appendix.
Then main change is made for the setup where island phase is realized. In the first version we provide a holographic Weyl transformed CFT which is nongravitational to realize island phase (the Setup 1). In the new version we provide two alternative setups, the gravitational Setup 1 and the DES model, where gravity is coupled to half of the effective theory and the application of the Island formula can be justified.
We also give more discussion in the last section on how the BPE is related to an optimization problem.
Serveral statements, for example the selfencoding property, the Setup 1, and the definition of the BPE, are presented in a more clear way.
Typos are fixed and several references are added.
See the reply to the referees for more details.
Current status:
Reports on this Submission
Anonymous Report 2 on 20231013 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202308_00014v1, delivered 20231013, doi: 10.21468/SciPost.Report.7937
Strengths
 Strong technical results
Weaknesses
 Claims on the applicability to nongravitational systems which are highly nontrivial
Report
I thank the Authors for their clarifications.
I think the manuscript has improved from the previous version, especially in the clarity of some explanations. Indeed, I do not have any complaint on the technical results found and on the matching between boundary and bulk quantities.
On the other hand, the authors emphasize that the results found can also be applied in the case of nongravitational systems, in particular the ones that display a "self encoding property" (referring to a previous paper of some of the Authors). In the manuscript, the Authors stress that "self encoding" systems comprise also nongravitating systems, thus conjecturing an extension of the Island formula. However, this is highly nontrivial, and it is possible that self encoding systems are mostly the boundary description of a holographic (thus gravitating in the bulk) systems. This matter becomes particularly relevant when analysing holographic setups which admit an intermediate picture (between boundary and bulk), like the one of the manuscript. I think a deeper discussion of this issue (maybe with some examples of "self encoding systems" that clearly do not have a semiclassical bulk descriptions) would greatly benefit the manuscript.
I would like to stress that the comment above is not to undermine the validity of the paper, which as emphasised before I believe is technically strong, but to spark some discussion (which could very well fit into future work) in the direction of "what entanglement/encoding structures distinguished holographic (thus gravitational) theories from non gravitational ones".
Requested changes
 Possible discussion on self encoding vs holographic (gravitational) systems, if not too long. Otherwise it can be regarded as future work (see Report).
Anonymous Report 1 on 2023914 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202308_00014v1, delivered 20230913, doi: 10.21468/SciPost.Report.7825
Strengths
1. The document has been improved with addition of more exposition and setups.
Weaknesses
1. The motivations for studying the selfencoding property are still somewhat unclear.
Report
I thank the authors for their response and their changes. I think the current document is an improvement.
My main remaining, rather small critique would be that that the overarching motivation for considering this selfencoding property is still somewhat buried in this version. Nonetheless, based on the authors' comments (particularly their response to my point 2 in the first round), they are trying to isolate some property of quantum information that yields the island rule, and selfencoding is their working proposal. The authors mention this briefly in the new introduction, but I think they should also add a clause in the abstract that explicitly states as much.
Furthermore, as stated in the author response, the way in which selfencoding emerges in a theory of gravity is not yet clear, but it seems like this an important question to pursue (at least in the future). I would urge the authors to say that somewhere, possibly in the Discussion.
Requested changes
1. Some additional motivation of the selfencoding property (outlined in the report).
Author: Qiang Wen on 20230922 [id 4003]
(in reply to Report 1 on 20230914)We thank the referee for the suggestion.
The referee: My main remaining, rather small critique would be that that the overarching motivation for considering this selfencoding property is still somewhat buried in this version. Nonetheless, based on the authors' comments (particularly their response to my point 2 in the first round), they are trying to isolate some property of quantum information that yields the island rule, and selfencoding is their working proposal. The authors mention this briefly in the new introduction, but I think they should also add a clause in the abstract that explicitly states as much.
Response: The selfencoding property was proposed in the reference [49], which was written by two of the authors in this paper. It is a result of combing the island formula with our standard understanding of quantum information. We used the selfencoding property to understand the physical meaning of the twopoint function of twist operators in our setups, and find that the twopoint function should not be understand as the entanglement entropy. Then we give our basic proposal 1 to interpret the twopoint function as a PEE, which is one of the cornerstone for all the calculations in this paper. The selfencoding property also helps us clarify the contribution structure to the entanglement entropy in island phases, and lead us to a new way to compute the PEE in island phases. As suggested by the referee, we will modify the abstract and some sentences in the discussion section, to clarify the role of the selfencoding property further.
The referee: Furthermore, as stated in the author response, the way in which selfencoding emerges in a theory of gravity is not yet clear, but it seems like this an important question to pursue (at least in the future). I would urge the authors to say that somewhere, possibly in the Discussion.
Response: We will mention this in the discussion section, and our group is really studying the selfencoding property of gravity by reproducing the island rules in certain setups from manipulating the Hilbert space.
By the way, we just replace the reference [49], which contains more discussion about why the island formula 1 is a special application of the island formula 2 in gravitational theories. Also, more discussion on why gravitational theory in island phases should selfencoded are added. Some of the statements, for example the two island formulas should be identical, are softened.