Advertisement

On the Noether charge and the gravity duals of quantum complexity

  • Zhong-Ying Fan
  • Minyong Guo
Open Access
Regular Article - Theoretical Physics
  • 37 Downloads

Abstract

The physical relevance of the thermodynamic volumes of AdS black holes to the gravity duals of quantum complexity was recently argued by Couch et al. In this paper, by generalizing the Wald-Iyer formalism, we derive a geometric expression for the thermodynamic volume and relate its product with the thermodynamic pressure to the non-derivative part of the gravitational action evaluated on the Wheeler-DeWitt patch. We propose that this action provides an alternative gravity dual of the quantum complexity of the boundary theory. We refer this to “complexity=action 2.0” (CA-2) duality. It is significantly different from the original “complexity=action” (CA) duality as well as the “complexity=volume 2.0” (CV-2) duality proposed by Couch et al. The latter postulates that the complexity is dual to the spacetime volume of the Wheeler-DeWitt patch. To distinguish our new conjecture from the various dualities in literature, we study a number of black holes in Einstein-Maxwell-Dilation theories. We find that for all these black holes, the CA duality generally does not respect the Lloyd bound whereas the CV-2 duality always does. For the CA-2 duality, although in many cases it is consistent with the Lloyd bound, we also find a counter example for which it violates the bound as well.

