On subregion action complexity in AdS3 and in the BTZ black hole

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We analytically compute subsystem action complexity for a segment in the BTZ black hole background up to the finite term, and we find that it is equal to the sum of a linearly divergent term proportional to the size of the subregion and of a term proportional to the entanglement entropy. This elegant structure does not survive to more complicated geometries: in the case of a two segments subregion in AdS3, complexity has additional finite contributions. We give analytic results for the mutual action complexity of a two segments subregion.

A preprint version of the article is available at ArXiv.


  1. [1]

    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].

  2. [2]

    J.D. Bekenstein, Black holes and entropy, Phys. Rev.D 7 (1973) 2333 [INSPIRE].

  3. [3]

    M. Rangamani and T. Takayanagi, Holographic entanglement entropy, Lect. Notes Phys.931 (2017) 1 [arXiv:1609.01287].

  4. [4]

    M. Headrick, Lectures on entanglement entropy in field theory and holography, arXiv:1907.08126 [INSPIRE].

  5. [5]

    L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys.64 (2016) 44 [Addendum ibid.46 (2016) 44] [arXiv:1403.5695] [INSPIRE].

  6. [6]

    D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev.D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].

  7. [7]

    L. Susskind, Entanglement is not enough, Fortsch. Phys.64 (2016) 49 [arXiv:1411.0690] [INSPIRE].

  8. [8]

    L. Susskind, Three lectures on complexity and black holes, 2018, arXiv:1810.11563 [INSPIRE].

  9. [9]

    M.A. Nielsen, A geometric approach to quantum circuit lower bounds, Quant. Inf. Comput.6 (2006) 213 [quant-ph/0502070].

  10. [10]

    M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, Quant. Inf. Comput.8 (2008) 861 [quant-ph/0701004].

  11. [11]

    R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP10 (2017) 107 [arXiv:1707.08570] [INSPIRE].

  12. [12]

    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].

  13. [13]

    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].

  14. [14]

    S. Chapman et al., Complexity and entanglement for thermofield double states, SciPost Phys.6 (2019) 034 [arXiv:1810.05151] [INSPIRE].

  15. [15]

    H.A. Camargo et al., Complexity as a novel probe of quantum quenches: universal scalings and purifications, Phys. Rev. Lett.122 (2019) 081601 [arXiv:1807.07075] [INSPIRE].

  16. [16]

    P. Caputa et al., Anti-de Sitter space from optimization of path integrals in conformal field theories, Phys. Rev. Lett.119 (2017) 071602 [arXiv:1703.00456] [INSPIRE].

  17. [17]

    P. Caputa et al., Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT, JHEP11 (2017) 097 [arXiv:1706.07056] [INSPIRE].

  18. [18]

    A. Bhattacharyya et al., Path-integral complexity for perturbed CFTs, JHEP07 (2018) 086 [arXiv:1804.01999] [INSPIRE].

  19. [19]

    P. Caputa and J.M. Magan, Quantum computation as gravity, Phys. Rev. Lett.122 (2019) 231302 [arXiv:1807.04422] [INSPIRE].

  20. [20]

    A. Belin, A. Lewkowycz and G. Sárosi, Complexity and the bulk volume, a new York time story, JHEP03 (2019) 044 [arXiv:1811.03097] [INSPIRE].

  21. [21]

    A.R. Brown et al., Holographic complexity equals bulk action?, Phys. Rev. Lett.116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].

  22. [22]

    A.R. Brown et al., Complexity, action and black holes, Phys. Rev.D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].

  23. [23]

    J. Couch, W. Fischler and P.H. Nguyen, Noether charge, black hole volume and complexity, JHEP03 (2017) 119 [arXiv:1610.02038] [INSPIRE].

  24. [24]

    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].

  25. [25]

    R.-G. Cai et al., Action growth for AdS black holes, JHEP09 (2016) 161 [arXiv:1606.08307] [INSPIRE].

  26. [26]

    S. Chapman, H. Marrochio and R.C. Myers, Complexity of formation in holography, JHEP01 (2017) 062 [arXiv:1610.08063] [INSPIRE].

