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Entanglement, tensor networks and black hole horizons

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Abstract

We elaborate on a previous proposal by Hartman and Maldacena on a tensor network which accounts for the scaling of the entanglement entropy in a system at a finite temperature. In this construction, the ordinary entanglement renormalization flow given by the class of tensor networks known as the Multi Scale Entanglement Renormalization Ansatz (MERA), is supplemented by an additional entanglement structure at the length scale fixed by the temperature. The network comprises two copies of a MERA circuit with a fixed number of layers and a pure matrix product state which joins both copies by entangling the infrared degrees of freedom of both MERA networks. The entanglement distribution within this bridge state defines reduced density operators on both sides which cause analogous effects to the presence of a black hole horizon when computing the entanglement entropy at finite temperature in the AdS/CFT correspondence. The entanglement and correlations during the thermalization process of a system after a quantum quench are also analyzed. To this end, a full tensor network representation of the action of local unitary operations on the bridge state is proposed. This amounts to a tensor network which grows in size by adding succesive layers of bridge states. Finally, we discuss on the holographic interpretation of the tensor network through a notion of distance within the network which emerges from its entanglement distribution.

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Notes

  1. An alternative way to find the canonical form of an MPS representation can be found in [33].

  2. In this paper, we will be mainly focused in the \(D=2\) case, for which the Eq. (2.13) reduces to, \(S_{A}=\mathrm{Length}(\gamma _{A})/4G^{(3)}_N\).

  3. In the following we use the notation \(\lbrace i \rbrace = \lbrace i_1,\ldots ,i_N\rbrace \), \(\lbrace j \rbrace = \lbrace j_1,\ldots ,j_N\rbrace \) and \(\lbrace i j\rbrace = \lbrace i \rbrace \cup \lbrace j \rbrace \). In case \(d=2\), the state can be written as \(\vert \Psi \rangle = \left( \frac{1}{\sqrt{2}}\, (\vert 0 0 \rangle + \vert 1 1 \rangle )\right) ^{\otimes N}\), i.e the purified MPS state resembles a system of \(N\) Bell pairs shared between the \(\left\{ i \right\} \) and \(\left\{ j \right\} \) sites.

  4. In the gravitational terminology this is a fiducial observer FIDO for which the local temperature diverges near the horizon.

  5. As has been pointed out in [16, 17], each tensor may represent the wavefunction of a region of AdS radius, so this notion of proximity could be appropriate between regions which size is similar to the AdS radius but not for smaller distances.

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Acknowledgments

The authors thank S. R. Clark, J. Rodríguez-Laguna and E. da Silva for giving very fruitful suggestions on the manuscript. JMV thanks the hospitality of Germán Sierra and Esperanza López at the Institute of Theoretical Physics CSIC-UAM in Madrid. This work has been funded by Ministerio de Economía y Competitividad Project No. FIS2012-30625.

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  1. Universidad Politécnica de Cartagena, C/Dr Fleming S/N, 30202, Cartagena, Spain

    J. Molina-Vilaplana & J. Prior

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Molina-Vilaplana, J., Prior, J. Entanglement, tensor networks and black hole horizons. Gen Relativ Gravit 46, 1823 (2014). https://doi.org/10.1007/s10714-014-1823-y

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