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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
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Submission summary
| Authors (as registered SciPost users): | Olga Papadoulaki · Suvrat Raju |
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| Preprint Link: | scipost_202010_00002v1 (pdf) |
| Date submitted: | 2020-10-06 04:06 |
| Submitted by: | Raju, Suvrat |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| 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 low-energy unitaries and make measurements in a small interval of time. The observers are not allowed to leave the near-boundary 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 back-reaction of a bulk excitation on the metric, and also relies on the entanglement-structure of the vacuum. For low-energy states, we show how the near-boundary observers can use this protocol to completely identify the bulk state. We explain why the protocol fails completely in theories without gravity, including non-gravitational 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 semiclassical gravity.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2021-1-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202010_00002v1, delivered 2021-01-19, doi: 10.21468/SciPost.Report.2433
Strengths
The paper states clearly what it tries to achieve. The proposed answer is appealingly simple.
Weaknesses
A strong claim is made on the basis of a (perhaps) deceptively simple proposal. The devil may hide in the details.
Report
This paper suggests a protocol to determine the quantum state of a gravitational system using only measurements in a small boundary band of AdS (or alternatively a small annular region near the boundary of a Cauchy slice that runs through the bulk).
The paper is thought provoking and appealing in its simplicity. The basic algorithm to determine the boundary state has a similar flavour as basic quantum-informational protocols.
On the other hand its simplicity is perhaps deceptive. Before recommending publication I would like the authors to clarify a few points and comments.
1. Practicality of their protocol a): a typical observer (inasmuch as an observer in AdS can be called typical) only has access to a single AdS space. I am not convinced by the authors’ suggestion of using a super observer who instead lives in a spacetime with many, presumably a very large or even infinite number of, identical copies. For the protocol to be useful, it should be practically be implementable in a single copy of the Universe they live in.
2. The protocol involves measuring the local value of the metric. This seems to be a gauge-variant quantity. Could the authors be clearer about what exact measurements the observer needs to make?
3. Practicality of the protocol b): as the authors mention, in practice the fraction of measuring zero energy can only be exactly determined by performing an infinite number of experiments. This means the protocol naturally involves a tolerance parameter $\delta$. Throughout the paper we meet various other notions of cutoffs (e.g. to define low-energy), semi-classical parameters (1/N) and so on. It would be highly useful if the authors could clearly state at some point in their manuscript how all the relevant scales interact, and where exactly the allowed window for the protocol is to be found. Does this effectively lead to a bound on the allowed non-locality the protocol seems to suggest?
4. The protocol relies heavily on the energy being quantised. Are the authors saying that their protocol would not work for example in the Poincaré path of AdS? If so, how are we to think of this? Furthermore this seems to pose a strong possible obstruction of implementing a similar protocol for more general spacetimes, where energy may very well not be quantised.
Report #1 by Samir Mathur (Referee 1) on 2020-10-17 (Invited Report)
- Cite as: Samir Mathur, Report on arXiv:scipost_202010_00002v1, delivered 2020-10-17, doi: 10.21468/SciPost.Report.2083
Strengths
This paper is clearly written; this is very useful for a topic which relates to the area of information and black holes, as there are many diverse opinions in this area of work.
Weaknesses
I do not find myself in agreement with the main physics ideas presented in the paper.
Report
This is a clearly written paper, but I do not find myself in agreement with the main claims. It is possible that I have not understood the arguments, so I will try to give a detailed explanation of what I think are the problems with the general approach of this and similar papers.
The paper argues that theories with gravity are special in that they allow a determination of low energy states of a system from measurements at the boundary. The process of determination involves some extensions of the idea that if all states are assumed to have different energies, then measuring this energy from infinity will identify the state. I list my difficulties below:
(1) Suppose I have a particle in a box, but I do not know if it is a proton or an electron. From measurements far away, I can measure the small electric field produced by the particle, and find out which particle is in the box. This is not surprising; it is just a consequence of having a long range potential sourced by the charged particles.
The way gravity is used in the paper does not look fundamentally different from this, though the authors argue that gravity is indeed different from gauge theories. It is true that with the electric field one cannot identify neutral excitations. Is it being argued that gravity is special because it couple to everything? (Also, in AdS3 the gravity theory can be recast as a Chern-Simmons gauge theory, and more generally gravity can be written as a gauge theory of local frame rotations.)
