Pentagons, hexagons, quantum gravity, AdS/CFT

[mitchell porter]

arxiv:1503.06237 proposes a 5-qubit error-correcting code as an analogy for AdS/CFT duality. One of the authors, Beni Yoshida, has an expository post at quantumfrontiers.com and motls.blogspot.com. It includes the idea of joining together many instances of the 5-qubit code (which as a diagram is five lines meeting at a point), to build a finer-grained model of the AdS "bulk".

Meanwhile, in category theory, pentagon and hexagon identities are important. What I'm wondering is whether this qubit-code tensor-network model of quantum gravity, has a relationship to some form of categorical quantum gravity.

Potentially there's continuity between this question and inconclusive previous discussions about AdS/LQG. But I also hope there might be crisp insight into whether or not this particular analogy is misguided.

 

[marcus]

Hi Mitchell, I don't recall the previous discussions involving LQG with AdS/CFT that you refer to. I expect they occurred in past threads here in BtSM forum and I simply didn't notice, or forgot them.
But your mentioning an analogy with LQG aroused my curiosity and I'm trying to guess how it might go (misguided or not )

A spin foam can arise from a 4d triangulation of a manifold into 4 simplexes. A 4-simplex has 5 sides.

Since the spin foam is dual to the triangulation, in the conventional sense, each 4-simplex is represented by a point with 5 lines emanating out from it.

So when you mention "five lines meeting at a point" as an elementary diagram...
and refer to many elementary diagrams joined together...
it does suggest a spinfoam history (the way Lqg describes evolving geometry within a bounded region, and gets transition amplitudes).

Since I don't know what you mean by a "code" or understand in detail, I can only point to similarity at this naive analogy.

 

[atyy]

It seems like a refinement of Physics Monkey's tensor network AdS/MERA proposal. Looking at the abstract, it's interesting that they find tensor networks where the Ryu-Takayanagi formula is satisfied exactly, whereas with the original argument in http://arxiv.org/abs/0905.1317 the MERA only gave an upper bound on the entanglement entropy.

The OP paper http://arxiv.org/abs/1503.06237 by Pastawski, Yoshida, Harlow, and Preskill is based on the earlier paper http://arxiv.org/abs/1411.7041 by Almheiri, Dong, and Harlow.

Another related paper is http://arxiv.org/abs/1501.06577 by Mintun, Polchinski and Rosenhaus.

Recent Perimeter talk by Harlow: http://pirsa.org/15030119
Bulk Locality and Quantum Error Correction in AdS/CFT
In this talk I will describe recent work with Almheiri and Dong, where we proposed a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction. Bulk notions such as Bogoliubov transformations, location in the radial direction, and the holographic entropy bound all have natural CFT interpretations in the language of quantum error correction. Time permitting, I will also discuss work in progress with Pastawski, Preskill, and Yoshida on a new class of stabilizer codes that explicitly realize many of the properties we argued the AdS/CFT error correcting code should have.

 

[mitchell porter]

marcus: I mean a quantum error-correcting code. It's a mapping from 5-qubit states to 1-qubit states with the property that a specified number of quantum 'errors' (bit flips, phase rotations) can be affect the starting state, and the mapping will still output the desired 1-qubit state, just as a classical error-correcting code allows a special subspace of e.g. bit-strings (if it's a binary code) to be reconstructed despite noise.

This has been proposed as an analogy for a type of redundancy in the AdS/CFT mapping. There, the operator for a specific quantum field, at a specific point in the holographically emergent space, can be expressed as an integral of field operators across a region of the 'boundary' (the edge of the emergent space, where an equivalent but lower-dimensional image of everything exists). But there are multiple ways to sum boundary operators in order to obtain the same emergent point, and this is the redundancy they mimic using the qubit code.

Then they concatenate instances of the code to describe going deeper and deeper into the emergent dimension by drawing on larger and larger regions of the lower-dimensional space - see diagrams in the blog post - and this is where the 'tensor network' or grid of lines comes from. But unlike the simplex case, this is still just a tesselation of a 2-d space (a spacelike slice of AdS3).

You mention spinfoam histories and this is actually one place where the other side of my speculative link, pentagon equations etc, does occur. For example see http://arxiv.org/abs/1101.3524 by Bonzom and Friedel, where a "pentagon identity" for some of the ... degrees of freedom? ... in 2+1 gravity, corresponds to the ... amplitude for? ... one of the Pachner moves that can act on the spinfoam to generate time evolution.

I also note that according to the wiki page on 6j-symbols, which are the degrees of freedom (?) above, they are "precisely the information that is lost when passing from a monoidal category to its Grothendieck group". Information loss suggests quantum codes, and monoidal categories seem to be the algebraic origin of pentagon equations (judging by a quick visit to Marni Sheppeard's thesis).

So in short, we have a theme of pentagons and information loss in 2+1 gravity, appearing in AdS/CFT *and* in Riemannian LQG. Unfortunately I suspect that this "connection" is spurious, that the concepts are playing a completely different role on each side.

Thanks to atyy for unearthing some relevant papers.