Ownerless island and partial entanglement entropy in island phases

First of all, the authors thank the referee for a careful reading of our manuscript and so many useful comments from various aspects. The report remind us that the setups where we carry out our analysis, as well as the presentation, need to be revised significantly before it can be published. In the revised version, we mark the main changes and new materials in blue. In the following, we present our response to all the comments and questions from the referee.

1, Referee: Above equation (3) is a brief summary of past work of some of the authors. Specifically, they define the notion of a "self-encoded system" by saying that the state of a subset that they call Is(A) is "encoded" by the state of another subset called A. They also say that this property holds when the system is "confined in a certain way." I think the authors should be more specific and rigorous about what they mean by all of this, for the sake of making their paper more self-contained and because the self-encoding property seems to be the main thing that makes their story work. For example, what do they mean by "confined in a certain way?" They should also try to provide some intuition as to why their equation (3) should hold in such systems.

Response: The referee is right, this part is a bit misleading.
We rewrite the paragraph around Eq. (3) to be more specific and rigorous about the self-encoded system (see the blue sentences), and add the footnote 1 to give a simple example.

2, Also above equation (3), the authors claim that their island formula holds in systems "with or without gravity". I believe that (3) is fine without gravity (assuming self-encoded-ness of course), but I am not clear on why it should hold with gravity. Fully gravitational systems without baths are qualitatively different. For example, it has been argued by Raju (arXiv:2110.05470) that the split property of quantum states no longer holds, and I am unsure of how to understand the self-encoded property in such a scenario since its definition seems to rely on the ability to factorize the full Hilbert space into Hilbert spaces of subregions. Of course, the authors in the present paper are only ever considering a non-gravitating system (the Weyl-transformed CFT) in applying (3), so my most minimal suggestion would be to get rid of the"(with or without gravity)" parenthetical.

Response: We removed the the"(with or without gravity)" parenthetical.

Nevertheless, we insist to assume that the island formula 2 (Eq. (3)) should work on theories with or without gravity provided the self-encoding property. One of the main proposal in our previous work [49] is that, the self-encoding property is the essential reason for the emergence of entanglement islands. Although, it is still not clear how explicitly the constraints are imposed in gravity to reduce the Hilbert space hence generate the self-encoding property. There are strong evidences for this claim. For example, the basic problem that why the degrees of freedom in the interior of the black hole described by the effective field theory vastly exceed the number characterized by the Bekenstein-Hawking entropy. This problem indicates a huge reduction of the Hilbert space of the semi-classical effective theory. The claim of the ``holography of information’’ developed by Raju etc, can be concluded as that “once all observables outside a bounded region have been specified there is no freedom to specify the state of the system inside the region”. This statement indeed can be classified as the self-encoding property.

In the revised version, besides the holographic Weyl transformed CFT, we also give two alternative setups where the Island formula applies. See our answer to question 5.

3, I am a bit confused by the equality in equation (13). Shouldn't I(A, BB1B2) equal sAA1A2(A)? As written, the authors definition would be true if I(A,A1A2)= (A,B) at the balancing point, but such a statement is not obvious to me.

Response: The referee was right, and the typo has been corrected.

4, Several steps of Section 3.1 are concerning to me. First off, the Weyl rescaling (17) used to define the system used in the calculations is not only not smooth, but it is also discontinuous (even divergent) at 0. This makes me skeptical of the authors subsequent attempts to apply the entropy formula (16) to the system. Indeed, the expression (19) is also confusing to me for two reasons. Simply calculating the entropy of (−∞,−a′]∪[a,∞) [which one might do by taking b,b′→∞ in (21)] does not yield (16), and the interval [−a′,a] that does produce (16) also runs over the 0 point where the Weyl rescaling diverges. I would appreciate some additional discussion of why this sort of Weyl rescaling is a valid thing to do, as well as more careful presentation of the entropy calculations.

Response: We thank the referee to point out this subtlety. Since the entropy formula of the Weyl transformed system only depends on the scalar field at the endpoints, we think this is not a fatal problem as long as we do not talk about the intervals ended on the $x=0$ point. One can also redefine the scalar field at the neighborhood of $x=0$ to retain smoothness there. This will not affect our following discussion. We added these information in the paragraph following Eq. (19).