Keywords

AdS-CFT Correspondence Gauge-gravity correspondence 

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]
    S. Lloyd, Ultimate physical limits to computation, Nature 406 (2000) 1047.ADSCrossRefGoogle Scholar
  2. [2]
    D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].ADSGoogle Scholar
  3. [3]
    A.R. Brown et al., Holographic complexity equals bulk action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    A.R. Brown et al., Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].ADSMathSciNetGoogle Scholar
  5. [5]
    Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    L. Lehner et al., Gravitational action with null boundaries, Phys. Rev. D 94 (2016) 084046 [arXiv:1609.00207] [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    S. Chapman, H. Marrochio and R.C. Myers, Complexity of formation in holography, JHEP 01 (2017) 062 [arXiv:1610.08063] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    D. Carmi, R.C. Myers and P. Rath, Comments on holographic complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    D. Carmi et al., On the Time Dependence of Holographic Complexity, JHEP 11 (2017) 188 [arXiv:1709.10184] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part I, JHEP 06 (2018) 046 [arXiv:1804.07410] [INSPIRE].
  11. [11]
    S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part II, JHEP 06 (2018) 114 [arXiv:1805.07262] [INSPIRE].
  12. [12]
    R.-G. Cai et al., Action growth for AdS black holes, JHEP 09 (2016) 161 [arXiv:1606.08307] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    R.-Q. Yang, C. Niu and K.-Y. Kim, Surface counterterms and regularized holographic complexity, JHEP 09 (2017) 042 [arXiv:1701.03706] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    R.-Q. Yang et al., Comparison of holographic and field theoretic complexities for time dependent thermofield double states, JHEP 02 (2018) 082 [arXiv:1710.00600] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    R.-Q. Yang, Strong energy condition and complexity growth bound in holography, Phys. Rev. D 95 (2017) 086017 [arXiv:1610.05090] [INSPIRE].ADSGoogle Scholar
  16. [16]
    R.-Q. Yang and S.-M. Ruan, Comments on joint terms in gravitational action, Class. Quant. Grav. 34 (2017) 175017 [arXiv:1704.03232] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    M. Moosa, Evolution of complexity following a global quench, JHEP 03 (2018) 031 [arXiv:1711.02668] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    M. Moosa, Divergences in the rate of complexification, Phys. Rev. D 97 (2018) 106016 [arXiv:1712.07137] [INSPIRE].ADSGoogle Scholar
  19. [19]
    B. Swingle and Y. Wang, Holographic complexity of Einstein-Maxwell-Dilaton gravity, arXiv:1712.09826 [INSPIRE].
  20. [20]
    M. Alishahiha et al., Complexity Growth with Lifshitz Scaling and Hyperscaling Violation, JHEP 07 (2018) 042 [arXiv:1802.06740] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    Y.-S. An and R.-H. Peng, Effect of the dilaton on holographic complexity growth, Phys. Rev. D 97 (2018) 066022 [arXiv:1801.03638] [INSPIRE].ADSGoogle Scholar
  22. [22]
    M. Alishahiha, Holographic complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].ADSMathSciNetGoogle Scholar
  23. [23]
    J.L.F. Barbon and E. Rabinovici, Holographic complexity and spacetime singularities, JHEP 01 (2016) 084 [arXiv:1509.09291] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    P.A. Cano, Lovelock action with nonsmooth boundaries, Phys. Rev. D 97 (2018) 104048 [arXiv:1803.00172] [INSPIRE].ADSGoogle Scholar
  25. [25]
    P.A. Cano, R.A. Hennigar and H. Marrochio, Complexity growth rate in Lovelock gravity, arXiv:1803.02795 [INSPIRE].
  26. [26]
    J. Couch, W. Fischler and P.H. Nguyen, Noether charge, black hole volume and complexity, JHEP 03 (2017) 119 [arXiv:1610.02038] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].
  28. [28]
    V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
  29. [29]
    I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    H. Lü, Y. Pang and C.N. Pope, AdS dyonic black hole and its thermodynamics, JHEP 11 (2013) 033 [arXiv:1307.6243] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    H.-S. Liu and H. Lü, Scalar charges in asymptotic AdS geometries, Phys. Lett. B 730 (2014) 267 [arXiv:1401.0010] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    H.-S. Liu, H. Lü and C.N. Pope, Thermodynamics of Einstein-Proca AdS black holes, JHEP 06 (2014) 109 [arXiv:1402.5153] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    H.-S. Liu and H. Lü, Thermodynamics of Lifshitz black holes, JHEP 12 (2014) 071 [arXiv:1410.6181] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    Z.-Y. Fan and H. Lü, SU(2)-colored (A)dS black holes in conformal gravity, JHEP 02 (2015) 013 [arXiv:1411.5372] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    H. Lü, C.N. Pope and Q. Wen, Thermodynamics of AdS black holes in Einstein-scalar gravity, JHEP 03 (2015) 165 [arXiv:1408.1514] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    Z.-Y. Fan and H. Lü, Thermodynamical first laws of black holes in quadratically-extended gravities, Phys. Rev. D 91 (2015) 064009 [arXiv:1501.00006] [INSPIRE].ADSGoogle Scholar
  37. [37]
    Z.-Y. Fan and H. Lü, Charged black holes in colored Lifshitz spacetimes, Phys. Lett. B 743 (2015) 290 [arXiv:1501.01727] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    B. Chen, Z.-Y. Fan and L.-Y. Zhu, AdS and Lifshitz scalar hairy black holes in Gauss-Bonnet gravity, Phys. Rev. D 94 (2016) 064005 [arXiv:1604.08282] [INSPIRE].ADSMathSciNetGoogle Scholar
  39. [39]
    X.-H. Feng et al., Black hole entropy and viscosity bound in Horndeski gravity, JHEP 11 (2015) 176 [arXiv:1509.07142] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    X.-H. Feng et al., Thermodynamics of charged black holes in Einstein-Horndeski-Maxwell theory, Phys. Rev. D 93 (2016) 044030 [arXiv:1512.02659] [INSPIRE].ADSMathSciNetGoogle Scholar
  41. [41]
    Z.-Y. Fan, Black holes with vector hair, JHEP 09 (2016) 039 [arXiv:1606.00684] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    Z.-Y. Fan, Black holes in vector-tensor theories and their thermodynamics, Eur. Phys. J. C 78 (2018) 65 [arXiv:1709.04392] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    H.-S. Liu, H. Lü and C.N. Pope, Holographic heat current as Noether current, JHEP 09 (2017) 146 [arXiv:1708.02329] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    Y.-Z. Li, H.-S. Liu and H. Lü, Quasi-topological Ricci polynomial gravities, JHEP 02 (2018) 166 [arXiv:1708.07198] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  45. [45]
    Z.-Y. Fan, Note on the Noether charge and holographic transports, Phys. Rev. D 97 (2018) 066013 [arXiv:1801.07870] [INSPIRE].ADSGoogle Scholar
  46. [46]
    P.A. González et al., Four-dimensional asymptotically AdS black holes with scalar hair, JHEP 12 (2013) 021 [arXiv:1309.2161] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    Z.-Y. Fan and H. Lü, Charged black holes with scalar hair, JHEP 09 (2015) 060 [arXiv:1507.04369] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    Z.-Y. Fan and B. Chen, Exact formation of hairy planar black holes, Phys. Rev. D 93 (2016) 084013 [arXiv:1512.09145] [INSPIRE].ADSMathSciNetGoogle Scholar
  49. [49]
    D. Astefanesei, R. Ballesteros, D. Choque and R. Rojas, Scalar charges and the first law of black hole thermodynamics, Phys. Lett. B 782 (2018) 47 [arXiv:1803.11317] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    H.-S. Liu, H. Lü and C.N. Pope, Generalized Smarr formula and the viscosity bound for Einstein-Maxwell-dilaton black holes, Phys. Rev. D 92 (2015) 064014 [arXiv:1507.02294] [INSPIRE].ADSMathSciNetGoogle Scholar
  51. [51]
    N. Margolus and L.B. Levitin, The Maximum speed of dynamical evolution, Physica D 120 (1998) 188 [quant-ph/9710043] [INSPIRE].
  52. [52]
    S.P. Jordan, Fast quantum computation at arbitrarily low energy, Phys. Rev. A 95 (2017) 032305 [arXiv:1701.01175] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    S. Chapman et al., Toward a definition of complexity for quantum field theory states, Phys. Rev. Lett. 120 (2018) 121602 [arXiv:1707.08582] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    R.-Q. Yang, Complexity for quantum field theory states and applications to thermofield double states, Phys. Rev. D 97 (2018) 066004 [arXiv:1709.00921] [INSPIRE].ADSGoogle Scholar
  56. [56]
    R. Khan, C. Krishnan and S. Sharma, Circuit complexity in fermionic field theory, arXiv:1801.07620 [INSPIRE].
  57. [57]
    L. Hackl and R.C. Myers, Circuit complexity for free fermions, JHEP 07 (2018) 139 [arXiv:1803.10638] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    R.-Q. Yang et al., Axiomatic complexity in quantum field theory and its applications, arXiv:1803.01797 [INSPIRE].
  59. [59]
    S. Chapman et al., Circuit complexity for thermofield double states, in preparation.Google Scholar
  60. [60]
    M.Y. Guo et al., Circuit complexity for coherent states, in preparation.Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Center for Astrophysics, School of Physics and Electronic EngineeringGuangzhou UniversityGuangzhouChina
  2. 2.Department of PhysicsBeijing Normal UniversityBeijingP.R. China
  3. 3.Perimeter Institute for Theoretical PhysicsWaterlooCanada

Personalised recommendations