Wednesday, June 11, 2014

Entanglement and networks of wormholes

The newly realized relationship between the geometric connections in the spacetime and the standard quantum entanglement has been the topic of exciting papers in recent years. One aspect of the papers that have been written down so far made them simple and too special: the entangled systems were always pretty much pairs of degrees of freedom and the wormhole correspondingly looked bipartite, like a cylindrical tunnel connecting two pretty much identical throats at the ends.

A newly published 65-page-long hep-th preprint
Multiboundary Wormholes and Holographic Entanglement
by Balasubramanian (I don't need a clipboard, Vijay!), Hayden, Maloney (hi, Alex!), Marolf, and Ross from Upenn/CUNY-Stanford-McGill/Harvard-UCSB-Durham (yes, seven affiliations for five authors, guess why!) was written in order to transcend this limitation.

In particular, they use a setup in \(AdS/CFT\). More precisely, it's \(AdS_3/CFT_2\) in the Euclidean context. The \(CFT\) looks much like the world sheet \(CFT\)s in perturbative string theory. But because it's a holographic boundary \(CFT\), there should be no two-dimensional gravity in it. Two-dimensional gravity has no local dynamical excitations but there is a difference because you shouldn't sum over topologies of the \(CFT_2\) etc.

Their boundary has the two-dimensional, Euclidean, connected geometry \(\Sigma\) – a Riemann surface – but this Riemann surface has several, namely \(n\), circular boundaries. The setup is a bit complicated so don't forget that this whole \(\Sigma\) is a boundary of something else, too.

The dynamical gravitational spacetime capping this two-dimensional Euclidean \(CFT\) boudary is Euclidean three-dimensional, some \(AdS_3\)-like geometry. They argue that they may find the gravitational i.e. geometric dual description to "non-maximally" entangled states similar to the GHZ state\[

{\ket\psi}_{GHZ} = \frac{ \ket{\uparrow\uparrow\uparrow} + \ket{\downarrow\downarrow\downarrow} }{ \sqrt{2} }

\] which is known from the discussions about the intrinsically non-classical or "paradoxical" character of the quantum entanglement. (We often call it the GHZM state and the dominant convention for the relative sign is "minus", but let's not be picky.)

What they find out is that the very topology of the three-dimensional interpolating \(AdS_3\)-based geometry isn't fixed by the discrete data about the two-dimensional Euclidean-signature Riemann surface \(\Sigma\). Instead, even the topology depends on the (shape i.e.) moduli of the Riemann surface \(\Sigma\).

(Once again, recall that these moduli are not integrated over because the boundary \(CFT_2\) isn't a gravitational theory. It is a boundary \(CFT\) which is, by general rules of holography, non-gravitational.)

For some values of the moduli, the connecting three-dimensional surface looks more bipartite while for others, it looks multipartite. The relevant geometry may be guessed from the appropriate – not uniquely determined – way to cut the surface \(\Sigma\) by scissors. The bulk spacetime topology is so dynamical and emergent that it seems to depend on lots of data that you wouldn't expect to matter if you thought that the spacetime topology may be decided in an a priori way.

This dependence of the topology on the point in the moduli space obviously generalizes the Hawking-Page transition. For decades, since the early days of holography, this phase transition between some gas and a black hole has been known to be holographically dual to the confinement/deconfinement transition in the boundary theory.

So in principle, similar topology changes as functions of unexpected parameters have been known except that the topology change in the newest paper is perhaps even more unexpected.

As I have emphasized since the early days of this entanglement-as-glue minirevolution, one should stop thinking about the spacetime topology in quantum gravity as something that is given by some good "topological quantum numbers" that may be decided at the very beginning so that everything else is constrained by these topological assumptions. Instead, the topological invariants in quantum gravity aren't even well-defined quantum numbers (they are not given by Hermitian operators) due to various ER-EPR-like dualities. And the most convenient topology for a given state actually requires some calculation.

The Hilbert space of the microscopic, e.g. the boundary \(CFT\), theory isn't "divided" to sectors of different bulk topologies in any easy way that you might immediately guess. This fact is a testimony of the fact that the spacetime geometry has become "really dynamical" or, if you won't misinterpret the adjective in a stupid Laughlinian way, "really emergent".


  1. What must Beltway lobotomites do to convince Americans that illegals are a good idea? Look around. We must all sacrifice for illegals and their advocates,

  2. I would gladly accept an amnesty for illegals if it were combined with an across-the-board moratorium on all further immigration into the United States until we have had time to assimilate and integrate the 40 million foreign-born who are already here, the vast majority of whom come from societies with no or very weak democratic traditions.

    To be sure, it is not just the values of our democratic culture that I am worried about but also the present and future living standards of the poorest, least skilled, most vulnerable of our citizens.

    To be honest it will be a long time -- a couple of generations at least -- before most of these immigrants are fully integrated and assimilated into our society. I think this interval might profitably be used to study the question of whether mass immigration helps or hurts the development of the poor countries from which most of them come.

    This whole issue should be about what is best for the majorities in both the sending and receiving countries. I These questions are amenable to empirical study.

  3. minor correction needed... Cantor raised $5 million but only spent about $1 million of those funds.

  4. [..the numerous illegals should become legal residents...]

    This would not help. You need to keep in mind, that US economy needs ILLEGAL immigrants, or more precisely ILLEGAL employees. There is one way how to avoid the minimum wage laws. You employ illegal immigrant, pay cash and avoid all the overhead (taxes, health insurance, minimum wage...). And you can be sure, that s/he will not complain to authorities, as if s/he did, s/he would get deported. The moment you legalize her/his stay, s/he will become as expensive as US citizen, and you will need to look for somebody else. ;-)

  5. cantor dusted. this result is perfect, his metrics were laughable, in the end all he had was a fractal of ambition and ego, all the diagonalizing he had been doing had left him totally disconnected from any kind of political coalition

  6. Cantor is the most right-wing guy in the leadership. He was the one who the Obama Admin liked the least, because he wouldn't go along with what they wanted. Cantor produced the sequester, which was designed to fail so Democrats could get more taxes, but then Republicans shot the hostage. It was Cantor who kept trying to get the rest of the leadership to pay attention to what the conservatives wanted.

    Ultimately, he lost not because he was too liberal on budget or other issues, but because Republicans don't trust the leadership on immigration. Mickey Kaus was a one-man Brat promoter. He previously ran on the issue himself and got a surprisingly high percentage of the vote against Barbara Boxer in the California primary.

  7. Is the GOP moving to the left, or is it being occupied by Tea Party?

  8. "I think that Cantor is a smart and sensible chap and I could disagree with some beliefs of Brat but as far as I am concerned, the positive emotions outweigh the negative ones."

    Sorry my friend but Cantor was a big-government military Keynesian who had no trouble bailing out the big banks and transferring tax revenues from workers and investors to the military complex. American voters are tired of the neoconservatives who have turned on the very principles that once made their country the best and freest in the world. After two presidents who had turned their backs on the Constitution and the rule of law many voters have had enough.

  9. The Tea Party in Virginia is not exactly the Tea Party that supports Ron Paul. It did not oppose Cantor and did not aid anyone running against him.

    The problems are not caused by left over right or right over left but authoritarian over libertarian.