Abstract
We provide an explicit connection between the differential generation of entan-glement entropy in a tensor network representation of the ground states of two field theories, and a geometric description of these states based on the Fisher information metric. We show how the geometrical description remains invariant despite there is an irreducible gauge freedom in the definition of the tensor network. The results might help to understand how spacetimes may emerge from distributions of quantum states, or more concretely, from the structure of the quantum entanglement concomitant to those distributions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement ‘thermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].
T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from entanglement in holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165] [INSPIRE].
J. Haegeman, T.J. Osborne, H. Verschelde and F. Verstraete, Entanglement renormalization for quantum fields in real space, Phys. Rev. Lett. 110 (2013) 100402 [arXiv:1102.5524] [INSPIRE].
B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
M. Nozaki, S. Ryu and T. Takayanagi, Holographic geometry of entanglement renormalization in quantum field theories, JHEP 10 (2012) 193 [arXiv:1208.3469] [INSPIRE].
A. Mollabashi, M. Nozaki, S. Ryu and T. Takayanagi, Holographic Geometry of cMERA for Quantum Quenches and Finite Temperature, JHEP 03 (2014) 098 [arXiv:1311.6095] [INSPIRE].
J. Molina-Vilaplana and J. Prior, Entanglement, tensor networks and black hole horizons, Gen. Rel. Grav. 46 (2014) 1823 [arXiv:1403.5395] [INSPIRE].
T. Hartman and J. Maldacena, Time evolution of entanglement entropy from black hole interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
H. Matsueda, M. Ishihara and Y. Hashizume, Tensor network and a black hole, Phys. Rev. D 87 (2013) 066002 [arXiv:1208.0206] [INSPIRE].
M. Miyaji and T. Takayanagi, Surface/state correspondence as a generalized holography, PTEP 2015 (2015) 073B03 [arXiv:1503.03542] [INSPIRE].
L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].
V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys. 74 (2002) 197 [INSPIRE].
S. Amari and H. Nagaoka, Methods of information geometry, American Mathematical Society, U.S.A. (2000).
A.M. Perelmonov, Coherent states for arbitrary Lie group, Commun. Math. Phys. 26 (1972) 222 [math-ph/0203002].
M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, cMERA as surface/state correspondence in AdS/CFT, arXiv:1506.01353 [INSPIRE].
F. Verstraete and J.I. Cirac, Continuous matrix product states for quantum fields, Phys. Rev. Lett. 104 (2010) 190405 [arXiv:1002.1824] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1503.07699
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Molina-Vilaplana, J. Information geometry of entanglement renormalization for free quantum fields. J. High Energ. Phys. 2015, 2 (2015). https://doi.org/10.1007/JHEP09(2015)002
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2015)002