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A physical protocol for observers near the boundary to obtain bulk information in quantum gravity
by Chandramouli Chowdhury, Olga Papadoulaki, and Suvrat Raju
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Olga Papadoulaki · Suvrat Raju 
Submission information  

Preprint Link:  scipost_202010_00002v2 (pdf) 
Date accepted:  20210428 
Date submitted:  20210129 06:07 
Submitted by:  Raju, Suvrat 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider a set of observers who live near the boundary of global AdS, and are allowed to act only with simple lowenergy unitaries and make measurements in a small interval of time. The observers are not allowed to leave the nearboundary region. We describe a physical protocol that nevertheless allows these observers to obtain detailed information about the bulk state. This protocol utilizes the leading gravitational backreaction of a bulk excitation on the metric, and also relies on the entanglementstructure of the vacuum. For lowenergy states, we show how the nearboundary observers can use this protocol to completely identify the bulk state. We explain why the protocol fails completely in theories without gravity, including nongravitational gauge theories. This provides perturbative evidence for the claim that one of the signatures of holography  the fact that information about the bulk is also available near the boundary  is already visible in the lowenergy theory of gravity.
List of changes
The list of changes is also provided in the reply to referee 2 and the changes are explained there.
1) The word "semiclassical" has been changed to "lowenergy" at various points in the text so as to be more precise about what we mean.
2) We have elaborated on the relationship of the various cutoffs in the subsection on "Tasks of the Observers"
3) We have inserted text in the subsection on "Abilities and Limitations.." emphasizing that the need for multiple copies is not special to our protocol and follows from the inherently probabilistic nature of quantum mechanics.
4) We have inserted text in the section on "Allowed manipulations" explaining why the second order term is not displayed. (Its contribution vanishes.)
5) In the section on "Allowed measurements", we have inserted a remark and a reference to a gaugeinvariant expression for the energy.
6) We have inserted a footnote in the section on "allowed measurements" explaining why the energyspectrum is expected to be discrete about the vacuum of global AdS.
7) In the section on "implications for black holes" we have noted that, within effective field theory in a blackhole background, the quantization of energy is nolonger visible because the gaps between energylevels are too small to seen perturbatively.
Published as SciPost Phys. 10, 106 (2021)
Reports on this Submission
Report 2 by Samir Mathur on 202148 (Invited Report)
 Cite as: Samir Mathur, Report on arXiv:scipost_202010_00002v2, delivered 20210408, doi: 10.21468/SciPost.Report.2767
Report
I remain puzzled about the results of this paper. However I have had a chance to understand the arguments better, so perhaps I can focus my concerns better as well. Notwithstanding these concerns, I think this paper should be published, since the authors have spent time thinking about these issues, and it would be useful to have the arguments available to the community to spur debate on these interesting issues.
From what I understand, the authors claim that in any theory with gravity, one can find the state inside some region $r < R$ by doing measurements only in a region $r>>R$; thus in particular light has not had time to bring any signals from the inner region to where the measurements are being done. Gravity can remain weak throughout, so no novel features of string theory or nonperturbative quantum gravity are being used.
The essence of the argument seems to be the following. The gravitational field h couples to the matter energy E. By doing careful measurements of the weak gravitational field near infinity, we can project the overall state to an eigenstate of the energy E. But energy eigenstates are typically travelling waves, which will keep oscillating all the way to infinity (rather than being localized wavepackets). Then we can observe this state near infinity, and thus infer what it must have been in the interior.
In particular, let the theory have two identical scale fields $\phi_1, \phi_2$. We make a localized wavepacket of one of these fields in the region $r<R$. We wish to detect from infinity which of the two fields we excited. We measure h at infinity, projecting to an energy eigenstate E for the scalar fields; then looking at the corresponding travelling wave of $\phi_i$ that reaches infinity in such an eigenstate, we will know which scalar field was excited in the interior.
One can worry about such an argument because of the simpler situation with just a scalar field in flat space. The vacuum has correlations at points that are spatial separated; this makes the Feynman propagator nonzero at spacelike separations. But an excitation at a point x cannot be detected at a spacelike  separated point y. To prove this one can do one of two things: (i) Check the vanishing of commutators of operators at these two points; this can be difficult in the gravity theory since one will need a good definition of a local operator first (ii) Put a source at one position and an Unruh de Wit detector at the other, checking that the latter is not excited; this however is a complicated computation due to cancellations between several terms in the amplitude, and the local operators would have to be made very carefully just as in (i).