  27. [27]

    D. Carmi et al., On the Time Dependence of Holographic Complexity, JHEP11 (2017) 188 [arXiv:1709.10184] [INSPIRE].

  28. [28]

    S. Chapman, D. Ge and G. Policastro, Holographic complexity for defects distinguishes action from volume, JHEP05 (2019) 049 [arXiv:1811.12549] [INSPIRE].

  29. [29]

    M. Moosa, Evolution of complexity following a global quench, JHEP03 (2018) 031 [arXiv:1711.02668] [INSPIRE].

  30. [30]

    M. Moosa, Divergences in the rate of complexification, Phys. Rev.D 97 (2018) 106016 [arXiv:1712.07137] [INSPIRE].

  31. [31]

    S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part I, JHEP06 (2018) 046 [arXiv:1804.07410] [INSPIRE].

  32. [32]

    S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part II, JHEP06 (2018) 114 [arXiv:1805.07262] [INSPIRE].

  33. [33]

    J.L.F. Barbon and E. Rabinovici, Holographic complexity and spacetime singularities, JHEP01 (2016) 084 [arXiv:1509.09291] [INSPIRE].

  34. [34]

    S. Bolognesi, E. Rabinovici and S.R. Roy, On some universal features of the holographic quantum complexity of bulk singularities, JHEP06 (2018) 016 [arXiv:1802.02045] [INSPIRE].

  35. [35]

    M. Flory and N. Miekley, Complexity change under conformal transformations in AdS 3/CFT 2, JHEP05 (2019) 003 [arXiv:1806.08376] [INSPIRE].

  36. [36]

    M. Flory, WdW-patches in AdS 3and complexity change under conformal transformations II, JHEP05 (2019) 086 [arXiv:1902.06499] [INSPIRE].

  37. [37]

    M. Alishahiha, A. Faraji Astaneh, A. Naseh and M.H. Vahidinia, On complexity for F(R) and critical gravity, JHEP05 (2017) 009 [arXiv:1702.06796] [INSPIRE].

  38. [38]

    A. Akhavan, M. Alishahiha, A. Naseh and H. Zolfi, Complexity and behind the horizon cut off, JHEP12 (2018) 090 [arXiv:1810.12015] [INSPIRE].

  39. [39]

    M. Alishahiha et al., Complexity growth with Lifshitz scaling and hyperscaling violation, JHEP07 (2018) 042 [arXiv:1802.06740] [INSPIRE].

  40. [40]

    M. Ghodrati, Complexity growth in massive gravity theories, the effects of chirality and more, Phys. Rev.D 96 (2017) 106020 [arXiv:1708.07981] [INSPIRE].

  41. [41]

    R. Auzzi, S. Baiguera and G. Nardelli, Volume and complexity for warped AdS black holes, JHEP06 (2018) 063 [arXiv:1804.07521] [INSPIRE].

  42. [42]

    R. Auzzi et al., Complexity and action for warped AdS black holes, JHEP09 (2018) 013 [arXiv:1806.06216] [INSPIRE].

  43. [43]

    H. Dimov, R.C. Rashkov and T. Vetsov, Thermodynamic information geometry and complexity growth of a warped AdS black hole and the warped AdS 3/CFT 2correspondence, Phys. Rev.D 99 (2019) 126007 [arXiv:1902.02433] [INSPIRE].

  44. [44]

    M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP12 (2014) 162 [arXiv:1408.6300] [INSPIRE].

  45. [45]

    M. Alishahiha, Holographic complexity, Phys. Rev.D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].

  46. [46]

    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP07 (2007) 062 [arXiv:0705.0016] [INSPIRE].

  47. [47]

    D. Carmi, R.C. Myers and P. Rath, Comments on holographic complexity, JHEP03 (2017) 118 [arXiv:1612.00433] [INSPIRE].

  48. [48]

    O. Ben-Ami and D. Carmi, On volumes of subregions in holography and complexity, JHEP11 (2016) 129 [arXiv:1609.02514] [INSPIRE].

