## Wednesday, January 28, 2015

### The holographic Da Vinci code and quantum error correction

Since the wrong 2012 papers, I have been encouraging
Dr Polchinski, tear down this firewall!
He's an exceptional physicist who has done cool things and can do many more so the earlier he can escape from the firewall, or extinguish it, the better. Finally, there is a fun paper about the holographic code co-authored by Polchinski that doesn't mention the firewall and it is pretty cool and, in my opinion, at least morally correct:
Bulk-Boundary Duality, Gauge Invariance, and Quantum Error Correction
by Mintun, Polchinski, Rosenhaus (Santa Barbara). Their primary question is how to reconstruct bulk operators from the boundary operators in AdS/CFT. But I actually think that closely analogous claims are valid for the microstates of the black holes and the black hole interior, too.

They try to clarify some previous ideas about the relationship between the holographic dictionary in quantum gravity on one side; and quantum error correction in quantum information science on the other side.

The paper is only four two-column pages long which is why you may want to read it in its entirety and I won't try to reproduce the technicalities here.

But a broader point is that the gauge symmetries of the boundary theory used to be considered "completely irrelevant" for the dual bulk description but it is becoming very likely that they actually do play a role in the bulk physics, too. And the precise mechanism that allows these gauge symmetries to play a role resembles the "quantum erasure correction" and "quantum secret sharing", concepts that are known and studied by the folks in the quantum information science, except that quantum gravity automatically seems to do these things in a smarter way than what they were able to find out so far!

In particular, Mintun et al. say that the bulk operator $$\Phi(0)$$ at the center of the AdS (or any point) has to commute with the spatially separated operators, thanks to the bulk locality. Via the AdS/CFT dictionary, it means that $$\Phi(0)$$ has to commute with all boundary local operators $${\mathcal O} (x_i)$$, those restricted to a single site.

That condition may sound weird – it looks as if the bulk had nothing to do with the boundary and required "new" degrees of freedom. Except that this troublesome conclusion isn't right. No one is saying that the bulk operators have to be constructed from the local, one-site operators on the boundary. If you allow to contract operators from several sites at the boundary, you will find out that solutions exist – and there are actually many of them. Those solutions are referred to as the "precursor" operators.

Note that Polchinski has been intrigued by precursors in AdS/CFT for more than 15 years.

In the usual picture of AdS/CFT, both sides have some local or gauge symmetries. The boundary theory has an $$SU(N)$$ Yang-Mills symmetry. The SUGRA description of the bulk has diffeomorphisms, gauge symmetries for the $$p$$-form gauge fields, and so on. Normally, we say that only the physical states on both states – all the kinematical states modulo all the gauge symmetries – are supposed to match. And the unphysical states are some junk that depends on the description.

However, this paper says that even the unphysical states, especially the gauge-non-invariant states on the boundary, have some important application on the other side, especially in the bulk, because the bulk operators may be more naturally written using unphysical operators on the boundary etc. Those non-gauge-invariant operators would carry slightly more information but if you have just slightly gauge-non-invariant operators, you may find the corresponding right gauge-invariant operator and the procedure is analogous to quantum error correction.

This new paper has already made me think about various connections with ideas in my research or research of others but most of the details remain confusing. So let me sketch just two of these relationships and do so briefly.

Connections between the paper and some seemingly inequivalent ideas

First, two weeks ago, I wrote a blog post about the monster group in quantum gravity and I also mentioned a relationship between black hole microstates and the monodromies around the Euclideanized horizon. Some people think that I was completely kidding but I was not. ;-)

The strongest claim is that the volume of the whole (infinite-dimensional etc.) gauge group reduced to the event horizon will be nothing else than $$\exp(S_{BH})$$, the exponentiated black hole entropy – essentially because the trip around the thermal circle is allowed to return you to the same state, up to any gauge transformation. (Well, the monodromies actually don't label all pure microstates but some particular mixed or ER-EPR states but I don't want to go into details here.) So every theory with gravitational degrees of freedom has to have a certain amount of symmetries. There are no black hole horizons in the paper by Mintun et al. but I do think that the degree of ambiguity of the precursor operators they acknowledge is mathematically analogous to the microstates of a black hole – or an event horizon, to be more general – and their work also implies something about the black hole code, too.

