Journal of High Energy Physics

, 2018:48 | Cite as

Holographic spacetimes as quantum circuits of path-integrations

  • Tadashi TakayanagiEmail author
Open Access
Regular Article - Theoretical Physics


We propose that holographic spacetimes can be regarded as collections of quantum circuits based on path-integrals. We relate a codimension one surface in a gravity dual to a quantum circuit given by a path-integration on that surface with an appropriate UV cut off. Our proposal naturally generalizes the conjectured duality between the AdS/CFT and tensor networks. This largely strengthens the surface/state duality and also provides a holographic explanation of path-integral optimizations. For static gravity duals, our new framework provides a derivation of the holographic complexity formula given by the gravity action on the WDW patch. We also propose a new formula which relates numbers of quantum gates to surface areas, even including time-like surfaces, as a generalization of the holographic entanglement entropy formula. We argue the time component of the metric in AdS emerges from the density of unitary quantum gates in the dual CFT. Our proposal also provides a heuristic understanding how the gravitational force emerges from quantum circuits.


AdS-CFT Correspondence Gauge-gravity correspondence Holography and condensed matter physics (AdS/CMT) 


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.


  1. [1]
    G. ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308 (1993) 284 [gr-qc/9310026] [INSPIRE].
  2. [2]
    L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    D. Bigatti and L. Susskind, TASI lectures on the holographic principle, in Strings, branes and gravity. Proceedings, Theoretical Advanced Study Institute, TASI99, Boulder, U.S.A., May 31 – June 25, 1999, pp. 883–933, hep-th/0002044 [INSPIRE].
  4. [4]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    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].CrossRefzbMATHGoogle Scholar
  6. [6]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  10. [10]
    M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, Lect. Notes Phys. 931 (2017) pp.1 [arXiv:1609.01287] [INSPIRE].
  11. [11]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic Entanglement Entropy: An Overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  12. [12]
    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].Google Scholar
  13. [13]
    M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys. 352 (2017) 407 [arXiv:1604.00354] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    J.I. Cirac and F. Verstraete, Renormalization and tensor product states in spin chains and lattices, J. Phys. A 42 (2009) 504004 [arXiv:0910.1130] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  18. [18]
    G. Evenbly and G. Vidal, Tensor network states and geometry, J. Stat. Phys. 145 (2011) 891 [arXiv:1106.1082].MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    G. Vidal, A class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101 (2008) 110501 [quant-ph/0610099].
  20. [20]
    G. Vidal, Entanglement Renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165] [INSPIRE].
  21. [21]
    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].CrossRefGoogle Scholar
  22. [22]
    J. Cotler, M.R. Mohammadi Mozaffar, A. Mollabashi and A. Naseh, Renormalization Group Circuits for Weakly Interacting Continuum Field Theories, arXiv:1806.02831 [INSPIRE].
  23. [23]
    J. Cotler, M.R. Mohammadi Mozaffar, A. Mollabashi and A. Naseh, Entanglement Renormalization for Weakly Interacting Fields, arXiv:1806.02835 [INSPIRE].
  24. [24]
    M. Nozaki, S. Ryu and T. Takayanagi, Holographic Geometry of Entanglement Renormalization in Quantum Field Theories, JHEP 10 (2012) 193 [arXiv:1208.3469] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    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].CrossRefGoogle Scholar
  26. [26]
    C. Beny, Causal structure of the entanglement renormalization ansatz, New J. Phys. 15 (2013) 023020 [arXiv:1110.4872] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  27. [27]
    B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral Geometry and Holography, JHEP 10 (2015) 175 [arXiv:1505.05515] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    B. Czech, L. Lamprou, S. McCandlish and J. Sully, Tensor Networks from Kinematic Space, JHEP 07 (2016) 100 [arXiv:1512.01548] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    N. Bao, C. Cao, S.M. Carroll and A. Chatwin-Davies, de Sitter Space as a Tensor Network: Cosmic No-Hair, Complementarity and Complexity, Phys. Rev. D 96 (2017) 123536 [arXiv:1709.03513] [INSPIRE].
