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
An alternative way to find the canonical form of an MPS representation can be found in [33].
In this paper, we will be mainly focused in the \(D=2\) case, for which the Eq. (2.13) reduces to, \(S_{A}=\mathrm{Length}(\gamma _{A})/4G^{(3)}_N\).
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.
In the gravitational terminology this is a fiducial observer FIDO for which the local temperature diverges near the horizon.
References
Aharony, O., Gubser, S.S., Maldacena, J.M., Ooguri, H., Oz, Y.: Large N field theories. String theory and gravity. Phys. Rep. 323, 183 (2000). hep-th/9905111
Maldecena, J.M.: The Large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998). hep-th/9711200
Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B 428, 105 (1998). hep-th/9802109
Witten, E.: Anti-de Sitter space, thermal phase transition, and confinement in gauge theories. Adv. Theor. Math. Phys. 2, 505 (1998). hep-th/9803131
Douglas, M.R., Mazzucato, L., Razamat, S.S.: Holographic dual of free field theory. Phys. Rev. D 83, 071701 (2011). arXiv:1011.4926 [hep-th]
de Boer, J., Verlinde, E., Verlinde, H.: On the Holographic renormalization group. JHEP 0008, 003 (2000). hep-th/9912012
Fukuma, M., Matsuura, S., Sakai, T.: Holographic renormalization group. Prog. Theor. Phys. 109, 489–562 (2003). hep-th/0212314
White, S.R.: Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863 (1992)
Perez-Garcia, D., Verstraete, F., Wolf, M.M., Cirac, J.I.: Matrix Product State Representations. Quantum Inf. Comput. 7, 401 (2007). quant-ph/0608197
Verstraete, F., Cirac, J.I., Murg, V.: Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems. Adv. Phys. 57, 143 (2008). arXiv:0907.2796 [quant-ph]
Vidal, G.: Entanglement renormalization. Phys. Rev. Lett. 99, 220405 (2007). cond-mat/0512165 [cond-mat.str-el]
Levin, M., Nave, C.P.: Tensor renormalization group approach to 2D classical lattice models, Phys. Rev. Lett. 99, 120601 (2007). cond-mat/0611687 [cond-mat.stat-mech]
Gu, Z., Wen, X.-G.: Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order. Phys. Rev. B. 80, 155131 (2009). arXiv:0903.1069 [cond-mat.str-el]
Denny, S.J., Biamonte, J.D., Jaksch, D., Clark, S.R.: Algebraically contractible topological tensor network states. J. Phys. A Math. Theor. 45, 015309 (2012). arXiv:1108.0888 [quant-ph]
Evenbly, G., Vidal, G.: Tensor network states and geometry. J. Stat. Phys. 145, 891–918 (2011). arXiv:1106.1082 [quant-ph]
Swingle, B.: Entanglement Renormalization and Holography. Phys. Rev. D 86, 065007 (2012). arXiv:0905.1317 [cond-mat.str-el]
Hartman, T., Maldacena, J.: Time evolution of entanglement entropy from black hole interiors. JHEP 05, 014 (2013). arXiv:1303.1080 [hep-th]
Molina-Vilaplana, J.: Holographic Geometries of one-dimensional gapped quantum systems from Tensor Network States. JHEP 1305, 024 (2013) arXiv:1210.6759 [hep-th]
Matsueda, H., Ishihara, M., Hashizume, Y.: Tensor network and black hole. Phys. Rev. D 87, 066002 (2013). arXiv:1208.0206 [hep-th]
Mollabashi, A., Nozaki, M., Ryu, S., Takayanagi, T.: Holographic geometry of cMERA for quantum quenches and finite temperature. arXiv:1311.6095 [hep-th]
Molina-Vilaplana, J., Sodano, P.: Holographic view on quantum correlations and mutual information between disjoint blocks of a quantum critical system. JHEP. 10, 011 (2011). arXiv:1108.1277 [quant-ph]
Balasubramanian, V., McDermott, M.B., Van Raamsdonk, M.: Momentum-space entanglement and renormalization in quantum field theory. Phys. Rev. D 86, 045014 (2012). arXiv:1108.3568 [hep-th]
