Monday, April 04, 2016

Holographic principle from threads and bottlenecks

Matt Headrick and Michael Freedman have released an interesting paper on holography and entanglement:
Bit threads and holographic entanglement
First, I must say that Matt is at faculty of Brandeis (we've known each other in Greater Boston). Michael Freedman is the most senior person at Station Q (stands for "quantum computing", I guess) of Microsoft Research in Santa Barbara. He's the same man who received a Fields medal for his contributions to the Poincaré conjecture (now theorem) in the 1980s and who co-discovered things like exotic 4-spheres.

We could say that they have proposed a new way to visualize the Ryu-Takayanagi (RT) formula as an emergent consequence of a picture with threads connecting points at the boundary (or perhaps a holographic surface).

To remind you, the RT formula is a quantitative result within the Van Raamsdonk et al. (who doesn't get cited) "entanglement is glue" minirevolution. This RT formula formally looks like the Bekenstein-Hawking law\[

S = \frac{A}{4G}

\] except that the precise meaning of the variables is special and optimized for the "geometrical connectedness of space is entanglement" paradigm. Here, we have a spatial region on the boundary \(R\) and want to determine the entanglement entropy \(S\) between this region and the rest of the boundary. The result is given by the area \(A\) of the minimum surface in the bulk that terminates at the region \(R\).

If you formulate it in this way, it looks like that you first need to list all possible surfaces, find the minimum one, and only when you measure its area, the fundamental form of the RT formula may be formulated. And you don't really know where it came from.

Freedman and Headrick manage to decompose this RT result to smaller ones – to "derive it" from some more elementary axioms. You may see that Freedman has worked on the "Riemannian flows" in the context of the Poincaré conjecture because these flows are everywhere in this paper, too.

They try to attach a collection of threads to the region \(R\) in all possible ways. Each thread is an "information flux tube" that carries one bit. I suppose that they should have called them wires, and not threads, because such strings should better be conductors to transfer information. ;-)

According to their story, the "minimum surface" \(A\) appears in the entanglement entropy because the minimum is the "bottleneck" where the lowest possible number of wires may be squeezed and "a chain is only as strong as its weakest link", but one must consider the "bottleneck" of the "best" way to connect \(A\) with other surfaces in the same homology class. So the relevant area enumerating the entanglement entropy results from some "minimum of maxima" or vice versa procedures and I don't want to get confused too much.

But the point is that assuming the legitimacy of their wires, one may derive why this particular structure of "minimum of maxima" emerges and why it's related to the minimal surfaces of the RT formula. The RT formula is basically derived from 1) the assumption of the wires, and 2) generally valid considerations about the flow of information and entanglement.

The appearance of the "bottlenecks" makes it irresistible to mention Raphael Bousso's covariant [holographic] entropy bounds. In the late 1990s, Bousso generalized the Bekenstein bounds to more general, time-dependent spacetimes, using some (null) light sheets and the minimum areas within them.

First, I can't get rid of the feeling that because Headrick and Freedman talk about the bottlenecks in a "proof of holography", they are using a trick that first appeared in Bousso's head because he had to consider the area of the "bottlenecks" of this light sheets, too. Second, Freedman and Headrick don't talk about any (null) light sheets so this detail is different from Bousso's methodology. But maybe they should orient the wires along some null surfaces at the end because it's the superior thing to do.

The new Headrick-Freedman way to understand the RT formula is "newer" than things like Bousso's bounds because the RT formula itself is newer. But I have not yet decided whether the overall picture of Headrick and Freedman is conceptually newer or more accurate than the older "pictures of holography". Is it clear that positive progress is being made?

At the end, I do believe that results such as the RT formula, the ER-EPR correspondence, and the PR formulae for the black hole interior fields etc. (and maybe things like the fuzzball representation of black hole microstates or additional ideas that are currently known or unknown), among other things, will be pretty much "derived" from some more elementary starting point (in this respect, it will be similar to Headrick-Freedman) and this derivation will make it much clearer why all these things are true (or why some of them are false if they are false).

On the other hand, I am not sure whether these wires are really the new "elementary building blocks", especially because we haven't been shown how they may be reconciled with the detailed mathematics of string theory yet. If all of string theory could be reformulated using a dual formalism based on the "wires connecting points at the boundary" and if the evidence (or proof) of this duality were given, that would be an entirely different story.

Incidentally, it seems also plausible to me (but this is even more speculative than all the things above) that if black holes are added to the configuration, the Headrick-Freedman "threads" (which I called "wires") are basically the same kind of one-dimensional objects (solution-generating strings) that are also used by Mathur and pals to construct fuzzballs. The Mathur fuzz could be made out of the Freedman-Headrick threads, although one has to pick a closed topology of them to turn them into a "fuzz". ;-)

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