  30. [30]
    N. Bao et al., Consistency conditions for an AdS multiscale entanglement renormalization ansatz correspondence, Phys. Rev. D 91 (2015) 125036 [arXiv:1504.06632] [INSPIRE].MathSciNetGoogle Scholar
  31. [31]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    X.-L. Qi and Z. Yang, Space-time random tensor networks and holographic duality, arXiv:1801.05289 [INSPIRE].
  34. [34]
    M. Miyaji, T. Takayanagi and K. Watanabe, From path integrals to tensor networks for the AdS/CFT correspondence, Phys. Rev. D 95 (2017) 066004 [arXiv:1609.04645] [INSPIRE].MathSciNetGoogle Scholar
  35. [35]
    P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 071602 [arXiv:1703.00456] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  36. [36]
    P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT, JHEP 11 (2017) 097 [arXiv:1706.07056] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    B. Czech, Einstein Equations from Varying Complexity, Phys. Rev. Lett. 120 (2018) 031601 [arXiv:1706.00965] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  38. [38]
    A. Bhattacharyya, P. Caputa, S.R. Das, N. Kundu, M. Miyaji and T. Takayanagi, Path-Integral Complexity for Perturbed CFTs, JHEP 07 (2018) 086 [arXiv:1804.01999] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    J. Molina-Vilaplana and A. Del Campo, Complexity Functionals and Complexity Growth Limits in Continuous MERA Circuits, JHEP 08 (2018) 012 [arXiv:1803.02356] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    G. Evenbly and G. Vidal, Tensor Network Renormalization, Phys. Rev. Lett. 115 (2015) 180405 [arXiv:1412.0732].MathSciNetCrossRefGoogle Scholar
  41. [41]
    G. Evenbly and G. Vidal, Tensor network renormalization yields the multi-scale entanglement renormalization ansatz, Phys. Rev. Lett. 115 (2015) 200401 [arXiv:1502.05385].CrossRefGoogle Scholar
  42. [42]
    A. Milsted and G. Vidal, Tensor networks as path integral geometry, arXiv:1807.02501 [INSPIRE].
  43. [43]
    A. Milsted and G. Vidal, Tensor networks as conformal transformations, arXiv:1805.12524 [INSPIRE].
  44. [44]
    M. Miyaji and T. Takayanagi, Surface/State Correspondence as a Generalized Holography, PTEP 2015 (2015) 073B03 [arXiv:1503.03542] [INSPIRE].
  45. [45]
    M. Miyaji, S. Ryu, T. Takayanagi and X. Wen, Boundary States as Holographic Duals of Trivial Spacetimes, JHEP 05 (2015) 152 [arXiv:1412.6226] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Continuous Multiscale Entanglement Renormalization Ansatz as Holographic Surface-State Correspondence, Phys. Rev. Lett. 115 (2015) 171602 [arXiv:1506.01353] [INSPIRE].CrossRefGoogle Scholar
  47. [47]
    Y. Nomura, N. Salzetta, F. Sanches and S.J. Weinberg, Spacetime Equals Entanglement, Phys. Lett. B 763 (2016) 370 [arXiv:1607.02508] [INSPIRE].CrossRefzbMATHGoogle Scholar
  48. [48]
    Y. Nomura, N. Salzetta, F. Sanches and S.J. Weinberg, Toward a Holographic Theory for General Spacetimes, Phys. Rev. D 95 (2017) 086002 [arXiv:1611.02702] [INSPIRE].MathSciNetGoogle Scholar
  49. [49]
    Y. Nomura, P. Rath and N. Salzetta, Pulling the Boundary into the Bulk, Phys. Rev. D 98 (2018) 026010 [arXiv:1805.00523] [INSPIRE].MathSciNetGoogle Scholar
  50. [50]
    L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with \( T\overline{T} \), JHEP 04 (2018) 010 [arXiv:1611.03470] [INSPIRE].CrossRefzbMATHGoogle Scholar
  51. [51]
    D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].Google Scholar
  52. [52]
    L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].CrossRefGoogle Scholar
  55. [55]
    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].MathSciNetGoogle Scholar
  56. [56]
    L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries, Phys. Rev. D 94 (2016) 084046 [arXiv:1609.00207] [INSPIRE].MathSciNetGoogle Scholar
  57. [57]
    M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Distance between Quantum States and Gauge-Gravity Duality, Phys. Rev. Lett. 115 (2015) 261602 [arXiv:1507.07555] [INSPIRE].CrossRefGoogle Scholar
  58. [58]
    M. Alishahiha, Holographic Complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].MathSciNetGoogle Scholar
  59. [59]
