Journal of High Energy Physics

, 2018:44 | Cite as

Holographic complexity and volume

  • Josiah Couch
  • Stefan Eccles
  • Ted JacobsonEmail author
  • Phuc Nguyen
Open Access
Regular Article - Theoretical Physics


The previously proposed “Complexity=Volume” or CV-duality is probed and developed in several directions. We show that the apparent lack of universality for large and small black holes is removed if the volume is measured in units of the maximal time from the horizon to the “final slice” (times Planck area). This also works for spinning black holes. We make use of the conserved “volume current”, associated with a foliation of spacetime by maximal volume slices, whose flux measures their volume. This flux picture suggests that there is a transfer of the complexity from the UV to the IR in holographic CFTs, which is reminiscent of thermalization behavior deduced using holography. It also naturally gives a second law for the complexity when applied at a black hole horizon. We further establish a result supporting the conjecture that a boundary foliation determines a bulk maximal foliation without gaps, establish a global inequality on maximal volumes that can be used to deduce the monotonicity of the complexification rate on a boost-invariant background, and probe CV duality in the settings of multiple quenches, spinning black holes, and Rindler-AdS.


AdS-CFT Correspondence Black Holes 


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]
    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]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    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].
  4. [4]
    L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014)126007 [arXiv:1406.2678] [INSPIRE].
  7. [7]
    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].ADSCrossRefGoogle Scholar
  8. [8]
    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].
  9. [9]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    D. Carmi, R.C. Myers and P. Rath, Comments on holographic complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    S. Chapman, H. Marrochio and R.C. Myers, Complexity of formation in holography, JHEP 01 (2017) 062 [arXiv:1610.08063] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    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
  14. [14]
    D. Harlow and P. Hayden, Quantum computation vs. firewalls, JHEP 06 (2013) 085 [arXiv:1301.4504] [INSPIRE].
  15. [15]
    M. Headrick and V.E. Hubeny, Riemannian and Lorentzian flow-cut theorems, Class. Quant. Grav. 35 (2018) 10 [arXiv:1710.09516] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    R. Cleve, An introduction to quantum complexity theory, quant-ph/9906111 [INSPIRE].
  17. [17]
    J. Watrous, Quantum computational complexity, arXiv:0804.3401.
  18. [18]
    S. Aaronson, The complexity of quantum states and transformations: from quantum money to black holes, arXiv:1607.05256 [INSPIRE].
  19. [19]
    M. Moosa, Evolution of complexity following a global quench, JHEP 03 (2018) 031 [arXiv:1711.02668] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    M. Moosa, Divergences in the rate of complexification, Phys. Rev. D 97 (2018) 106016 [arXiv:1712.07137] [INSPIRE].
  21. [21]
    L. Susskind and Y. Zhao, Switchbacks and the bridge to nowhere, arXiv:1408.2823 [INSPIRE].
  22. [22]
    R. Laflamme, Geometry and thermofields, Nucl. Phys. B 324 (1989) 233 [INSPIRE].
  23. [23]
    A.O. Barvinsky, V.P. Frolov and A.I. Zelnikov, Wavefunction of a black hole and the dynamical origin of entropy, Phys. Rev. D 51 (1995) 1741 [gr-qc/9404036] [INSPIRE].
  24. [24]
    J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    A.R. Brown, L. Susskind and Y. Zhao, Quantum complexity and negative curvature, Phys. Rev. D 95 (2017) 045010 [arXiv:1608.02612] [INSPIRE].
  26. [26]
    A.R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE].