  49. [49]

    R. Abt et al., Topological complexity in AdS 3/CFT 2, Fortsch. Phys.66 (2018) 1800034 [arXiv:1710.01327] [INSPIRE].

  50. [50]

    R. Abt et al., Holographic subregion complexity from kinematic space, JHEP01 (2019) 012 [arXiv:1805.10298] [INSPIRE].

  51. [51]

    C.A. Agón, M. Headrick and B. Swingle, Subsystem complexity and holography, JHEP02 (2019) 145 [arXiv:1804.01561] [INSPIRE].

  52. [52]

    M. Alishahiha, K. Babaei Velni and M.R. Mohammadi Mozaffar, Black hole subregion action and complexity, Phys. Rev.D 99 (2019) 126016 [arXiv:1809.06031] [INSPIRE].

  53. [53]

    E. Cáceres, J. Couch, S. Eccles and W. Fischler, Holographic purification complexity, Phys. Rev.D 99 (2019) 086016 [arXiv:1811.10650] [INSPIRE].

  54. [54]

    P. Roy and T. Sarkar, Note on subregion holographic complexity, Phys. Rev.D 96 (2017) 026022 [arXiv:1701.05489] [INSPIRE].

  55. [55]

    P. Roy and T. Sarkar, Subregion holographic complexity and renormalization group flows, Phys. Rev.D 97 (2018) 086018 [arXiv:1708.05313] [INSPIRE].

  56. [56]

    E. Bakhshaei, A. Mollabashi and A. Shirzad, Holographic subregion complexity for singular surfaces, Eur. Phys. J.C 77 (2017) 665 [arXiv:1703.03469] [INSPIRE].

  57. [57]

    A. Bhattacharya, K.T. Grosvenor and S. Roy, Entanglement entropy and subregion complexity in thermal perturbations around pure-AdS spacetime, arXiv:1905.02220 [INSPIRE].

  58. [58]

    R. Auzzi et al., Subsystem complexity in warped AdS, JHEP09 (2019) 114 [arXiv:1906.09345] [INSPIRE].

  59. [59]

    B. Chen et al., Holographic subregion complexity under a thermal quench, JHEP07 (2018) 034 [arXiv:1803.06680] [INSPIRE].

  60. [60]

    R. Auzzi et al., On volume subregion complexity in Vaidya spacetime, JHEP11 (2019) 098 [arXiv:1908.10832] [INSPIRE].

  61. [61]

    M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett.69 (1992) 1849 [hep-th/9204099] [INSPIRE].

  62. [62]

    M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev.D 48 (1993) 1506 [Erratum ibid.D 88 (2013) 069902] [gr-qc/9302012] [INSPIRE].

  63. [63]

    E. Caceres et al., Complexity of Mixed States in QFT and Holography, arXiv:1909.10557 [INSPIRE].

  64. [64]

    E. Poisson, A relativist’s toolkit: the mathematics of black-hole mechanics, Cambridge University Press, Cambridge U.K. (2004).

  65. [65]

    B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a density matrix, Class. Quant. Grav.29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].

  66. [66]

    V.E. Hubeny and M. Rangamani, Causal holographic information, JHEP06 (2012) 114 [arXiv:1204.1698] [INSPIRE].

  67. [67]

    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].

  68. [68]

    V. Balasubramanian et al., Holographic thermalization, Phys. Rev.D 84 (2011) 026010 [arXiv:1103.2683] [INSPIRE].

  69. [69]

    A. Akhavan and F. Omidi, On the role of counterterms in holographic complexity, JHEP11 (2019) 054 [arXiv:1906.09561] [INSPIRE].

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Correspondence to Roberto Auzzi.

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ArXiv ePrint: 1910.00526

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Auzzi, R., Baiguera, S., Legramandi, A. et al. On subregion action complexity in AdS3 and in the BTZ black hole. J. High Energ. Phys. 2020, 66 (2020) doi:10.1007/JHEP01(2020)066

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  • AdS-CFT Correspondence
  • Black Holes