(2) In their example on pg 3, the authors start with a state |g> peaked near the center of AdS, act on this with an operator X near the boundary and ask for the amplitude to get the vacuum |0>. They write the amplitude for this as <X|g>, and ask of this is close to <X|X>; if it is, then one has shown that the state in the interior is |g>.
I would have expected <X|g> to be very small (i.e. not equal to <X|X> ~1), since X operates in a region where |g> has a small tail. What is the physical picture that the authors have in mind here? Are they arguing for nonlocal effects here? For example, suppose we are allowed to make local unitary operations on a sphere with a large radius. Are they arguing that such operations can instantaneously create an object at the center of the sphere? (They are doing all their operations close to a time t=0, so there would not be time for signals to travel from the sphere to its center. ) Is it being argued that causality as we normally understand it is violated in a large way when the theory contains gravity?
(3) The authors are focused on whether they can identify a state localized near the center by measurements at the boundary. While they do not focus on issues connected to black hole physics, they do mention that the questions they are looking at related to quantum information questions in black holes. With the traditional picture of the hole (i.e., when the hole has a horizon) we can have both positive and negative energy excitations in the region inside the horizon. In this situation one can make an arbitrarily large number of states that are arbitrarily close to each other in the energy that is observed at infinity. Thus I do not understand how measuring the energy at infinity can help with the problem of black holes. (With the fuzzball paradigm there is no region that is inside a horizon, so there are no negative energy modes; but in this paradigm the black hole microstate is just like a piece of coal, so there is no information paradox to think about.)
(4) The authors are concerned with trying to identify the state |g> near the center, from measurements far away. But such a measurement is not pertinent to the puzzle associated with black holes. This puzzle is termed the 'information paradox', but this is a misnomer; what we have is actually an 'entanglement problem' between the radiation quanta and the hole.
To understand this point in the context of the present paper, consider an electron at the center and another at the boundary, entangled in a singlet state 01-10. We can certainly know the state of the electron in the interior if we measure the state of the electron at the boundary. But this does not mean that there is no entanglement between the two electrons!
Further, In line with what this paper proposes, we can measure the spin of the electron at the center by looking at the small frame dragging effects on the metric near the boundary. Again, this does not mean that there is no entanglement between the two electrons. The black hole puzzle is that if the electron at the center disappears from the universe (when the black hole in which it is contained evaporates) then the remaining electron at the boundary has no well defined state. This problem has nothing to do with 'knowing' the state of the electron at the center. The entanglement of the electron at the boundary implies that if we pass this electron through a Stern Gerlach placed at any angle, there will be an equal probability for the electron to show up in either branch of the apparatus. By contrast, if the electron at the boundary was in a non-entangled state like 0+1, then a Stern-Gerlach placed along the x-axis would see the electron always pass through the +x branch of the apparatus. Thus entanglement is a simple measurable property of an electron, and has nothing to do with 'knowing' the state of the 'other' electron. The authors talk about the uncertainty related to entanglement in the standard language of Von Neumann entropy. But this uncertainty or lack of information is not an uncertainty about the state of the other electron (the one in the interior). Rather, it is the uncertainty of the boundary electron itself in the sense that we will never be able to say which branch of the Stern-Gerlach it will pass through.
(5) The authors base their arguments on a few computations; let me list my difficulties with these computations:
(i) In eq. 2.5 they expand the unitary operator to first order as 1+iJX. I am worried about not keeping the second order term here, since it is these second order terms that help give causality in field theory. Take a free scalar field \phi in 1+1 dimensions. Apply a unitary O_1=e^{i JX} at (t,x)=(0,0), and O_2=e^{-iJX} at (t,x)=(\epsilon, D). Thus the two points of application are spacelike separated. Now compute <0|T[O_2 O_1 ]|0>. This gives J^2<0\phi(\epsilon,D)\phi(0,0)|0> which is nonvanishing. It would therefore appear that a unitary operation at the first point can be detected at the second point, which is of course in conflict with causality. To resolve this conflict one has to expand the exponentials to one more order. The authors should check that a similar effect is not leading to apparently acausal effects in their computations.
