Advertisement

Journal of High Energy Physics

, 2019:69 | Cite as

Beyond toy models: distilling tensor networks in full AdS/CFT

  • Ning Bao
  • Geoffrey Penington
  • Jonathan SorceEmail author
  • Aron C. Wall
Open Access
Regular Article - Theoretical Physics
  • 8 Downloads

Abstract

We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with non­ intersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the “holographic entanglement of purification” conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.

Keywords

AdS-CFT Correspondence Gauge-gravity correspondence Conformal Field Theory 

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]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [hep-th/9711200] [INSPIRE].
  2. [2]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS jCFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].
  3. [3]
    S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP08 (2006) 045 [hep-th/0605073] [INSPIRE].
  4. [4]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
  5. [5]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
  6. [6]
    X. Dong, A. Lewkowycz and M. Rangamani, Deriving covariant holographic entanglement, JHEP11 (2016) 028 [arXiv:1607.07506] [INSPIRE].
  7. [7]
    B. Swingle, Entanglement renormalization and holography, Phys. Rev.D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
  8. [8]
    B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].
  9. [9]
    F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: toy models for the bulk/ boundary correspondence, JHEP06 (2015) 149 [arXiv:1503.06237] [INSPIRE].
  10. [10]
    P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks, JHEP11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
  11. [11]
    A.J. Ferris and D. Poulin, Tensor networks and quantum error correction, Phys. Rev. Lett.113 (2014) 030501 [arXiv:1312.4578].
  12. [12]
    T. Kohler and T. Cubitt, Toy models of holographic duality between local Hamiltonians, JHEP08 (2019) 017 [arXiv:1810.08992] [INSPIRE].
  13. [13]
    A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS jCFT, JHEP04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
  14. [14]
    Z. Yang, P. Hayden and X.-1. Qi, Bidirectional holographic codes and sub-AdS locality, JHEP01 (2016) 175 [arXiv:1510.03784] [INSPIRE].
  15. [15]
    T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality, Nature Phys.14 (2018) 573 [arXiv:1708.09393] [INSPIRE].
  16. [16]
    P. Nguyen, T. Devakul, M.G. Halbasch, M.P. Zaletel and B. Swingle, Entanglement of purification: from spin chains to holography, JHEP01 (2018) 098 [arXiv:1709.07424] [INSPIRE].
  17. [17]
    X. Dong, The gravity dual of Renyi entropy, Nature Commun. 7 (2016) 12472 [arXiv:1601.06788] [INSPIRE].
  18. [18]
    P. Hayden, B. Swingle and M. Walter, One-shot information theory in quantum field theory, unpublished.Google Scholar
  19. [19]
    G. Vidal, Entanglement renormalization, Phys. Rev. Lett.99 (2007) 220405 [cond-mat/0512165] [INSPIRE].
  20. [20]
    F. Verstraete, V. Murg and J. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys.57 (2008) 143 [arXiv:0907.2796].
  21. [21]
    F. Verstraete and J.I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, cond-mat/0407066.
  22. [22]
    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
  23. [23]
    N. Engelhardt and A.C. Wall, Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime, JHEP01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
  24. [24]
    M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys.352 (2017) 407 [arXiv:1604.00354] [INSPIRE].
  25. [25]
    P. Hayden and A. Winter, Communication cost of entanglement transformations, Phys. Rev.A 67 (2003) 012326 [quant-ph/0204092].
  26. [26]
    B. Czech, P. Hayden, N. Lashkari and B. Swingle, The information theoretic interpretation of the length of a curve, JHEP06 (2015) 157 [arXiv:1410.1540] [INSPIRE].
  27. [27]
    H. Wilming and J. Eisert, Single-shot holographic compression from the area law, Phys. Rev. Lett.122 (2019) 190501 [arXiv:1809.10156] [INSPIRE].
  28. [28]
    M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev.D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
  29. [29]
    C. Akers and P. Rath, Holographic Renyi entropy from quantum error correction, JHEP05 (2019) 052 [arXiv:1811.05171] [INSPIRE].
  30. [30]
    X. Dong, D. Harlow and D. Marolf, Flat entanglement spectra in fixed-area states of quantum gravity, JHEP10 (2019) 240 [arXiv:1811.05382] [INSPIRE].
  31. [31]
