SciPost Submission Page
The Mechanism behind the Information Encoding for Islands
by Hao Geng
Submission summary
| Authors (as registered SciPost users): | Hao Geng |
| Submission information | |
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| Preprint Link: | scipost_202510_00007v1 (pdf) |
| Date submitted: | Oct. 6, 2025, 8:14 p.m. |
| Submitted by: | Geng, Hao |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Entanglement islands are subregions in a gravitational universe whose information is fully encoded in a disconnected non-gravitational system away from it. In the context of the black hole information paradox, entanglement islands state that the information about the black hole interior is encoded in the early-time Hawking radiation. Nevertheless, it was unclear how this seemingly nonlocal information encoding emerges from a manifestly local theory. In this paper, we provide an answer to this question by uncovering the mechanism behind this information encoding scheme. As we will see, the early understanding that graviton is massive in island models plays an essential role in this mechanism. As an example, we will discuss how this mechanism works in detail in the Karch-Randall braneworld. This study also suggests the potential importance of this mechanism to the ER=EPR conjecture.
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
Author comments upon resubmission
We note that the question regarding the state-dependence from Referee 1 seems to be answered by Referee 2. Referee 1 was wondering how to see that the Goldstone boson dressed operator is state-dependent. Referee 2 pointed out that the dressed operator is for some reason state-dependent.
The question about the locality of the dressed operator by Referee 2 is articulated in the footnote 9 on Page 14. As suggested by the Referee 2, in the semiclassical limit with $G_{N}ll1$ this dressed operator is sensibly localized at point x.
List of changes
I added the footnote 9 on Page 14 to address the question asked by Referee 2.
Current status:
Reports on this Submission
Strengths
- Interesting idea
Weaknesses
- Shoddy execution, ignoring many important counterarguments.
Report
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First of all, let me apologise for the delay. I spent some time doing background reading.
I think this paper is at best mis-named and at worst completely wrong.
The author claims to have uncovered "the" mechanism for the non-local encoding of islands into a bath. But the actual result of the work is merely to exhibit an operator that is roughly localised to an island region which has a non-zero commutator with the bath Hamiltonian. There is no evidence provided that this is the operator that is encoded in the bath. Without this evidence, how can the paper be called "The mechanism?"
Let me back up a bit to explain this point. Given an operator $O(x)$ in a non-gravitational QFT there are many dressed operators $O_{dr} (x)$ in semiclassical gravity such that $\lim_{G_N \to 0} O_{dr} (x) = O(x)$. The author has taken $O(x)$ to be an operator in the island and found one possible dressed operator $O_{dr,G} (x) = O(x + V)$.
Let us call the operator encoded in the bath $O_{dr,I} (x)$. The question of whether $O_{dr,G} \overset{?}{=} O_{dr,I}$ is, I believe, not even considered by the author. (There might not be a unique $O_{dr,I}$; the refined question is whether there is any $O_{dr,I}$ that agrees with $O_{dr,G}$.) This is why they cannot claim that they have uncovered the mechanism for non-local encoding. If the answer to the questio is plausibly yes, it is only the name of the paper that is wrong.
However, I don't believe that $O_{dr,G}$ is a plausible candidate for $O_{dr,I}$. The reason is that we actually know somehting about $O_{dr,I}$: it is whatever is constructed by the Petz map. But, since the Petz map involves replica wormholes, I expect that $[O_{dr,I}, H_{bath}] = O (e^{-1/G_N})$ whereas $[O_{dr,G}, H_{bath}] = O (G_N^\alpha)$ (See e.g. 3.18,4.74 which assert that $\alpha = 0$).
The absolute minimum this work needs, then, is (a) a change of title and (b) a qualitative argument that $O_{dr,G}$ is plausibly a candidate for $O_{dr,I}$. The appendix added does not come even close to making this argument.
As a stretch goal, the author may try to genuinely show that these operators are the ones we recover from entanglement wedge reconstruction, thereby proving my concerns wrong; this would make this paper amazing. Apart from the Petz map, https://arxiv.org/abs/1912.02210 had a relatively concrete way of reconstructing island operators.