So I tried to make a counterexample to their claim as follows. Suppose we can make two solutions of the wavefunctional, satisfying the gauge constraints, with the following property: In the region $r<R$, the two are different, while in the region $r>2R$, they are identical. In this case I believe that no experiment at $r>2R$ can detect the difference between the two solutions at $r<R$. I believe the authors would claim that two such solutions cannot be made, but it was not clear to me that such was the case.
In the attached pdf file I sketch how I would make such a counterexample for the simpler case of the electromagnetic theory; I am guessing that the gravitational case is more messy to construct but similar. Thus my question would be
(i) Would such a counterexample, if it exists, disprove the authors' claim? Or have I completely misunderstood the problem?
(ii) If yes to (i), is there an error in the gauge theory example I have tried to make?
(iii) If the gauge theory counterexample is correct, is the gravity case very different? (The gravity case does have an extra spatial Laplacian term, but it is not clear why this should change the existence of such a construction)
To summarize, I remain confused by the claim of this paper. But as mentioned above, I think the paper should be published. In a complicated field like quantum gravity, it cannot be the authors' responsibility to convince another person (the referee) of their ideas; the correctness of the ideas should emerge by further investigations in the general gravity community.
Anonymous Report 1 on 2021321 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202010_00002v2, delivered 20210321, doi: 10.21468/SciPost.Report.2722
Strengths
see my first report
Weaknesses
see my first report
Report
I would like to thank the authors for replying to my comments and questions and for introducing modifications to their manuscript where necessary. I recommend publishing the paper in the present form.
Author: Suvrat Raju on 20210416 [id 1367]
(in reply to Report 2 by Samir Mathur on 20210408)We would like to thank the referee for the latest report and for recommending publication of the paper.
Our response to the second referee report is available in the attached pdf file (p.1 to p. 3). A simple textual summary is also provided below.
Aside from this scientific response, we would like to explain some background that, we believe, is important for readers and editors to understand this exchange.
Background
The question being discussed here is whether gravity localizes quantum information differently from other local quantum field theories.
The referee's perspective on this issue is known to be different from ours. There has been extensive discussion on this issue already in the literature, and some of this difference in perspective can be traced back to alternate resolutions of the black hole information paradox. In arXiv:0909.1038, the referee proposed a "small corrections theorem" that is used to argue that fuzzballs are necessary to resolve the information paradox. In arXiv:1211.6767, it was pointed out that this result tacitly assumes that information is stored locally in gravity, just as it is in local QFTs. Without this assumption, the "small corrections theorem" would fail.
The paper under consideration adds to a growing and already significant body of evidence that gravity does localize quantum information unusually. Some recent independent evidence comes from an analysis of operator algebras in gravity (SciPost Phys. 10, 041 (2021)). But this paper points out that this effect is visible even in the low energy theory of gravity. This paper is not about black holes but the obvious implication of our results is that the qubit models of black hole evaporation used in arXiv:0909.1038 miss an essential aspect of the physics.
Therefore, it is natural that the referee is opposed to the results. To be clear: we have no objection if scientists who are opposed to a program are chosen as referees for a paper that advances the program. This may lead to a productive scientific discourse, as we have had with the referee. But it is very important that, in fairness to the authors, this difference in perspective be accounted for when it comes to making a final judgement on the validity of a paper.
Previous Correspondence
The referee's second report emerges from an extended discussion between the authors and the referee. With the referee's kind permission (p.32 and p.39), we are attaching a transcript of this discussion to this reply. The transcript is available in the attached pdf file, after our response to the referee report. We believe that this correspondence, which covered a number of issues, will be helpful for readers.
The attached transcript runs to over 50 pages, which is considerably longer than our original manuscript.
But we would like to draw the reader's attention to some of the salient features of this transcript. We have repeatedly asked the referee, in our reply to the previous referee report and also in the correspondence (p. 27 and p. 34) to point out any error in any equation or line of text in our paper. No such error has been found or even hinted at.
