Wednesday, March 13, 2019

AdS bulk is a neural network, entanglement is a quantum gauge field

I want to mention three new papers. One of them has a cool title and the other two have some cool ideas in their bodies.

First, F. F. Faria wrote a hep-ph paper with a Conformal theory of everything. The number of papers with the ambitious titles involving "a theory of everything" is still small enough so if you want to be sure that people like me would spend at least 0.2 seconds with each page of your paper, call it "a theory of everything".

It's my policy to quickread papers with "theories of everything" in titles because it has worked for me – such a paper (not very good one) was the first paper from which I learned about Matrix theory which was rather important for me, and still is. ;-)

Sadly, the paper just writes down some action as a sum of some Standard Model, conformal gravity, and dilaton actions, with no sign of a unification or anything else that would be new and interesting. Still, good for a paper written in Brazil (an Amazon researcher is shown on the picture above).

OK, more seriously, I picked two papers about quantum information within quantum gravity, a subfield where the excitement kept on rising in the recent decade, I would say. First, Koži Hašimoto (Osaka) whom I know from Harvard wrote a hep-th paper
AdS/CFT as a deep Boltzmann machine
which elaborates upon a great idea. You know the AdS/CFT duality, right? So the AdS is the boundary and it looks like peripherals that may read the data. The funny idea is that the AdS bulk inside is a neural network, a special kind of it named as "a deep Boltzmann machine", that can be trained to understand the generating functional of the boundary CFT.

So what looks like a "real and natural" spacetime may also be visualized almost as a "man-made" circuit that may be adjusted to predict what's happening in the only real world governed by the boundary observables. The weights of the "neural cells" are functions of the bulk metric at the given point and Koži gives you a dictionary translating in between some sexy computer science of machine learning on one side, and the geometric physics concepts of the AdS bulk on the other side. Einstein's action is given some machine learning interpretation, too. As another example, the "holographic renormalization" is translated as "autoencoder" into computer science.

Well, the discrete character of Koži's paper is a clear defect (it looks like the causal dynamical triangulations) which breaks the local Lorentz symmetry or AdS isometry – the conformal group – but there could exist a smoother improvement of his paradigm.

The first hep-th paper today – which was really posted at 18:00:00 UTC, the first possible second (great job, guys!) – is one by Czech, de Boer, Ge, and Lamprou.
A Modular Sewing Kit for Entanglement Wedges
Although you might fail to guess, Bartoloměj Czech is a Pole in China. ;-) That's despite the fact that the St Bartholomew Cathedral here in Pilsen has the tallest tower in Czechoslovakia. They work on the entanglement-in-quantum-gravity minirevolution. Just like Hašimoto connects Einstein's action to some quantities in the machine learning setup, Czech et al. give an interpretation to the Riemann curvature tensor – in terms of some quantities calculated for a general "system with quantum information".

A part of their toolkit is the "modular Hamiltonian" \(H_{\rm mod}\) which is defined as \(\rho=\exp(-H)\) for a given density matrix \(\rho\). Such a Hamiltonian is meaningful for states that are close to that density matrix. You may immediately see that this is the kind of thinking that was introduced by our friends Papadodimas and Raju, so often discussed on this blog. Sadly, the paper by Czech et al. chooses to emphasize our non-friend Daniel Harlow instead.

The connection with the Papadodimas-Raju is cool and they extend the ideas in an interesting direction. Take two qubits and entangle them maximally. So the reduced density matrix is proportional to the unit matrix for both and there's an \(SU(2)\) group for each qubit that keeps the density matrix, and therefore the "modular Hamiltonian", invariant. However, the correlators involving both qubits aren't invariant under both \(SU(2)\) groups because the entanglement picks a link between them.

So the entanglement tells you how to move from one qubit to another. It's just like moving from one point to a nearby point in gauge theory. The entanglement hiding in the density matrix is a gauge field of a sort, a connection. And they see an example of the Papadodimas-Raju "mirror operators" right here, too. At the same moment, in quantum gravity, entanglement is the glue that connects two nearby regions. So one may think of the entanglement as of a gauge field and evaluate the monodromies in the "purely quantum information" setup. More precisely, these monodromies are the so-called "Berry phase of the aforementioned modular Hamiltonian".

Their most specific class of examples involves the entanglement wedges in particular cases of AdS/CFT. Each wedge is fully specified by a density matrix and they connect them, thus telling you how wedges may be connected to a whole spacetime.

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