Another clarification is the role of state-dependence. $O_{dr,G} = O(x + V)$, where $V$ is an operator. If one takes $x$ inside the island, $O_{dr,G} (x) |\psi\rangle = \sum_i \langle V_i | \psi \rangle O(x+V_i) | V_i \rangle$, where $V_i$ are eigenvalues. It could be that some of the values $x+V_i$ do not lie in the island! This is a standard subtlety one has to worry about when working with dressed operators.
The other referee correctly pointed out that $V_i = O (\sqrt{G_N})$ and so one does not need to worry about it for that reason. The author doesn't seem to have made even a rudimentary effort to understand either my comment or the referee's reply (as evidenced by the author comments in the resubmission). The author's words were
Referee 1 was wondering how to see that the Goldstone boson dressed operator is state-dependent. Referee 2 pointed out that the dressed operator is for some reason state-dependent. [Emphasis mine]
I would suggest that the author spend a little more time thinking about referee comments. While I admit that my description of the question was a bit quick, referee 2 was quite clear about the question as well as the answer.
I still don't understand eq 3.18. If I understand correctly $z \neq \epsilon$, $V^\mu (x,z) \neq \epsilon^{d+2} U^\mu (x)$. But 3.18 asserts that $[V^\mu (x,z), H_{bath}] = [\epsilon^{d+2} U^\mu (x), H_{bath}]$. 3.18 thus implies $[V^\mu (x,z) - \epsilon^{d+2} U^\mu (x), H_{bath}] = 0$. This can either be true if $z \neq \epsilon$, $V^\mu (x,z) \neq \epsilon^{d+2} U^\mu (x)$ (but this is not the case) or there is some non-trivial reason that the difference commutes with $H_{bath}$. Shouldn't this non-trivial reason be explained, if it exists?
If this non-trivial reason doesn't exist, then I have the following objection. If $(x,z)$ is in the exterior of the black hole, then it is clear (via HKLL) that $[V^\mu (x,z), H_{bath}] = O(G_N^0)$ as in 3.18. However, if $(x,z)$ is inside the black hole, then it seems to this reviewer that the $G_N$ scaling of $[V^\mu (x,z), H_{bath}]$ needs to be found via a non-trivial calculation.
Requested changes
The absolute minimum this work needs is (a) a change of title and (b) a qualitative argument that $O_{dr,G}$ is plausibly a candidate for $O_{dr,I}$. The appendix added does not come even close to making this argument.
As a stretch goal, the author may try to genuinely show that these operators are the ones we recover from entanglement wedge reconstruction, thereby proving my concerns wrong; this would make this paper amazing. Apart from the Petz map, https://arxiv.org/abs/1912.02210 had a relatively concrete way of reconstructing island operators.
Recommendation
Ask for major revision
Author: Hao Geng on 2025-11-11 [id 6012]
(in reply to Report 2 on 2025-11-09)I thank the referee for the report and comments. I have a few questions regarding the comments and requests from the referee, which I think should be resolved before I go ahead and revise my draft according to the referee's comments.
Firstly, two minor comments:
1) I agree with the referee Equ. (3.18) can be easily misunderstood by the readers . The reason for $[V^{\mu}(x,z)-\epsilon^{d+2}U^{\mu}(x), H_{\text{bath}}]=0$ has been explained in Section 3.1. I will write an Appendix to provide another derivation which doesn't rely on the relevant discussions in Section 3.1.
2) I think the state-dependence the referee expanded out is not the state-dependence people usual talk in the context of black hole interior. What the referee means is called field-dependence (see for example the first paragraph on page 3 of https://arxiv.org/pdf/1506.01337). This field-dependence is not a trouble of quantum mechanics and not a sharp feature of black holes. Discussions about state-dependence in black hole interior are huge in the literature so let me not expand it here for this minor point.