(ii) On pg 8 the authors say that energy gaps in AdS are of order the inverse AdS radius. But consider AdS3 X S^3. Consider a graviton with a large angular momentum j\gg 1 on the sphere. The radial energy levels in the AdS for this graviton are quantized in units of 1/j times the inverse AdS3 radius. Thus the authors should check carefully what energy gaps are allowed for massive particles in AdS, where the mass can come from rotation on the sphere or because we considered a heavy string state.
(iii) The authors give a computation in their Appendix A which seems to say that knowing a field in a small region near the boundary can allow a determination of its value in the interior. I am concerned about this, in view of the following toy example. Consider functions on the line interval (-1, 1). Let f1=1 be the constant function on this interval. Let f2=1 for x in the range (0,1), and f2=1-Exp[1/x] for x in the range [-1,0]. Thus f1 and f2 are the same for x>1 but differ in a completely smooth (C_\infty) way in the region x<1. If we are given only access to the range of x given by (1-\epsilon, 1), then we will not be able to distinguish these functions. Why is the situation different for the case with Hilbert space states that the authors consider? If I take a fermion field, then can I not make exactly the above two wavefunctions with local operators, and therefore be unable to distinguish the states from the boundary?
(7) In summary, I am not in agreement with the general idea of the paper that there are interesting effects at the level of perturbative gravity that imply a revision of our thinking about locality and causality. Of course it is possible that I have not understood some of the arguments, and if this is the case, I apologize in advance for being critical of them.
Author: Suvrat Raju on 2020-10-20 [id 1012]
(in reply to Report 1 by Samir Mathur on 2020-10-17)
We are grateful to Prof. Mathur for his detailed report, and also for signing the report, which helped us understand the referee's perspective.
It is true that our paper reaches a novel and striking conclusion. So we are delighted to have the opportunity to answer Prof. Mathur's questions since other readers may have similar questions.
We believe that the main strength of this paper is that its conclusions are based on a simple and unimpeachable technical calculation. So we would like to start by addressing point (5) in Prof. Mathur's report, which is the only point where he mentions any technical objections. The other points are more general, and so we will answer them later.
(5)
(i) Prof. Mathur suggests that we are "not keeping the second order term here." But this is incorrect. In fact, we are very careful about this second order term in our analysis. It is displayed explicitly in 1.1, and the reason it is not displayed in eqn. (2.5) is mentioned immediately below that equation and again below eqn. (3.1): the reason is that this term drops out of the analysis.
Let us reiterate the reason that this term drops out. The unitary, U, always acts on states $|g \rangle$ that satisfy $\langle 0 | g \rangle = 0$. We are always interested in measuring $P_0$ after acting with the unitary. If we take the unitary to coincide with $e^{i J X}$ to second order in $J$, its action is $U |g \rangle = |g \rangle + i J X |g \rangle - (J^2 X^2/2) |g \rangle$. Now consider what happens when we sandwich $P_0=|0 \rangle\langle 0|$ between the state and its dual bra. The answer is clearly $\langle g|U^*|0 \rangle\langle 0|U|g \rangle = |\langle 0|U|g \rangle|^2$. Now $\langle 0|U|g \rangle = i J \langle X|g \rangle - J^2 \langle 0|X^2|g \rangle/2$ where the leading "1" in the unitary goes away because $\langle 0|g \rangle = 0$. When we now consider $|\langle 0|U|g \rangle|^2$ we see that the second order term in $J$ only contributes to $J^3$ and higher order terms!
Note that, for this second order term to drop out, it was important that $\langle 0|g \rangle = 0$. This is the reason we act with the unitary $\mathfrak{U}^z$ in step 1 of our protocol (Figure 2) and this unitary is the subject of Appendix B.
So far from ignoring the second order term, we believe that we have devoted significant attention to ensure that it does not enter the final answer. We can emphasize this even more in our revised paper.
The example that Prof. Mathur gives where one expands both exponentials to first order and then reaches an incorrect result when the exponentials are multiplied (because both exponentials also contain a factor of 1), is not what is happening in our calculation. There is no analogue of this effect in local field theories.