For the other confusion, we agree that we will not reproduce the entropy of (−∞,−a′]∪[a,∞) by taking the limit b and b’ to $\infty$. But this has nothing to do with the x=0 point. Actually this disagreement also happens for the holographic CFT_2 without Weyl transformation. More explicitly you will not reproduce the entanglement entropy of (−∞,−a′]∪[a,∞) by taking the limit b and b’ to $\infty$ , because the RT surface connecting b and b’ is abandoned when we calculate the entanglement entropy of (−∞,−a′]∪[a,∞).

5, On a conceptual note about the entropy calculation in the Weyl-rescaled CFT, I am not convinced that the authors are actually doing a QES calculation. By their own words (footnote 4, page 9), gravity is not turned on anywhere in this CFT. As such, why should it have a holographic description as the theory on a brane coupled to a half-space CFT? I would say that this calls into question whether it even makes sense to compare their answers to those of a bulk AdS3 calculation (which is essentially the second half of their paper).

Response: The referee is right about our original set-up (Set-up 1 in the revised version). We understand that it has not been justified for directly applying the island formula (or QES formula) to non-gravitational systems. We give more details about this setup in page 9 and give the long footnote 5 in page 10 to support applying the island formula in this set-up.

Furthermore, we give two alternative set-ups. One (first paragraph in page 10) is to couple the Weyl transformed part of the CFT in Set-up 1 to gravity, hence the effective theory is a gravitational CFT in AdS2 background coupled to a bath CFT (called the gravitational Set-up 1). The Set-up 2 (see the appendix A) is a deformed version of the AdS/BCFT, with additional conformal matter added to the EoW brane. It was proposed in [91] and is named as the defect extremal surface (DES) model. The key property we need in our setups is to have well-defined two-point functions for non-symmetrical intervals.

6, In Section 4, the authors seek to compare their BPE calculations in their 2d CFT to entanglement wedge cross section (EWCS) calculations in AdS3 with an end-of-the-world brane. However, I do not think that the authors are as clear in laying out the parameters specifying the bulk geometry as they should be. Specifically, end-of-the-world branes in AdS3 are often supposed by a tension term in the action. This tension in turn influences the answer in any bulk calculations involving entanglement surfaces that intersect the brane, which is basically how we get island terms out of classical Ryu--Takayanagi. So how are they parameterizing the 3d bulk metric and the end-of-the-world brane, and what is the brane tension in their formulae? At the very least, the quantity closest to a tension might be the κ parameter that they introduce in the previous sections through the Weyl rescaling. However, going back to my concern in the previous bullet point, this makes it seem like they are forcibly equating the AdS3 setup to a completely non-gravitating CFT with a discontinuous and non-regular Weyl rescaling.

Response: We thank the referee for pointing out this confusion. In the revised version we give a more explicit description on the holographic Weyl transformed CFT in page 9 and 10. Especially we give the figure 2 to depict the entanglement structure of the Weyl transformed CFT. As was pointed out by the referee, when applying island formula we are equating the AdS/BCFT setup to the non-gravitational holographic Weyl CFT set-up in some sense. The cutoff brane determined by the parameter $\kappa$ plays the same role as the EoW brane determined by the tension. We give more support for this claim in the footnote 5.

For readers who are not convinced by our Set-up 1, they can skip this part and start from the gravitational Set-up 1 and Set-up 2 in the appendix. The calculations in all these setups are almost the same.

7, The authors should add a reference to the sentence, Later, it was found that the PEE can be interpreted as an additive two-body correlation..." above equation (7).

Response: Thanks. We have added a reference there.

8, The ordering of their discussion of BPE is odd to me because they introduce the balancing conditions before actually writing the definition of BPE in terms of PEE. I suggest the authors move equation (13) before they enumerate the balancing and minimal requirements.

Response: The order of presentation is changed accordingly.

At first the authors thank the referee for a careful reading of our manuscript and many useful comments from various aspects. Also, we are very grateful for the relative positive report from the referee. We have revised the manuscript significantly according to the referee’s report. The changes and new materials are marked blue in the paper. In the following, we list all the changes according to the comments and replies to all the referee’s questions.

1, The referee wrote: In my opinion, the most unclear point of the paper is why the authors stress that the Weyl-transformed CFT2 is non-gravitational. If the CFT is non-gravitational, then there is no reason to apply the island formula (eq. (19) in the manuscript). Depending on the position of twist operators, the result of the entanglement entropy is fixed by standard results for 2D CFTs. The authors comment on this point in footnote 4, where they say that the area term in the island formula is missing because the theory is non-gravitational. However, in that case also the extremization should not be taken into account. Vice-versa, considering dynamical gravity in the Weyl-transformed CFT, one is forced to extremize, but the area term can be absent due to the fact that the boundary of the island is a measure-zero point. For this reason I believe that the results of the paper are valid only in the presence of dynamical gravity.