I think the major question I had is about the Petz map proposal for entanglement wedge reconstruction. Based on my understanding, the Petz map entanglement wedge reconstruction requires HKLL map between the bulk field and the CFT operators. For example, Equ. (36) and Equ.(37) in https://arxiv.org/pdf/1902.02844 spelled this out explicitly. I was wondering if I understand this point correctly. If this understanding is wrong then I have to think further. Otherwise please let me go as following.
First, in the standard AdS/CFT the HKLL reconstruction is not just solving the eom of a free massive field in AdS. The full HKLL map takes into account the gravitational dressing. See https://arxiv.org/abs/1212.3788 from the K and one of the L in HKLL. Thus, the dressing is really a property of the HKLL map and it enters in to the Petz map reconstruction formula for a bulk operator.
Second, in the case of island we have AdS coupled with a non-gravitational bath. In this case, the HKLL map from island to the bath is not so obvious and as a fair comment nobody worked this out before. I think the referee's comment that "the replica wormhole tells us that the commutator between the reconstructed operator and the bath Hamiltonian is non-perturbatively small" ignored the fact that the HKLL map itself could have some dressing. The dressings are perturbative effects.
Third, I don't agree with the referee that there are many possible dressings for operators inside the island in the case I considered in this draft. In this case, I worked out everything explicitly and we found that there are only two types of dressings a) dress to the boundary of the AdS using Donnelly-Giddings gravitational Wilson line; b) dress to the bath using the Goldstone vector boson I uncovered. By the holographic interpretation of island we only have the choice b) for operators inside the island. There are no other possibilities. This is a clean and solvable example with all details carefully worked out. This also explains why the HKLL map in this case necessarily has dressing, why it is consistent with Petz map reconstruction proposal and why the commutator between operators inside the island and the bath is not non-perturbatively small.
Anonymous on 2025-11-12 [id 6013]
(in reply to Hao Geng on 2025-11-11 [id 6012])I reread section 3.1 and I continue to not understand 3.18. Perhaps the author could point me to the specific equation/paragraph where they explain why this commutator is 0?
I agree that this is not Papadodimas-Raju state-dependence. I also agree with the cited reference that the operator in the paper is linear. I was saying that "the question of whether the operator is in the island is state-dependent." I fully admit I didn't explain it well. Referee 2 explained it well though, I felt. Regardless of these historical details, referee 2's explanation of why the operator is reliably localised in the island belongs in the paper.
In eq 37 of the linked paper, I believe that they are considering a case without a horizon as in figure 1 and they say you can use the global HKLL reconstruction (which is usually opposed to subregion HKLL reconstruction); the author can compare with similar statements in arXiv:1704.05839 (see e.g. between eqs 11 and 12). I admit the phrasing of this paper is ambiguous from the current viewpoint, since it appeared before islands. Petz map in island context is well-explained in the Penington-Shenker-Stanford-Yang paper (main text as well as appendix).
In general, HKLL only works for the exterior of a black hole, since it is an application of the timelike tube theorem. So HKLL is not even an option for an interior operator.
I would appreciate it if the author could point me to their proof of there being only two possible dressings. My understanding of the literature is that we don't have a complete classification of all dressings in even the simplest cases. Apart from gravitational Wilson lines and the author's dressing, I would have thought for example that dressing to the extremal surface was an option a la arXiv:2311.09403. In general, my understanding is that the number of possible dressings is constrained only by cleverness. So, if the author has a classification result of the sort mentioned in their reply, I would find that the most interesting result in the work.
Anonymous on 2025-11-12 [id 6014]
(in reply to Anonymous Comment on 2025-11-12 [id 6013])I want to thank the referee again for the comments.
Here are my understandings and comments on the questions from the referee.
