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
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".
A newly published 65-page-long hep-th preprint
Multiboundary Wormholes and Holographic Entanglementby 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".
Entanglement and networks of wormholes
![Entanglement and networks of wormholes]()
Reviewed by MCH
on
June 10, 2014
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