Ishihara, M., Lin, F.-L., Ning, B.: Refined holographic entanglement entropy for the AdS solitons and AdS black holes. Nucl. Phys. B 872, 392426 (2013) arXiv:1203.6153 [hep-th]
Nozaki, M., Ryu, S., Takayanagi, T.: Holographic geometry of entanglement renormalization in quantum field theories. JHEP 10, 193 (2012). arXiv:1208.3469 [hep-th]
Haegeman, J., Osborne, T.J., Verschelde, H., Verstraete, F.: Entanglement renormalization for quantum fields in real space. Phys. Rev. Lett. 110, 100402 (2013) arXiv:1102.5524 [hep-th]
Swingle, B.: Constructing holographic spacetimes using entanglement renormalization (2012). arXiv:1209.3304 [hep-th]
Singh, S., Vidal, G.: Symmetry protected entanglement renormalization, arXiv:1303.6716 [cond-mat.str-el]
Israel, W.: Thermofield dynamics of black holes. Phys. Lett. A 57, 107 (1976)
Maldacena, J.M.: Eternal black holes in anti-de Sitter. JHEP 0304, 021 (2003). hep-th/0106112
Van Raamsdonk, M.: Comments on quantum gravity and entanglement. arXiv:0907.2939 [hep-th]
Van Raamsdonk, M.: Building up spacetime with quantum entanglement. Gen. Relativ. Gravit. 42, 2323 (2010)
Van Raamsdonk, M.: Building up spacetime with quantum entanglement. Int. J. Mod. Phys. D 19, 2429 (2010) arXiv:1005.3035 [hep-th]
Vidal, G.: Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003) arXiv:quant-ph/0301063
Ryu, S., Takayanagi, T.: Holographic derivation of entanglement entropy from the antide Sitter space/conformal field theory correspondence. Phys. Rev. Lett. 96, 181602 (2006). hep-th/0603001
Ryu, S., Takayanagi, T.: Aspects of holographic entanglement entropy. JHEP 0608, 045 (2006). hep-th/0605073
Nishioka, T., Ryu, S., Takayanagi, T.: J. Phys. A 42, 504008 (2009). arXiv:0905.0932 [hep-th]
Takayanagi, T.: Entanglement entropy from a holographic viewpoint. Class. Quantum Grav. 29, 153001 (2012). arXiv:1204.2450 [gr-qc]
Verstraete, F., Garcia-Ripoll, J.J., Cirac, J.I.: Matrix product density operators: simulation of finite-T and dissipative systems. Phys. Rev. Lett. 93, 207204 (2004). arXiv:cond-mat/0406426 [cond-mat.other]
Brown, J.D., Henneaux, M.: Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity. Commun. Math. Phys. 104(2), 177–352 (1986)
Headrick, M.: Entanglement Renyi entropies in holographic theories. Phys. Rev D82, 126010 (2010) arXiv:1006.0047 [hep-th]
Calabrese, P., Cardy, J.L.: Evolution of entanglement entropy in one-dimensional systems. J. Stat. Mech. P04010 (2005) cond-mat/0503393
Wolf, M.M., Verstraete, F., Hastings, M.B., Cirac, J.I.: Area laws in quantum systems: mutual information and correlations. Phys. Rev. Lett. 100, 070502 (2008) arXiv:0704.3906 [quant-ph]
Fannes, M., Nachtergaele, B., Werner, R.F.: Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144, 443 (1992)
Ostlund, S., Rommer, S.: Thermodynamic limit of density matrix renormalization. Phys. Rev. Lett. 75, 3537 (1995)
Rommer, S., Ostlund, S.: Class of ansatz wave functions for one-dimensional spin systems and their relation to the density matrix renormalization group. Phys. Rev. B 55, 2164 (1997)
Vidal, G.: A class of quantum many-body states that can be efficiently simulated. Phys. Rev. Lett. 101, 110501 (2008) arXiv:quant-ph/0610099
Rizzi, M., Montangero, S., Vidal, G.: Simulation of time evolution with the MERA. Phys. Rev. A 77, 052328 (2008). arXiv:0706.0868 [quant-ph]
Maldacena, J., Susskind, L.: Cool horizons for entangled black holes. Fortsch. Phys. 61, 781–811 (2013). arXiv:1306.0533 [hep-th]
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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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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DOI: https://doi.org/10.1007/s10714-014-1823-y