  27. [27]
    I. Bengtsson and J.M.M. Senovilla, The region with trapped surfaces in spherical symmetry, its core and their boundaries, Phys. Rev. D 83 (2011) 044012 [arXiv:1009.0225] [INSPIRE].
  28. [28]
    V. Balasubramanian et al., Thermalization of strongly coupled field theories, Phys. Rev. Lett. 106 (2011)191601 [arXiv:1012.4753] [INSPIRE].
  29. [29]
    V. Balasubramanian et al., Holographic thermalization, Phys. Rev. D 84 (2011) 026010 [arXiv:1103.2683] [INSPIRE].
  30. [30]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    R. Emparan, AdS/CFT duals of topological black holes and the entropy of zero energy states, JHEP 06 (1999) 036 [hep-th/9906040] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    R. Wald, General relativity, University of Chicago Press, Chicago, IL, U.S.A., (1984).Google Scholar
  33. [33]
    J. Couch, S. Eccles, W. Fischler and M.-L. Xiao, Holographic complexity and noncommutative gauge theory, JHEP 03 (2018) 108 [arXiv:1710.07833] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part I, JHEP 06 (2018) 046 [arXiv:1804.07410] [INSPIRE].
  35. [35]
    S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part II, JHEP 06 (2018) 114 [arXiv:1805.07262] [INSPIRE].
  36. [36]
    R. Bousso and N. Engelhardt, New area law in general relativity, Phys. Rev. Lett. 115 (2015) 081301 [arXiv:1504.07627] [INSPIRE].
  37. [37]
    R. Bousso and N. Engelhardt, Proof of a new area law in general relativity, Phys. Rev. D 92 (2015)044031 [arXiv:1504.07660] [INSPIRE].
  38. [38]
    N. Engelhardt and A.C. Wall, Decoding the apparent horizon: a coarse-grained holographic entropy, arXiv:1706.02038 [INSPIRE].
  39. [39]
    N. Engelhardt and A.C. Wall, Coarse graining holographic black holes, arXiv:1806.01281 [INSPIRE].
  40. [40]
    Y. Zhao, Uncomplexity and black hole geometry, Phys. Rev. D 97 (2018) 126007 [arXiv:1711.03125] [INSPIRE].
  41. [41]
    V.E. Hubeny and H. Maxfield, Holographic probes of collapsing black holes, JHEP 03 (2014) 097 [arXiv:1312.6887] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    S.W. Hawking, C.J. Hunter and M. Taylor, Rotation and the AdS/CFT correspondence, Phys. Rev. D 59 (1999) 064005 [hep-th/9811056] [INSPIRE].
  43. [43]
    I. Bengtsson and E. Jakobsson, Black holes: their large interiors, Mod. Phys. Lett. A 30 (2015)1550103 [arXiv:1502.01907] [INSPIRE].
  44. [44]
    M.J. Duncan, Maximally slicing a black hole with minimal distortion, Phys. Rev. D 31 (1985)1267.Google Scholar
  45. [45]
    B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, Rindler quantum gravity, Class. Quant. Grav. 29 (2012) 235025 [arXiv:1206.1323] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    M. Parikh and P. Samantray, Rindler-AdS/CFT, JHEP 10 (2018) 129 [arXiv:1211.7370] [INSPIRE].CrossRefGoogle Scholar
  47. [47]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    M. Headrick, General properties of holographic entanglement entropy, JHEP 03 (2014) 085 [arXiv:1312.6717] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    E. Witten, Canonical quantization in anti de Sitter space, conference talk at 20 years later: the many faces of AdS/CFT,, (2017).
  51. [51]
    F. Estabrook, H. Wahlquist, S. Christensen, B. DeWitt, L. Smarr and E. Tsiang, Maximally slicing a black hole, Phys. Rev. D 7 (1973) 2814 [INSPIRE].
  52. [52]
    I. Gelfand and S. Fomin, Calculus of variations, Dover Books on Mathematics, Dover Publications, U.S.A., (2012).Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Josiah Couch
    • 1
  • Stefan Eccles
    • 1
  • Ted Jacobson
    • 2
    Email author
  • Phuc Nguyen
    • 2
  1. 1.Theory Group, Department of PhysicsThe University of Texas at AustinAustinU.S.A.
  2. 2.Maryland Center for Fundamental Physics and Department of PhysicsUniversity of MarylandCollege ParkU.S.A.

Personalised recommendations