(ii) In AdS/CFT, the energy levels in global AdS that we are referring to are dual to the spectrum of operators primaries in the CFT. Prof. Mathur is right that the spectrum of operator primaries is not integrally quantized but our assumption is only that the spectrum of primaries is discrete and the number of primaries below a fixed cutoff is finite. We believe that this is a very reasonable assumption, but we would be glad to mention it explicitly.
(iii) We believe that Prof. Mathur has misunderstood the point of Appendix A. We certainly do not claim that knowing any function in a small interval allows one to reconstruct it in a larger interval, which would be obviously incorrect as Prof. Mathur points out.
The argument of appendix A is that any state created by smearing an operator with a function in the time-band $[0, \pi]$ can be reproduced by smearing the same operator with a different function in the time-band $[0, \epsilon]$.
We have already referred to the appropriate literature in the text but for convenience, we reiterate the argument here. Consider the state $\int_0^{\pi} O(t) f(t) | 0 \rangle$ . Expanding both O and f in Fourier modes, this state is just $\sum_{n > 0} O_{n}^{\dagger} f_{n} | 0 \rangle$. It is important that the sum above runs only over positive $n$ since the energy of the vacuum cannot be lowered, and so all the negative frequency parts of $f$ drop out. Therefore if we want to find another function $g(t)$ with support in $[0, \epsilon]$ so that $\int_0^{\epsilon} O(t) g(t) |0 \rangle = \int_0^{\pi} O(t) f(t) |0 \rangle$, we only need to make sure that the positive frequency part of $g$ coincides with the positive frequency part of $f$.
Given a function with support in a larger interval, one can always find a function with support in a smaller interval, so that the two functions agree arbitrarily well in their *positive-frequency parts*. This can be proved using elementary complex analysis. In the Appendix, we have also verified this result numerically, and our code is publicly available.
We would like to emphasize, once more, that the argument of Appendix A holds even in local QFTs and so, by itself, it does not tell us that we can completely identify the state from a small time band. It only tells us that we can produce all states by acting with operators from a small time band. (Note: "produce state" $\neq$ "identify state"). To be able to identify states, we need to combine this argument with the projector on the vacuum, which is the subject of the main text of the paper.
We believe that the above explanations completely address the technical points raised by Prof. Mathur. We hope that Prof. Mathur will agree that our technical arguments are correct, and if he has any doubt about the correctness of any equation in the paper, we would be glad to address it.
We now turn to the other questions in the referee report, which are of a more general nature.
(1) Gravity is very different from gauge theories because the projector on the vacuum selects a unique state. In gauge theories, one can project onto states of zero charge but there are an infinite number of such states. This is why one can obtain information about the bulk from the boundary, whereas one cannot do so in non-gravitational gauge theories.
This is a physical fact that utilizes the positivity of energy, which does not change if one formulates the theory differently.
In Prof. Mathur's language this is indeed because there are no localized neutral excitations in gravity, whereas there are such excitations in gauge theories.
(2) $X$ is indeed an operator near the boundary but, as we explained above, $X |0 \rangle$ can create an arbitrary state including the state $|g \rangle$. This does not involve any nonlocality and is true in an LQFT.
The magic in gravity is that the matrix element $\langle 0|X|g \rangle$ becomes physically accessible. This matrix element is not accessible to a near-boundary observer in any theory without gravity.
Points (3) and (4) are both about black holes. The focus in this paper is not on black holes and --- even though we recognize that Prof. Mathur may not agree with our perspective on black holes --- we would like to keep this discussion brief in order to keep the discussion on track. We answer (4) before (3).
(4) In the example of an EPR pair, that Prof. Mathur gives, it is possible to act with a unitary operator on the electron in the center. In a non-gravitational theory, the action of this unitary is completely invisible to the observer near the boundary. Since the near-boundary observer has no way of detecting whether or not the observer in the center has acted with this unitary , in the language of this paper, we would say that the observer near the boundary cannot obtain "information" about the electron in the center.
In this paper we have shown that, in the presence of gravitational effects, the observer near the boundary can determine whether or not the unitary was applied in the center. There is no analogue of this in qubit systems with factorized Hilbert spaces.
In the context of black holes, this means that a copy of the information in the middle of the black hole interior remains in the exterior. When the black hole evaporates away, this copy still remains and this is why there is no information loss.