Second, I believe that there are intense overlooked relationships between the work by Mintun et al. (and older papers they built upon) and the formalism of Papadodimas and Raju. To be slightly more specific, look at and below equation 5 in Mintun et al.. A commutator isn't zero but it is "effectively" zero – in this case, when acting on gauge-invariant states, for some reason. This is similar to comments in Papadodimas-Raju in and below equation 9 in which the commutator of $${\mathcal O}$$ and $$\tilde{\mathcal O}$$ "effectively" vanishes – when included in a product of operators with a small enough number of factors.

In both cases, some commutators are claimed to be "much smaller than generic" because of certain special circumstances. If the claims are basically analogous, there is a relationship between the gauge invariance of (a priori more general, gauge-non-invariant) boundary operators; and between operators that keep the pre-existing bulk (black hole...) microstate the same (because they only allow a limited number of local field excitations).

Another general point. Both your humble correspondent and (later) Polchinski have been nervous about the main ambiguity in the Papadodimas-Raju construction. There are many ways how the black hole interior operators may be represented in terms of the boundary CFT operators. There's this state dependence (which I am not anxious about but Polchinski is) but it is not the only source of ambiguity: one has to pick a convention on ensembles and related things, too.

Now, Mintun, Polchinski, and Rosenhaus see something similar in their construction. In the boundary CFT (the same side), they also see many precursor operators to represent the bulk fields. They're equivalent when acting on gauge-invariant operators. However, it could be an important unexpected point that
the black hole interior operators are dual to gauge-non-invariant operators in the boundary CFT
and if that's the case, much of the new content in the papers by Mintun et al. and Papadodimas-Raju could actually be the same, at least morally! If the quote above is right, we face a potential "intermediate" answer to the question whether the AdS/CFT correspondence contains predictions for the observations performed by the infalling observer. It may be the case that the precise predictions for the black hole interior may be extracted from the boundary CFT – but only if you add something like a "gauge choice", too.

Many details remain confusing, as you can see, but I do think that we are getting closer to a picture that teaches us a great deal about the character of the "holographic code" in quantum gravity as well the way how the black hole interior (and its local operators) are interpreted in terms of the boundary CFT (a non-gravitational or bulk-non-local description of the microstates). Effectively or approximately vanishing operators due to gauge symmetries and boundary gauge symmetries themselves seem to play a vital role in this final picture that is going to be found rather soon.

And that's my prophesy.

1. Lubos, thanks for this post.

Error correcting codes also appear in S. J. Gates Jr.'s work, "

Codes and Supersymmetry in One Dimension".
Justing pointing that out.

2. Hi Python Programmer (did I get it right?),
thanks for the comment.

Gates' papers are very fun but as far as I understand, the way how error correction is linked to the physics is very different and much less direct there.

Those Gates' - or Duff's etc. - papers talk about the mathematics of a few qubits and what quantum computing guys are doing them, and find something analogous with the fate of a small number of components or indices that enter fields in SUGRA, and in multilinear invariants constructed from those fields. They are not talking about the real huge S = A / 4G information - exp(A/4G) states - that is included in the black holes or in spacetime. It's just some mathematical analogy of a sort.

On the other hand, Polchinski and pals, and many others who talk about AdS/CFT-related things, holography, black hole information puzzles etc. *do* talk about the detailed huge information stored in the real physical objects.

http://arxiv.org/pdf/1307.7080v1.pdf

4. Apologies, I don't understand this paper. That doesn't mean that I am convinced it is right. I genuinely don't understand it.