Second, this correspondence has gone well beyond the traditional confines of peerreview which is usually limited to checking the "correctness" of a paper. Instead, after a few initial questions about the paper itself, the referee has presented us with a sequence of puzzles, drawn from contexts outside of the setup of our paper.
We have been able to resolve all of these puzzles. Thus, it can be seen, in the attached correspondence, that we have resolved puzzles about the difference between gauge and gravity theories, resolved a puzzle about a spin in AdS interacting with a magnetic field and also a puzzle about global symmetries. The most recent discussion has focused on how our picture can be understood from the perspective of wavefunctionals. An explicit error was found in the previous version of the referee's argument. (See p. 37,38) The second referee report attempts to refine this latest argument from the correspondence.
Summary of Response to Report
The referee's "counterexample" is presented in the context of electromagnetism although our paper is about theories of gravity. We are surprised that the referee terms this construction a "counterexample" since the analysis presented is not in contradiction with our results. In fact, we explicitly explain this physical point in our paper. We discuss nongravitational gauge theories in the Introduction and devote an entire subsection, subsection 4.2, to explaining the difference between such theories and gravity. Here we explain that electromagnetism has a a "split property" just like local quantum field theories. Therefore, in such theories, information about a state is not available at infinity. (The only special case, which we describe in section 4.2.1, is where strong priors are given to the observers.) This was also explained in the attached correspondence (p. 8 and p.25 and 26) and the referee notes this explicitly in the referee's own note in the correspondence (p. 41). In fact, the referee's construction in the report aligns with a construction that we ourselves outlined (p. 37).
Perhaps the referee terms this a "counterexample" because of the claim towards the end of the report that this construction can be generalized to gravity. But no justification is given for this claim, and no explicit construction is provided either. Instead the referee only states ''It is not clear to me that a construction similar to the electromagnetic case will not work.''
It is rather simple to see that such a generalization is impossible.
The difference between gravity and electromagnetism is the following. In the electromagnetic case, once we specify that the integral of the electric field at infinity is $Q$, the integral of the bulk current density, $\rho(x)$, is constrained to be $Q$. However, this is not a significant constraint since the current density at different points commutes: $[\rho(x), \rho(x')] = 0$. So $\rho(x)$ can be chosen independently at each point in the bulk and there are an infinite number of configurations of $\rho(x)$ that lead to the same charge as seen at infinity.
In gravity, the integral of the metric over the sphere at infinity again constrains the integral of the bulk energy density $H_{\rm{bulk}}(x)$. But this is now a significant constraint. For instance, in global AdS, if the metric at infinity indicates that the energy is $0$ then there is a unique bulk configuration that is consistent with this constraint, and there is no freedom left in the choice of the bulk wavefunction. Similarly, if the metric at infinity indicates that the energy is $E$, in AdS units, there are a finite number of bulk configurations that are consistent with this constraint since there are a finite number of bulk states with energy $E$.
One might wonder why one cannot repeat the trick used in electromagnetism by choosing $H_{\rm bulk}(x)$ independently at each bulk point while only keeping its integral constant. The difference is that the commutator $[H_{\rm bulk}(x), H_{\rm bulk}(x')]$ leads to the socalled ''Schwinger terms'' that are proportional to $\delta'(x  x')$. In a lattice regularization, this would mean that the energy density at one lattice point does not commute with the energy density at an adjacent lattice point and so the energy density operators at different points in space cannot be diagonalized simultaneously. And this, of course, is why the ground state of the system cannot be found by independently minimizing the energy at each lattice site.
So an attempt to generalize this construction of a split state from electromagnetism to gravity fails since the specification of the metric at infinity does not allow an arbitrary prescription of $H_{\rm bulk}(x)$ in the bulk.
As mentioned above, we have explained this detail in the beginning of the attached pdf file in the note titled ''Author response to the second referee report.''
Summary
To summarize: our paper has undergone intensive scrutiny. We have successfully addressed a large number of questions and puzzles, including the latest "counterexample" presented by the referee even though these discussions have taken us well beyond the original scope of the paper. We understand that the referee may still have reservations about the results. But we would like to respectfully point out  and the reader can easily verify this by perusing the correspondence above  that not a single error has been found in our manuscript in this entire process.
We believe that this is a strong indicator of the correctness and quality of the manuscript.
Attachment:
correspondence_with_notes.pdf