Thursday, November 19, 2015

Entanglement + spacetime - can someone clue me in here?

There is a feature article in the current issue of Nature titled "The quantum source of space-time", and I can't decide if I'm just not smart enough to appreciate some brilliant work, or if this is some really bad combination of hype and under-explanation.   

The article talks about the AdS-CFT correspondence - the very pretty mathematical insight that sometimes you can take certain complicated "strong coupling" problems (say gravitational problems) in 3d and map them mathematically to simpler (weakly coupled) problems about variables that live on the 2d boundary of that 3d volume.  I've mentioned this before as a trendy idea that's being applied to some condensed matter problems, though this is not without criticism.

Anyway, the article then says that there is deep high energy theory work going on looking at what happens if you mess with quantum entanglement of the degrees of freedom on that boundary.  The claim appears to be that, in some abstract limit that I confess I don't understand, if you kill entanglement on the boundary, then spacetime itself "falls apart" in the 3d bulk.  First question for my readers:  Can anyone point to a genuinely readable discussion of this stuff (tensor networks, etc.) for the educated non-expert?

Then things really go off the deep end, with claims that entanglement between particles is equivalent to an Einstein-Rosen wormhole connecting the particles.  Now, I'm prepared to believe that maybe there is some wild many-coordinate-transformations way of making the math describing entanglement look like the math describing some wormhole.  However, the theorists quoted here say things that sound stronger than that, and that's completely crazy.  I can create entangled photons in a lab with a low-power laser and a nonlinear crystal, and there is no way that this is physically equivalent to creating highly curved regions of spacetime and nontrivially altering the topology of spacetime.    Can someone explain to me whether the theoretical claims are like the former (there is some formal mathematical similarity between entangled particles and wormholes) or the much more extreme statement?


Don Monroe said...

I'm not sure it's helpful, but New Scientist also just had a story on this.

Anil Ananthaswamy writes that "At first glance, the ER=EPR hypothesis would mean quantum systems that become entangled, and therefore enter a superposition, suddenly gain a wormhole--a conjuring trick the superposition principle doesn't obviously allow." He says Joe Polchinski, who knows all about string theory and the firewall, isn't buying it for that reason. So I guess it is early days for this idea, but I'm not sure our intuitions (mine agrees with yours) are much use for wormholes in spacetime. I think fundamentally it is very hard to reconcile our intutive feel for any kind of geometry with a quantum indeterminacy. Geometry just feels too "real." Any reconciliation of general relativity and quantum mechanics is probably going to require us to give up on that.

Anonymous said...

I can also refer you to a nice post some time ago by John Preskill, although I don't think it will quite answer your questions: .

Looking briefly at the Susskind/Maldacena paper (and certainly not with an expert eye!), I think perhaps a more precise phrasing of their claim is that entanglement between two particles is something like the building block for a wormhole. They certainly agree that it is not a smooth classical wormhole, which as you say would presumably have some large energy cost. Instead, it would be some unspecified, highly quantum form of a "proto-wormhole" (my words, not theirs). But they suggest, for example, that if one were to take a collection of particles in Bell states, and group them pairwise so they collapsed into two black holes, these black holes would not only be entangled but would also be connected by a wormhole, and that these are essentially the same statement.

Shauna Kravec said...

Hi Doug,

I enjoy your blog! Maybe I can offer some words on this.

With holography (of which AdS/CFT is the most fleshed out realization) you have a quantum field theory, living in D spacetime dimensions, which is claimed equivalent to a theory of quantum gravity in D+1 spacetime. Equivalent in that the Hilbert spaces are equal and the partition functions are equal. This is a bold claim and we're far from understanding it because we're far from understanding quantum gravity. The example we understand best is when the field theory has a conformal symmetry and the gravity theory is (asymptotically) AdS.

Anyway there's a lot to say about AdS/CFT but from your other post you have the idea. These notes are good if you'd like to read more:

One question you could ask: "How can I compute entanglement entropy of the field theory using holography?" Pick a region of space in the field theory and call it A. To compute it in field theory we'd pick a state (the ground state for instance) and compute a reduced density matrix \rho by tracing out the complement of A. Then S(A)= -Tr[\rho \log \rho]

In AdS/CFT there is a formula due to Ryu and Takayanagi: S(A) \propto \Sigma_A where \Sigma_A is the area of the minimal surface in the bulk which has A as a boundary. This is similar to the Bekenstein-Hawking entropy of a black hole. It's been computed in several circumstances and reproduces known results from field theory as well as general features of the entanglement entropy like the area law and strong sub-additivity. This is a review:

The fact it is so geometric is probably the most concrete insight into all of this business. There's the thought that maybe the entire bulk spacetime is "built up" out of these minimal surfaces. For example you could ask, "What is the gravity dual of a product state vs an entangled state?"

Shauna Kravec said...

[Part 2 of the above since it was cut short]

A lot of this idea, and the connection to MERA which I'll mention in a bit, is earliest described here:

Well for a product state it should be a pair of disconnected spacetimes. So then superpositions of disconnected spacetimes should make up other states. As it turns out these superpositions can sometimes be understood as a simple connected spacetime.

For instance consider two identical non-interacting copies of a CFT with energy eigenstates \ket{n}_{L/R}. The Hilbert space is a tensor product of the pair. The state \ket{\psi} = \sum_{n} e^{-\beta E_n /2} \ket{n}_L \otimes \ket{n}_R is dual to a pair of blackholes with a common time. The horizons are what keeps the CFTs from talking to each other but the entanglement between the states is realized in this wormhole bridge.

This point about 'tuning the entanglement' is simple. Suppose you compute the mutual information I(A,B) between two regions, each from one copy of the CFT. Assuming the Ryu-Takayanagi formula how does the geometry of the holographic spacetime change as I goes to 0 ? (This is \beta \rightarrow \infty) From the geometry this implies an increase in the length of the shortest geodesic connecting the regions. In the blackhole geometry the throat closes.

This essay is readable and discusses these points:
This is a recent paper trying to take this ideas further:
This is a blog post by John Preskill with an encouraging review of the 1306 paper:

Finally there's the connection to tensor networks, MERA in particular. These tensor networks give you an ansatz for (usually) groundstate wavefunctions of many-body systems. MERA is a particular type which was built to describe groundstates of critical theories which have scale/conformal symmetry. It consists of tensors which disentangle pairs of spins and those which 'course-grain' like block transformations, implementing something like an RG flow. In this case the network graph looks like a discretized version of AdS and reproduces things like the Ryu-Takayanagi formula.

See here:
Another paper by Brian Swingle on this:

Douglas Natelson said...

Thanks, all. Anon, Shauna, your comments were especially helpful. Hmm. So, if I form two black holes, each containing one of two mutually entangled particles, the holes are/could be/should be linked by a wormhole? That's not at all obvious to me. Indeed, I could make an argument that, from the point of view of an observer outside both blackholes, the entangled particles never actually make it into the blackholes, right? They "come to rest" on the horizons, and so it's fine for their entanglement to remain....

Anonymous said...

I'm a bit late to the party here, but I have always found the Quanta Magazine articles a brilliant (popular) read. They had a series of 3 articles on this subject.

David Brown said...

Are the majority of theorists correct about quantum gravity? I say that Milgrom is the Kepler of contemporary cosmology. Are string theorists wise to ignore MOND? Consider 2 articles without references to MOND:
”Building up spacetime with quantum entanglement" by Mark Van Raamsdonk, 2010
”String Theory: Progress and Problems" by John H. Schwarz, 2007
Now listen to McGaugh:
”Dark Matter or Modified Gravity?" - Stacy McGaugh - YouTube, 2015