    J.L.F. Barbon and E. Rabinovici, Holographic complexity and spacetime singularities, JHEP 01 (2016) 084 [arXiv:1509.09291] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    O. Ben-Ami and D. Carmi, On Volumes of Subregions in Holography and Complexity, JHEP 11 (2016) 129 [arXiv:1609.02514] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    S. Chapman, H. Marrochio and R.C. Myers, Complexity of Formation in Holography, JHEP 01 (2017) 062 [arXiv:1610.08063] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    D. Carmi, R.C. Myers and P. Rath, Comments on Holographic Complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    A. Reynolds and S.F. Ross, Divergences in Holographic Complexity, Class. Quant. Grav. 34 (2017) 105004 [arXiv:1612.05439] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    D. Carmi, S. Chapman, H. Marrochio, R.C. Myers and S. Sugishita, On the Time Dependence of Holographic Complexity, JHEP 11 (2017) 188 [arXiv:1709.10184] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    R.-Q. Yang, C. Niu, C.-Y. Zhang and K.-Y. Kim, Comparison of holographic and field theoretic complexities for time dependent thermofield double states, JHEP 02 (2018) 082 [arXiv:1710.00600] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    Y. Zhao, Uncomplexity and Black Hole Geometry, Phys. Rev. D 97 (2018) 126007 [arXiv:1711.03125] [INSPIRE].MathSciNetGoogle Scholar
  67. [67]
    B. Swingle and Y. Wang, Holographic Complexity of Einstein-Maxwell-Dilaton Gravity, JHEP 09 (2018) 106 [arXiv:1712.09826] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    Z. Fu, A. Maloney, D. Marolf, H. Maxfield and Z. Wang, Holographic complexity is nonlocal, JHEP 02 (2018) 072 [arXiv:1801.01137] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    L. Susskind, Why do Things Fall?, arXiv:1802.01198 [INSPIRE].
  70. [70]
    S. Bolognesi, E. Rabinovici and S.R. Roy, On Some Universal Features of the Holographic Quantum Complexity of Bulk Singularities, JHEP 06 (2018) 016 [arXiv:1802.02045] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    B. Chen, W.-M. Li, R.-Q. Yang, C.-Y. Zhang and S.-J. Zhang, Holographic subregion complexity under a thermal quench, JHEP 07 (2018) 034 [arXiv:1803.06680] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    D.S. Ageev, I. Ya. Aref’eva, A.A. Bagrov and M.I. Katsnelson, Holographic local quench and effective complexity, JHEP 08 (2018) 071 [arXiv:1803.11162] [INSPIRE].
  73. [73]
    C.A. Agón, M. Headrick and B. Swingle, Subsystem Complexity and Holography, arXiv:1804.01561 [INSPIRE].
  74. [74]
    K. Hashimoto, N. Iizuka and S. Sugishita, Thoughts on Holographic Complexity and its Basis-dependence, Phys. Rev. D 98 (2018) 046002 [arXiv:1805.04226] [INSPIRE].MathSciNetGoogle Scholar
  75. [75]
    S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part I, JHEP 06 (2018) 046 [arXiv:1804.07410] [INSPIRE].
  76. [76]
    S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part II, JHEP 06 (2018) 114 [arXiv:1805.07262] [INSPIRE].
  77. [77]
    S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Toward a Definition of Complexity for Quantum Field Theory States, Phys. Rev. Lett. 120 (2018) 121602 [arXiv:1707.08582] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  78. [78]
    R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  79. [79]
    K. Hashimoto, N. Iizuka and S. Sugishita, Time evolution of complexity in Abelian gauge theories, Phys. Rev. D 96 (2017) 126001 [arXiv:1707.03840] [INSPIRE].MathSciNetGoogle Scholar
  80. [80]
    R. Khan, C. Krishnan and S. Sharma, Circuit Complexity in Fermionic Field Theory, arXiv:1801.07620 [INSPIRE].
  81. [81]
    L. Hackl and R.C. Myers, Circuit complexity for free fermions, JHEP 07 (2018) 139 [arXiv:1803.10638] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  82. [82]
    J.M. Magán, Black holes, complexity and quantum chaos, JHEP 09 (2018) 043 [arXiv:1805.05839] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  83. [83]
    D.W.F. Alves and G. Camilo, Evolution of complexity following a quantum quench in free field theory, JHEP 06 (2018) 029 [arXiv:1804.00107] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  84. [84]
    P. Caputa and J.M. Magán, Quantum Computation as Gravity, arXiv:1807.04422 [INSPIRE].
  85. [85]
    H.A. Camargo, P. Caputa, D. Das, M.P. Heller and R. Jefferson, Complexity as a novel probe of quantum quenches: universal scalings and purifications, arXiv:1807.07075 [INSPIRE].
  86. [86]
    M. Guo, J. Hernandez, R.C. Myers and S.-M. Ruan, Circuit Complexity for Coherent States, JHEP 10 (2018) 011 [arXiv:1807.07677] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  87. [87]