I will add an Appendix about another derivation in the revised draft. A quick reply to referee regarding the relationship between Section 3.1 and Equ. (3.18) is as follows. We should first notice that Equ.(3.18) is in fact a summary of our results from the discussions in Sec.3.1. From Sec.3.1, we noted that the transformation of $U^{\mu}(x)$ deduced from the bulk large gauge transforms Equ.(3.16) is proportional to the transformations of $U^{\mu}(x)$ generated by the bath stress-energy tensor as in Equ.(3.14). Thus, the first equality in Equ.(3.18) is summarizing the above observation and saying that \textit{the bulk large gauge transform can be understood as generated by the bath stress-energy tensor}, and it is not suggesting that $V^{\mu}(x,z)=\epsilon^{d+2}U^{\mu}(x)$ for all $z$.
I thank the referee for the clarification. I think it is fair to say that this point is resolved?
I think I agree that there are detailed discussions about the Petz map in PSSY paper. I think my confusion is that I was wondering if it is correct that to operate the Petz map we need the quantum channel that maps bulk to boundary+bath. If so I think this quantum channel is not chosen arbitrarily and it should obey the bulk gravitational constraint.
a) I think the paper https://arxiv.org/abs/2311.09403 didn't construct any dressing to the extremal surface. The closest statement from this paper to the referee's comment I could see is that in the eternal black hole case one can dress operators on the left exterior to the right asymptotic boundary. This dressing is not worked out in that paper as that paper only discussed the classical situation. At the quantum level, this dressing is done using the gravitational Wilson line that goes through the bifurcation horizon.
b) I think to see that in the case I considered there are no other choices for the dressing, it is easier to consider the empty AdS in the usual AdS/CFT. Let's consider the global AdS in that case. The boundary CFT has a gapped spectrum and the gap is 1/l_AdS which is not nonperturbatively suppressed. Thus, any bulk operator would have a perturbatively nonzero commutator with the boundary Hamiltonian. This commutator is [$O_{\text{bulk}}(t)$, H$_{\partial}$]=$i\partial_t O_{\text{bulk}}$ and this can only be achieved by the gravitational Wilson line. This is because $H_{\partial}=H_{ADM}$ which is some component of the bulk metric fluctuation near the boundary. Thus, using canonical quantization one can see that this nonzero commutator is achieved only if the bulk operator is dressed by a line that connects the bulk point to the boundary and the line operator should be constructed from the bulk metric fluctuation. In the case I considered, i.e. empty+bath, the only difference is that the Hamiltonian constraint is modified by an extra term from the graviton mass. This extra terms provides another possibility as dressing using the Goldstone vector boson.
Anonymous on 2025-11-13 [id 6026]
(in reply to Anonymous Comment on 2025-11-12 [id 6014])I don't understand this still. 3.16 is only near $z \to 0$ right? How does it imply 3.18 which is supposed to be at all $z$?
Here's a different perspective. If $(x,z)$ is in the exterior, we can surely write $V^\mu (x,z) = \int dy K^{\mu}_{\ \nu} (x,z; y) U^\nu (y)$ correct? Is the author claiming that $U^\mu (x)$ and $\int dy K^{\mu}_{\ \nu} (x,z; y) U^\nu (y)$ have the same commutator with $H_{bath}$?
The point is resolved as long as it is included in the revision.
We need a channel that maps from bulk to boundary not boundary+bath. PSSY discuss how one can use the path integral to create this channel.
a) Folkestad doesn't construct a dressing to the extremal surface, it is true; he has a non-constructive argument. As said in the abstract, "We find that extremal surfaces, non-perturbative lumps of matter, and generic trapped surfaces are structures that enable dressing and subregion independence." It is also true that Folkestad works at classical level. How does the author disprove that this fails at perturbative level? The arguments of Raju et al (including the author) only say that the commutators are non-zero at finite $G_N$, as far as I can tell; so a non-perturbative commutator is completely consistent with these arguments.
b) The author has explained that his two dressings are allowed. This does not constitute a proof that no others are allowed. Concretely, they say "using canonical quantization one can see that this nonzero commutator is achieved only if the bulk operator is dressed by a line that connects the bulk point to the boundary." This statement needs a non-trivial proof.