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Holographic duality from random tensor networks

  • Patrick Hayden
  • Sepehr Nezami
  • Xiao-Liang Qi
  • Nathaniel Thomas
  • Michael Walter
  • Zhao Yang
Open Access
Regular Article - Theoretical Physics

Abstract

Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.

Keywords

AdS-CFT Correspondence Black Holes in String Theory Gauge-gravity correspondence Random Systems 

Notes

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.

References

  1. [1]
    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].ADSGoogle Scholar
  2. [2]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    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].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149 [arXiv:1503.06237] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282 [INSPIRE].
  8. [8]
    Z. Yang, P. Hayden and X.-L. Qi, Bidirectional holographic codes and sub-AdS locality, JHEP 01 (2016) 175 [arXiv:1510.03784] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    X. Dong, The Gravity Dual of Renyi Entropy, Nature Commun. 7 (2016) 12472 [arXiv:1601.06788] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    B. Collins and I. Nechita, Random matrix techniques in quantum information theory, J. Math. Phys. 57 (2016) 015215 [arXiv:1509.04689].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    B. Collins, I. Nechita and K. Życzkowski, Random graph states, maximal flow and fuss-catalan distributions, J. Phys. A 43 (2010) 275303 [arXiv:1003.3075].ADSMathSciNetMATHGoogle Scholar
  15. [15]
    B. Collins, I. Nechita and K. Życzkowski, Area law for random graph states, J. Phys. A 46 (2013) 305302 [arXiv:1302.0709].ADSMathSciNetMATHGoogle Scholar
  16. [16]
    M.B. Hastings, Random MERA States and the Tightness of the Brandao-Horodecki Entropy Bound, arXiv:1505.06468.
  17. [17]
    B. Collins, C.E. Gonzalez Guillen and D. Pérez García, Matrix product states, random matrix theory and the principle of maximum entropy, Commun. Math. Phys. 320 (2013) 663 [arXiv:1201.6324].
  18. [18]
    S.X. Cui, M.H. Freedman, O. Sattath, R. Stong and G. Minton, Quantum Max-flow/Min-cut, J. Math. Phys. 57 (2016) 062206 [arXiv:1508.04644] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].
  20. [20]
    F. Verstraete and J.I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, cond-mat/0407066 [INSPIRE].
  21. [21]
    A.W. Harrow, The Church of the Symmetric Subspace, arXiv:1308.6595.
  22. [22]
    N. Bao, S. Nezami, H. Ooguri, B. Stoica, J. Sully and M. Walter, The Holographic Entropy Cone, JHEP 09 (2015) 130 [arXiv:1505.07839] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, Bulk and Transhorizon Measurements in AdS/CFT, JHEP 10 (2012) 165 [arXiv:1201.3664] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    S. El-Showk and K. Papadodimas, Emergent Spacetime and Holographic CFTs, JHEP 10 (2012) 106 [arXiv:1101.4163] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    N. Benjamin, S. Kachru, C.A. Keller and N.M. Paquette, Emergent space-time and the supersymmetric index, JHEP 05 (2016) 158 [arXiv:1512.00010] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
  30. [30]
    J. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888 [cond-mat/9403051].
  32. [32]
    E. Lubkin and T. Lubkin, Average quantal behavior and thermodynamic isolation, Int. J. Theor. Phys. 32 (1993) 933.MathSciNetCrossRefGoogle Scholar
  33. [33]
    J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    M.A. Nielsen and D. Poulin, Algebraic and information-theoretic conditions for operator quantum error correction, Phys. Rev. A 75 (2007) 064304 [quant-ph/0506069].
  35. [35]
    M. Leifer, N. Linden and A. Winter, Measuring polynomial invariants of multiparty quantum states, Phys. Rev. A 69 (2004) 052304 [quant-ph/0308008].
  36. [36]
    C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
  38. [38]
