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The Mechanism behind the Information Encoding for Islands

by Hao Geng

Submission summary

Authors (as registered SciPost users): Hao Geng
Submission information
Preprint Link: https://arxiv.org/abs/2502.08703v2  (pdf)
Date submitted: March 12, 2025, 4 a.m.
Submitted by: Geng, Hao
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
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
Current status:
In refereeing

Reports on this Submission

Report #1 by Anonymous (Referee 1) on 2025-7-17 (Invited Report)

Strengths

  • Interesting calculation and idea
  • Explicit

Weaknesses

  • Probably probing other physics than what is claimed

Report

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I think the calculations and main idea in this paper are very interesting.

The author poses the following tension: if island operators are in the bath, then they should fail to commute with the bath hamiltonian.

The answer they come up with is that, in massive gravity models, the dressing of the island operator becomes a physical operator and it extends out to the bath, thus failing to commute with the bath hamiltonian.

While an interesting idea, there is a crucial sanity check that this proposal seems to fail. The commutator of $\hat{V}_\mu$ with the bath Hamiltonian is polynomially small in $G_N$ (as the braneworld model clearly shows), whereas I would expect the commutator to be exponentially small. The reason for this expectation is that the GPI calculation of the commutator includes Euclidean wormholes. This indicates that the author is looking at the wrong set of operators: there are many ways of promoting a bulk QFT operator to a dressed operator, and it is not clear that this is the dressing reconstructed by e.g. the Petz map.

I could be wrong about this point, but I believe that the author has to make a case that these operators are in fact the ones reconstructed by the Petz map for the central claim of this paper to have validity.


Another point I felt should be treated more carefully, is that, since $\hat{V}$ is an operator, the question of whether $\phi (x + \hat{V})$ is in the island seems state-dependent.

This is of course related to the point above that it is unclear whether these are the right interior operators.


As to the actual calculation, I find myself completely stumped by equation 3.18, whose first equality states

$$[V_\mu (x,z), H_b] = \epsilon^{d+2} [U^\mu (x), H_b]$$

The only way I can make sense of this equation is to assume that $V_\mu$ inside the island can be HKLL reconstructed. (And even then, there should be a kernel on the RHS.)

But this is tantamount to assuming the result. Perhaps I have missed a step; if so, the author should explain carefully what is going on here.

Recommendation

Ask for major revision

  • validity: low
  • significance: ok
  • originality: good
  • clarity: high
  • formatting: good
  • grammar: good

Author:  Hao Geng  on 2025-07-17  [id 5647]

(in reply to Report 1 on 2025-07-17)

We thank the referee for the interesting comments. Here are a few points which we think we should clarify before the editor-in-charge has made the recommendation.

1) We believe that the question we are trying to address has been correctly summarized by the referee as "if island operators are in the bath, then they should fail to commute with the bath hamiltonian."

Though we believe that the answer we found is in the general context of entanglement islands, i.e. a gravitational AdS coupled to a nongravitational bath. We'd rather call this model island model as we have been doing, instead of massive gravity model as the referee alluded to. It is true that graviton is massive in this setup but the mass is rather from the Higgs mechanism as discussed in detail in this draft. Thus, as opposed to massive gravities which are not guaranteed to be UV complete, the setup we are considering has a UV complete description through holography.

2) We don't agree with the referee that the wormhole in the computation of the commutator indicates any contradiction with the results we had. As the referee already mentioned, that is a nonperturbative effect which can only be seen in certain toy models of gravity and its status is unclear in general. Thus, this wormhole effect is interesting and it deserves a better understanding to be on a firm ground. However, the basic reason that such wormhole effect is not contradicting our results is that one has to start with the correct observables and then do the computations. In gravitational theories, the correct observables should be diffeomorphism invariant, this is a basic consistency condition and this consistency condition together with the holographic interpretation of entanglement island implies the "polynomially small commutator in G_N". This is a perturbative effect which is already sitting in front of the much more suppressed nonperturbative wormhole effect. Hence, one shouldn't use the existence of a much more suppressed effect to argue the non-existence of an even bigger effect.

3) We agree with the referee that the details of how the "Petz map" proposal for the bulk reconstruction works in the context of entanglement island deserves a better understanding. However, in this draft we are not yet touching this question as we have discussed in the draft we are intended to understand a more universal question as the referee pointed out in 1) and we leave the details of how the operator phi(x+V) is reconstructed in the bath for future work.

4) There are various different notions of state-dependence. Since the referee mention "interior", we will take it as the Papadodimas-Raju type state-dependence.

This type of state-dependence is a large breakdown of linearity and in the black hole context it is due to the horizon, which is a sharp geometric structure, for a single-sided black hole. Therefore, the point is similar to 3) that to see whether we have such kind of state-dependence we need the explicit reconstruction map for the operators in island. Thus, we don't agree with the referee that by just looking at phi(x+V) one can claim that this operator is not state-dependent. The Goldstone boson inside phi only guarantees that this is the correct bulk observable and it doesn't give a full reconstruction map for this operator from the bath. Although, given the fact that this dressed operator doesn't commute with the bath Hamiltonian, this operator might be practically reconstructable using the protocol in https://scipost.org/10.21468/SciPostPhys.10.5.106.

We think the existence of such a protocol and the explicit example of the Karch-Randall braneworld and JT gravity for the existence of entanglement island in empty AdS in fact brings up an interesting question on whether one should think of the reconstruction in the context of entanglement islands as strongly state-dependent. We believe this deserves a better understanding.

5) About Equation (3.18), it is intended as a summary of the results (3.15), (3.16) and (3.17) in a reverse order. The referee brings up an interesting point which we are investigating in an on-going project that how one can derive (3.18) directly from HKLL type formula. This question is not super trivial as the Goldstone vector field and the bulk graviton are mixed in the equation of motion. A simple way to simplify the calculation is to fix the unitary gauge setting the Goldstone vector field to zero and studying the HKLL for a massive graviton. Though, this strategy wouldn't serve the purpose to derive (3.18) as for that we need the HKLL expression for the Goldstone vector field. We hope to report on this study in the near future.

In summary, we believe there are many interesting questions to be understood to put the insights from the referee on a firm ground. We hope that this draft opens the door to this fruitful research direction.

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