5. There is a bit of anthropological nonsense here in talking about error correcting codes because it suggests a telos to the Universe or a designer. I don't doubt that the math/physics is correct but I take this error correction to be simply a redundancy perhaps or a copying mechanism similar to de coherence that is just a part of the structure of nature or if it is seemingly extraneous to the structure then is just a manifestation of probabilistic nature. But this sort of programmer error correction talk leads to the woo woo of new age abuse of physics.

6. It's a very interesting potential objection but I personally don't think that your criticism is justified.

Indeed, the words "error correction" indicate some "purpose" - of someone "intelligent" who wants to achieve something, namely learn some information from some error-laden data source.

But I do think it's right that it's so. The intelligent agent is not "Nature" but the observer who is extracting information about the bulk field observables etc.

What matters mathematically is that the information is encoded in the same way as in some framework designed by an intelligent designer that has a purpose. You agree that the mathematics may be isomorphic but you seem to disagree with the suggestion that there's the purpose, right?

I think that there is a purpose here, too. The purpose is for the observer to learn the observable that is encoded through the error-correcting codes etc. The observer wants things to make sense - to fit into his picture of the spacetime.

In quantum mechanics, one may try to minimize the role of the observer's perspective but at the end, it can never be eliminated completely because the observer defines which observables are "measured" or "perceived". It's possible to talk about the error correcting in the usual quantum-computation applications while verbally minimizing the role of the "human designers", too. At the end, I think that the situations are fully analogous.

7. Thanks a lot for the reply, I have to think about it although I have a flu and my brain is foggy.

but here are some references for the interested people.

http://en.wikipedia.org/wiki/Born_reciprocity

and the latest paper

http://arxiv.org/pdf/1405.3949.pdf

8. Excellent point. I agree 100% about the intelligent observer. I just might say brace yourself for a DaVinci Code movie full of "creativity/bs" where a Cosmologist finds proof of God in the laws of physics and is chased by the Catholic Church or a bunch of conspirators or something.

Still waiting for a misleading article in a popular science publication.

9. Python programmer is correct (if you ever have to do python development, the ide pycharm is a nice one to work with). Thank you for the extra information in response and giving context to these new developments in the main post ... I'll make one more note. Wikipedia says that the Mathieu group M24 is the " permutation automorphism group of the extended binary golay code." In general, I am very interested to see how the stories of all these objects and theories will unfold.

10. Hi! The Matthieu group M24 has been linked, via string theory's Matthieu moonshine, to partition functions on the K3 surface

https://en.wikipedia.org/wiki/Monstrous_moonshine#Mathieu_moonshine

so a similar fascinating story exists here as it does for the monster group.

11. whenever someone talk about science like he is Big Brother of science to prevented from becoming spoiled with god idea i become irritated , after all first natural explanation that we could come up with after seeing beauty and complexity life was that there must be a god aside from devastating consequence from people believing in this idea ( that allow many jackass crocks become their god by claiming to be in contact with god) our ancestors wasn't stupid to think so ,and until darwin evolution theory science had agreement with religions that there must be a god and beside physics we had metaphysics that science couldn't explore with scientific method so most physicists were part time philosopher to try understand metaphysical aspect of reality .what quantum mechanics introduced 90 years ago bring back metaphysical aspect of reality that's why neil bohr was shocked by quantum mechanics .you can not with scientific method investigate observer in quantum mechanics you can not say who has subjective experience of reality ,who is observer and who is not ,who is real and who is not real. Evolution is just top down logic of time in our perception not origin of life just like big bang is top down logic of time in our perception of universe not origin of universe these are just a initial state of things not origin . as lubos say only real thing is our perception of this moment.

12. I just browsed through the article by Almheiri et al . I find it very ironic that many years ago, you were very dismissive of the relevance of continuous wavelet transforms to the AdS/CFT correspondence, and Rehren duality.

13. Sorry but your comment is complete nonsense. This paper by Polchinski and pals have nothing whatever to do with Rehren's vacuous essays.

Also, Mintun, Polchinski, and Rosenhaus don't refer to Rehren and they don't have the slightest reason to do so.