    D.V. Fursaev, Proof of the holographic formula for entanglement entropy, JHEP 09 (2006) 018 [hep-th/0606184] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    T. Faulkner, The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
  40. [40]
    N. Dutil and P. Hayden, One-shot multiparty state merging, arXiv:1011.1974.
  41. [41]
    L. Onsager, Crystal statistics. 1. A Two-dimensional model with an order disorder transition, Phys. Rev. 65 (1944) 117 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    B.M. McCoy and T.T. Wu, Theory of toeplitz determinants and the spin correlations of the two-dimensional ising model. IV, Phys. Rev. 162 (1967) 436.Google Scholar
  43. [43]
    W. Burton, N. Cabrera and F. Frank, Role of dislocations in crystal growth, Nature 163 (1949) 398.ADSCrossRefGoogle Scholar
  44. [44]
    W. Burton, N. Cabrera and F. Frank, The growth of crystals and the equilibrium structure of their surfaces, Phil. Trans. Roy. Soc. Lond. A 243 (1951) 299.ADSMathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    D.P. DiVincenzo et al., Entanglement of assistance, in Quantum Computing and Quantum Communications, Springer, Lect. Notes. Comput. Sci. 1509 (1999) 247 [quant-ph/9803033] [INSPIRE].
  46. [46]
    J.A. Smolin, F. Verstraete and A. Winter, Entanglement of assistance and multipartite state distillation, Phys. Rev. A 72 (2005) 052317 [quant-ph/0505038].
  47. [47]
    N. Dutil and P. Hayden, Assisted entanglement distillation, Quant. Inform. Comput. 11 (2011) 496 [arXiv:1011.1972].MathSciNetMATHGoogle Scholar
  48. [48]
    B. Czech, P. Hayden, N. Lashkari and B. Swingle, The Information Theoretic Interpretation of the Length of a Curve, JHEP 06 (2015) 157 [arXiv:1410.1540] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    M. Horodecki, J. Oppenheim and A. Winter, Quantum state merging and negative information, Commun. Math. Phys. 269 (2007) 107 [quant-ph/0512247].
  50. [50]
    N. Dutil, Multiparty quantum protocols for assisted entanglement distillation, Ph.D. Thesis, McGill University (2011), arXiv:1105.4657.
  51. [51]
    J. Renes, R. Blume-Kohout, C. Caves and A. Scott, Symmetric informationally complete quantum measurements, J. Math. Phys. 45 (2004) 2171 [quant-ph/0310075].
  52. [52]
    D. Gottesman, Stabilizer codes and quantum error correction, Ph.D. Thesis, California Institute of Technology (1997).Google Scholar
  53. [53]
    H.F. Chau, Unconditionally secure key distribution in higher dimensions by depolarization, IEEE Trans. Inform. Theor. 51 (2005) 1451 [quant-ph/0405016].
  54. [54]
    D. Gross and M. Walter, Stabilizer information inequalities from phase space distributions, J. Math. Phys. 54 (2013) 082201 [arXiv:1302.6990].ADSMathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    N. Linden, F. Matus, M. B. Ruskai and A. Winter, The Quantum Entropy Cone of Stabiliser States, in 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013), LIPICS 22 (2013) 270 [arXiv:1302.5453].MathSciNetGoogle Scholar
  56. [56]
    M. Headrick and T. Takayanagi, A Holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D 76 (2007) 106013 [arXiv:0704.3719] [INSPIRE].ADSMathSciNetGoogle Scholar
  57. [57]
    S. Nezami and M. Walter, Multipartite Entanglement in Stabilizer Tensor Networks, arXiv:1608.02595 [INSPIRE].
  58. [58]
    M.B. Hastings, The Asymptotics of Quantum Max-Flow Min-Cut, arXiv:1603.03717.
  59. [59]
    B. Schumacher and M.A. Nielsen, Quantum data processing and error correction, Phys. Rev. A 54 (1996) 2629 [quant-ph/9604022] [INSPIRE].
  60. [60]
    B. Schumacher and M.D. Westmoreland, Approximate quantum error correction, Quant. Inform. Process. 1 (2002) 5 [quant-ph/0112106].

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Patrick Hayden
    • 1
  • Sepehr Nezami
    • 1
  • Xiao-Liang Qi
    • 1
  • Nathaniel Thomas
    • 1
  • Michael Walter
    • 1
  • Zhao Yang
    • 1
  1. 1.Stanford Institute for Theoretical Physics, Department of PhysicsStanford UniversityStanfordU.S.A.

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