    A. Bhattacharyya, A. Shekar and A. Sinha, Circuit complexity in interacting QFTs and RG flows, JHEP 10 (2018) 140 [arXiv:1808.03105] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  88. [88]
    G. Hayward, Gravitational action for space-times with nonsmooth boundaries, Phys. Rev. D 47 (1993) 3275 [INSPIRE].Google Scholar
  89. [89]
    D. Brill and G. Hayward, Is the gravitational action additive?, Phys. Rev. D 50 (1994) 4914 [gr-qc/9403018] [INSPIRE].
  90. [90]
    J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  91. [91]
    V. Balasubramanian, B.D. Chowdhury, B. Czech, J. de Boer and M.P. Heller, Bulk curves from boundary data in holography, Phys. Rev. D 89 (2014) 086004 [arXiv:1310.4204] [INSPIRE].Google Scholar
  92. [92]
    S. Lloyd, Ultimate physical limits to computation, Nature 406 (2000) 1047.CrossRefGoogle Scholar
  93. [93]
    J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  94. [94]
    T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  95. [95]
    P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02 (2016) 004 [arXiv:1511.04021] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  96. [96]
    J. Cotler, C.-M. Jian, X.-L. Qi and F. Wilczek, Superdensity Operators for Spacetime Quantum Mechanics, JHEP 09 (2018) 093 [arXiv:1711.03119] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  97. [97]
    J. Dubail, Entanglement scaling of operators: a conformal field theory approach, with a glimpse of simulability of long-time dynamics in 1 + 1d, J. Phys. A 50 (2017) 234001 [arXiv:1612.08630] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  98. [98]
    T. Zhou and D.J. Luitz, Operator entanglement entropy of the time evolution operator in chaotic systems, Phys. Rev. B 95 (2017) 094206 [arXiv:1612.07327] [INSPIRE].CrossRefGoogle Scholar
  99. [99]
    L. Nie, M. Nozaki, S. Ryu and M.T. Tan, in preparation.Google Scholar
  100. [100]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].zbMATHGoogle Scholar
  101. [101]
    T. Okuda and T. Takayanagi, Ghost D-branes, JHEP 03 (2006) 062 [hep-th/0601024] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  102. [102]
    N. Evans, T.R. Morris and O.J. Rosten, Gauge invariant regularization in the AdS/CFT correspondence and ghost D-branes, Phys. Lett. B 635 (2006) 148 [hep-th/0601114] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  103. [103]
    T. Takayanagi, Holographic Dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602 [arXiv:1105.5165] [INSPIRE].CrossRefGoogle Scholar
  104. [104]
    M. Fujita, T. Takayanagi and E. Tonni, Aspects of AdS/BCFT, JHEP 11 (2011) 043 [arXiv:1108.5152] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  105. [105]
    P. Gao, D.L. Jafferis and A. Wall, Traversable Wormholes via a Double Trace Deformation, JHEP 12 (2017) 151 [arXiv:1608.05687] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  106. [106]
    T. Azeyanagi, T. Nishioka and T. Takayanagi, Near Extremal Black Hole Entropy as Entanglement Entropy via AdS 2 /CFT 1, Phys. Rev. D 77 (2008) 064005 [arXiv:0710.2956] [INSPIRE].Google Scholar
  107. [107]
    P. Hayden, M. Headrick and A. Maloney, Holographic Mutual Information is Monogamous, Phys. Rev. D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].Google Scholar
  108. [108]
    M. Nozaki, T. Numasawa and T. Takayanagi, Holographic Local Quenches and Entanglement Density, JHEP 05 (2013) 080 [arXiv:1302.5703] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  109. [109]
    T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  110. [110]
    J.M. Maldacena and L. Susskind, D-branes and fat black holes, Nucl. Phys. B 475 (1996) 679 [hep-th/9604042] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  111. [111]
    X. Dong, E. Silverstein and G. Torroba, de Sitter Holography and Entanglement Entropy, JHEP 07 (2018) 050 [arXiv:1804.08623] [INSPIRE].
  112. [112]
    K. Narayan, Extremal surfaces in de Sitter spacetime, Phys. Rev. D 91 (2015) 126011 [arXiv:1501.03019] [INSPIRE].MathSciNetGoogle Scholar
  113. [113]
    Y. Sato, Comments on Entanglement Entropy in the dS/CFT Correspondence, Phys. Rev. D 91 (2015) 086009 [arXiv:1501.04903] [INSPIRE].Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Center for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP)Kyoto UniversityKyotoJapan
  2. 2.Kavli Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan

Personalised recommendations