    C. Fuchs and J. van de Graaf, Cryptographic distinguishability measures for quantum-mechanical states, IEEE Trans. Inform. Theory45 (1999) 1216 [quant-ph/9712042].
  32. [32]
    G.C. Dorsch, S.J. Huber, T. Konstandin and J.M. No, A second Higgs doublet in the early universe: baryogene sis and gravitational waves, J CAP05 (2017) 052 [arXiv:1611.05874] [INSPIRE].
  33. [33]
    D. Harlow, The Ryu- Takayanagi formula from quantum error correction, Commun. Math. Phys.354 (2017) 865 [arXiv:1607.03901] [INSPIRE].
  34. [34]
    Y.Y. Shi, L.M. Duan and G. Vidal, Classical simulation of quantum many-body systems with a tree tensor network, Phys. Rev.A 74 (2006) 022320 [quant-ph/0511070] [INSPIRE].
  35. [35]
    L. Drescher and O. Fawzi, On simultaneous min-entropy smoothing, Proc. ISIT (2013) 161 [arXiv:1312.7642].
  36. [36]
    X. Dong, D. Harlow and A.C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Phys. Rev. Lett.117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
  37. [37]
    J. Cotler, P. Hayden, G. Penington, G. Salton, B. Swingle and M. Walter, Entanglement wedge reconstruction via universal recovery channels, Phys. Rev. X 9 (2019) 031011 [arXiv:1704.05839] [INSPIRE].
  38. [38]
    P. Hayden and G. Penington, Learning the alpha-bits of black holes, arXiv:1807.06041 [INSPIRE].
  39. [39]
    T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP07 (2017) 151 [arXiv:1704.05464] [INSPIRE].
  40. [40]
    P. Hayden and A. Winter, Weak decoupling duality and quantum identification, IEEE Trans. Inform. Theory58 (2012) 4914 [arXiv:1003.4994].
  41. [41]
    P. Hayden and G. Penington, Approximate quantum error correction revisited: introducing the alpha-bit, arXiv:1706.09434.
  42. [42]
    A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic represe ntation of local bulk operators, Phys. Rev.D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].
  43. [43]
    B.M. Terhal, M. Horodecki, D.W. Leung and D.P. DiVincenzo, The entangl ement of purification, J. Math. Phys.43 (2002) 4286 [quant-ph/0202044].
  44. [44]
    G. Vidal, On the characterization of entanglement, J. Mod. Opt.47 (2000) 355 [quant-ph/9807077] [INSPIRE].
  45. [45]
    N. Bao, Minimal purifications, wormhole geometries and the complexity=action proposal, arXiv:1811.03113 [INSPIRE].
  46. [46]
    N. Bao and I.F. Halpern, Holographic inequalities and entanglement of purification, JHEP03 (2018) 006 [arXiv:1710.07643] [INSPIRE].
  47. [47]
    N. Bao and I.F. Halpern, Conditional and multipartite entanglements of purification and holography, Phys. Rev.D 99 (2019) 046010 [arXiv:1805.00476] [INSPIRE].
  48. [48]
    K. Umemoto and Y. Zhou, Entanglement of purification for multipartite state s and its holographic dual, JHEP10 (2018) 152 [arXiv:1805.02625] [INSPIRE].
  49. [49]
    N. Bao, A. Chatwin-Davies and G.N. Remmen, Entanglement of purification and multiboundary wormhol e geometri es, JHEP02 (2019) 110 [arXiv:1811.01983] [INSPIRE].
  50. [50]
    M. Miyaji and T. Takayanagi, Surface/ state correspondence as a generalized holography, PTEP2015 (2015) 073B03 [arXiv:1503.03542] [INSPIRE].
  51. [51]
    X.-L. Qi, Z. Yang and Y.-Z. You, Holographic coherent states from random t ensor networks, JHEP08 (2017) 060 [arXiv:1703.06533] [INSPIRE].
  52. [52]
    M. Han and S. Huang, Discrete gravity on random tensor network and holographic Renyi entropy, JHEP11 (2017) 148 [arXiv:1705.01964] [INSPIRE].
  53. [53]
    A. Almheiri, X. Dong and B. Swingle, Linearity of holographic entanglement entropy, JHEP02 (2017) 074 [arXiv:1606.04537] [INSPIRE].
  54. [54]
    N. Bao and H. Ooguri, Distinguishability of black hole microstates, Phys. Rev.D 96 (2017) 066017 [arXiv:1705.07943] [INSPIRE].
  55. [55]
    M. Banados, C. Teitelboim and J. Zanelli, Black hole entropy and the dimensional continuation of the Gauss-Bonnet theorem, Phys. Rev. Lett.72 (1994) 957 [gr-qc/9309026] [INSPIRE].
  56. [56]
    S. Carlip and C. Teitelboim, Aspects of black hole quantum mechanics and thermodynamics in (2 +I)-dimensions, Phys. Rev.D 51 (1995) 622 [gr-qc/9405070] [INSPIRE].
  57. [57]
    D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entrop y, JHEP06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
  58. [58]
    N. Bao, C. Cao, S. Fischetti and C. Keeler, Towards bulk metric reconstruction from extremal area variations, Class. Quant. Grav.36 (2019) 185002 [arXiv:1904.04834] [INSPIRE].
  59. [59]
    D. Marolf and A.C. Wall, Eternal black holes and superselection in AdS/CFT, Class. Quant. Grav.30 (2013) 025001 [arXiv:1210.3590] [INSPIRE].