Response: Thank the referee for pointing out the set-up issue. We now understand that, despite the results on entanglement islands for non-gravitational systems in our previous work [49], it is still not justified to directly apply the island formula I in the non-gravitational Weyl transformed system. In the revised version, we give two additional set-ups. One is to let the Weyl transformed part of the CFT couple to gravity (called the gravitational Set-up 1), the other is to generalize the AdS/BCFT setup by adding conformal matter to the EoW brane, hence the effective description is a gravitational CFT coupled to a bath CFT with transparent boundary condition. The key property we need in our setups is to have well-defined two-point functions for non-symmetrical intervals.

See the blue paragraphs in page 3, 4, 9, 10 and the appendix, where we give more acceptable set-ups to analysis the PEE and BPE structure in island phases.

2, The referee wrote: The manuscript also lacks a clear a description of why studying PEE and BPE could be useful in the context of holography, and especially to study the unitary evolution of gravitational theories. The authors stress that PEE and BPE are more detailed descriptions of the structure of the entanglement than the entanglement entropy, but it's not specified how to apply these concepts, for example, in the context of the information paradox. I wonder, for personal curiosity, if the BPE could be used to study the evolution of the Hawking radiation of an evaporating black hole to a finer detail than "just" its Page curve.

Response: In AdS3/CFT2, the PEE corresponds to the bulk geodesic chords, which is a finer correspondence between quantum entanglement and geometry than the RT formula. One particular application of this correspondence is proposal of BPE as a quantum information dual of the EWCS. A more ambitious goal is to build up the spacetime based on the PEE structure of the boundary theory. Also in condensed matter theory, the time evolution of the PEE (or entanglement contour) has been studied in several scenarios to capture the entanglement spreading behavior. We added a paragraph below Eq. 6 to give more information about the impact on the PEE or entanglement contour in both high energy and condensed matter theories.

As pointed out by the referee, using the PEE structure to study unitary evolution of gravitational theories, especially the black hole evaporation, is quite interesting. This has been explored in [95,96], where they calculated the entanglement contour function for the radiation region following the ALC proposal and find that there are vanishing PEEs for certain regions. According to [96] this is a reflection of the protection of bulk island regions against erasures of the boundary state. We are very interested to take a deeper look at this problem in the future.

3, The referee wrote: Finally, another possibly unclear point is the connection between the BPE and the minimization of the EWCS. Indeed, while the latter is defined through an optimization problem, and thus it's natural to take the minimal value, the former is defined through some balance equations, and there is no clear reason to prefer one solution or another in the case of multiple choices. In particular, different solutions for the BPE arise assigning the ownerless island to different "island regions". However, from the point of view of the BPE, there is no reason to prefer one or the other. This point could be clarified connecting the BPE to an optimization problem as well, as the authors attempt for specific cases in Section 5.5. This I believe is an important point to establish the equality between BPE and EWCS, but likely goes beyond the scope of this work, since in the discussion is proposed as a future direction.

Response: Thanks. We believe that the minimization of the crossing PEE would eventually lead to a quantum-information dual for the EWCS through an optimization problem. So far, we only have some clues and a thorough demonstration will be our future task. We think that balance requirements could capture the property of the local saddle point of the target optimization problem. Solving the balance requirements in some sense gives us the local saddle points, and it is reasonable to choose the minimal saddle among all the saddles if we are actually solving a optimization problem. The different assignments for the ownerless island correspond to different partitions for the degrees of freedom determined by AB (including the island Is(AB)).

We give more discussion on this point in the last three paragraphs in the discussion section.

4, I would like the authors to consider changing the statements about non-dynamical gravity on the Weyl-transformed CFT.

Response: See our reply to the first question

5, the referee wrote: There are some typos in the paper, here are the ones I found: - Page 2, "Hawking radiation for An evaporating black hole after the Page time." - Page 4, "We find that the generalization involves" - Figure 2 is missing the label D1 - Figure 5 is missing the labels A1 and D1 - Eq. (74) and (75) maybe log(…) are missing?

Response: Thanks for pointing out the typos and missing labels, we have fixed them in the revised version.