We emphasize that our arguments involve no ad-hoc postulates about nonlocality and follow from a careful semiclassical analysis.
(3) We do not claim anywhere that a measurement of the energy is sufficient to completely distinguish between states. This cannot be the case since, clearly, there can be degeneracies in energy-levels even at low energies. Moreover, energy measurements, by themselves, cannot determine the relative phase in a linear combination of energy eigenstates. The claim is that measurements of the energy and also of correlators of the energy with other observables suffice to determine the state. In this paper, we have shown this explicitly at low energies. The formal arguments that we allude to imply that this continues to be true for high-energy states like black holes.
We hope that this comprehensively addresses Prof. Mathur's concerns and that the paper can proceed towards publication.
Author: Suvrat Raju on 2021-01-29 [id 1188]
(in reply to Report 2 on 2021-01-19)We would like to thank the referee for the helpful questions and comments.
Below, we have answered the referee's questions and also provided details of the changes made in response to the referee's comments.
1) Many copies
The necessity of many copies has no specific relation to our protocol but arises from the inherent features of quantum mechanics. Since the results of observations are probabilistic, in general, it requires multiple observations on identically prepared systems to obtain quantum information. This is true, not only for quantities like the entanglement entropy, but even for something as mundane as bulk correlation functions. Since correlators are expectation values of products of operators, to measure a correlator requires the observers to prepare many identical copies of the system, make measurements in each copy, and then average the results.
Therefore, even if the observers were to explore the bulk to obtain information, and not use our protocol at all, they would still require multiple identical copies in order to be able to fix the bulk state.
It is possible that by explicitly mentioning this mundane issue, we may may have unintentionally suggested that something unusual is going on. Therefore, we have inserted a remark to emphasize that there is nothing unusual about this aspect of our protocol and it follows from standard quantum mechanics.
2) Measurements of the metric
Thank you for this remark. The energy that we are measuring is the integrated component of $h_{00}$ in the Fefferman-Graham gauge. There is a corresponding gauge-invariant expression, but we wrote the expression in the Fefferman Graham gauge for simplicity. We have inserted a reference and a remark to clarify the issue.
3) Cutoff parameters
Thank you for the remark about cutoff parameters. We introduce two parameters to make the protocol precise. There is a UV-cutoff, $\Lambda$, and an accuracy scale, $\delta$, that can be set independently. But the UV cutoff is much smaller than the Planck scale, so that $\Lambda \ll N$ and the accuracy parameter is larger than the ratio of the cosmological scale to the Planck scale: $\delta \gg \frac{1}{N}$. We have inserted this clarification in the text. These cutoffs are meant to ensure the simplicity of our protocol, and can be relaxed if the observers are allowed to make more complicated measurements as is discussed in Ref. [1] of our paper.
4) Quantization of energy.
From the point of view of the observers near the boundary, there is not much difference between Poincare AdS and global AdS. Since a Cauchy slice for Poincare AdS also provides a Cauchy slice for global AdS, the Hilbert spaces are isomorphic. More specifically, observers near the boundary of ``Poincare AdS'' can always choose to classify states using a linear combination of the Poincare generators (see Luscher and Mack, Commun. math. Phys. 41,203--234 (1975)), obtain a discrete spectrum for these new generators, and then carry out precisely the protocol that we have indicated.
Nevertheless, the referee's question about infrared effects is an important one. Not only is it relevant for flat space, it is also important for extending our protocol to black holes in AdS. In effective-field theory about a black-hole background, the quantization of energy-levels is no longer apparent. This is a very interesting question that we hope to address in forthcoming work. We have added a remark to this effect in the paper mentioning this issue.
In addition to the changes described here, we have also added footnote 1 on the spectrum of operator dimensions. This is in response to a comment in Prof. Mathur's report, as indicated in our reply to that report. We have also added additional remarks to emphasize the fact that we are not ignoring the second-order term but that we do not display it explicitly at all times since it drops out of the analysis.
Finally, we have altered the word "semiclassical'' (which is understood in different ways by different researchers) to "low-energy theory'' so as to be more precise about what we mean.
We hope that this adequately addresses the concerns of both referees and the paper can now proceed towards publication.