  60. [60]
    A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
  61. [61]
    A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An apologia for firewalls, JHEP09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
  62. [62]
    S.D. Mathur, The information paradox: a pedagogical introduction, Class. Quant. Grav.26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
  63. [63]
    S.L. Braunstein, S. Pirandola and K. Zyczkowski, Better late than never: information retrieval from black holes, Phys. Rev. Lett.110 (2013) 101301 [arXiv:0907.1190] [INSPIRE].
  64. [64]
    K. Papadodimas and S. Raju, State-d ependent bulk-boundary maps and black hole complementarity, Phys. Rev.D 89 (2014) 086010 [arXiv:1310.6335] [INSPIRE].
  65. [65]
    J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys.61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
  66. [66]
    B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a density matrix, Class. Qua nt. Grav.29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].
  67. [67]
    V.E. Hubeny, Extremal surfaces as bulk probes in AdS/ CFT, JHEP07 (2012) 093 [arXiv:1203.1044] [INSPIRE].
  68. [68]
    N. Engelhardt and A.C. Wall, Extremal surface barriers, JHEP03 (2014) 068 [arXiv:1312.3699] [INSPIRE].
  69. [69]
    V. Balasubramanian, B.D. Chowdhury, B. Czech and J. de Boer, Entwinement and the emergence of spacetime, JHEP01 (2015) 048 [arXiv:1406.5859] [INSPIRE].
  70. [70]
    Y. Nomura, P. Rath and N. Salzetta, Pulling the boundary into the bulk, Phys. Rev.D 98 (2018) 026010 [arXiv:1805.00523] [INSPIRE].
  71. [71]
    V. Balasubramanian, P. Hayden, A. Maloney, D. Marolf and S.F. Ross, Multiboundary wormholes and holographic entanglement, Class. Quant. Grav.31 (2014) 185015 [arXiv:1406.2663] [INSPIRE].
  72. [72]
    D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev.D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
  73. [73]
    A. Peach and S.F. Ross, Tensor network models of multiboundary wormholes, Class. Quant. Grav.34 (2017) 105011 [arXiv:1702.05984] [INSPIRE].
  74. [74]
    P. Hayden, M. Headrick and A. Maloney, Holographic mutual information is monogamous, Phys. Rev.D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].
  75. [75]
    N. Bao, S. Nezami, H. Ooguri, B. Stoica, J. Sully and M. Walter, The holographic entropy cone, JHEP09 (2015) 130 [arXiv:1505.07839] [INSPIRE].
  76. [76]
    S.X. Cui, P. Hayden, T. He, M. Headrick, B. Stoica and M. Walter, Bit threads and holographic monogamy, arXiv:1808.05234 [INSPIRE].
  77. [77]
    V.E. Hubeny, Bulk locality and cooperative flows, JHEP12 (2018) 068 [arXiv:1808.05313] [INSPIRE].
  78. [78]
    C.A. Ag6n, J. De Boer and J.F. Pedraza, Geometric aspects of holographic bit threads, JHEP 05 (2019) 075 [arXiv:1811.08879] [INSPIRE].
  79. [79]
    O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
  80. [80]
    A. Mollabashi, N. Shiba and T. Takayanagi, Entanglement between two interacting CFTs and generalized holographic entanglement entropy, JHEP04 (2014) 185 [arXiv:1403.1393] [INSPIRE].
  81. [81]
    A. Karch and C.F. Uhlemann, Holographic entanglement entropy and the internal space, Phys. Rev.D 91 (2015) 086005 [arXiv:1501.00003] [INSPIRE].
  82. [82]
    X.-1. Qi and Z. Yang, Space-time random tensor networks and holographic duality, arXiv:1801.05289 [INSPIRE].
  83. [83]
    A. May, Tensor networks for dynamic spacetimes, JHEP06 (2017) 118 [arXiv:1611.06220] [INSPIRE].
  84. [84]
    A.C. Wall, Maximin surfaces and the strong subadditivity of the covariant holographic entanglement entropy, Class. Quant. Grav.31 (2014) 225007 [arXiv:1211.3494] [INSPIRE].
  85. [85]
    Y. Chen, X. Dong, A. 1ewkowycz and X.-1. Qi, Modular flow as a disentangler, JHEP12 (2018) 083 [arXiv:1806.09622] [INSPIRE].
  86. [86]
    T. Faulkner, M. Li and H. Wang, A modular toolkit for bulk reconstruction, JHEP04 (2019) 119 [arXiv:1806.10560] [INSPIRE].
  87. [87]
    R. Bousso, Holography in general space-time s, JHEP06 (1999) 028 [hep-th/9906022] [INSPIRE].
  88. [88]
    F. Sanches and S.J. Weinberg, Holographic entanglement entropy conjecture for general spacetimes, Phys. Rev.D 94 (2016) 084034 [arXiv:1603.05250] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Berkeley Center for Theoretical PhysicsUniversity of CaliforniaBerkeleyU.S.A.
  2. 2.Stanford Institute for Theoretical PhysicsStanford UniversityStanfordU.S.A.
  3. 3.Centre forM athenwtical SciencesCamhridge UniverrtityCamhridgeU.K.

Personalised recommendations