Holographic subregion complexity from kinematic space

  • Raimond Abt
  • Johanna ErdmengerEmail author
  • Marius Gerbershagen
  • Charles M. Melby-Thompson
  • Christian Northe
Open Access
Regular Article - Theoretical Physics


We consider the computation of volumes contained in a spatial slice of AdS3 in terms of observables in a dual CFT. Our main tool is kinematic space, defined either from the bulk perspective as the space of oriented bulk geodesics, or from the CFT perspective as the space of entangling intervals. We give an explicit formula for the volume of a general region in a spatial slice of AdS3 as an integral over kinematic space. For the region lying below a geodesic, we show how to write this volume purely in terms of entangling entropies in the dual CFT. This expression is perhaps most interesting in light of the complexity = volume proposal, which posits that complexity of holographic quantum states is computed by bulk volumes. An extension of this idea proposes that the holographic subregion complexity of an interval, defined as the volume under its Ryu-Takayanagi surface, is a measure of the complexity of the corresponding reduced density matrix. If this is true, our results give an explicit relationship between entanglement and subregion complexity in CFT, at least in the vacuum. We further extend many of our results to conical defect and BTZ black hole geometries.


AdS-CFT Correspondence Gauge-gravity correspondence Black Holes in String Theory 


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]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality, Nature Phys. 14 (2018) 573 [arXiv:1708.09393] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    P. Nguyen, T. Devakul, M.G. Halbasch, M.P. Zaletel and B. Swingle, Entanglement of purification: from spin chains to holography, JHEP 01 (2018) 098 [arXiv:1709.07424] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    S. Banerjee, J. Erdmenger and D. Sarkar, Connecting Fisher information to bulk entanglement in holography, JHEP 08 (2018) 001 [arXiv:1701.02319] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    N. Lashkari and M. Van Raamsdonk, Canonical Energy is Quantum Fisher Information, JHEP 04 (2016) 153 [arXiv:1508.00897] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  6. [6]
    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].ADSCrossRefGoogle Scholar
  7. [7]
    M. Alishahiha and A. Faraji Astaneh, Holographic Fidelity Susceptibility, Phys. Rev. D 96 (2017) 086004 [arXiv:1705.01834] [INSPIRE].
  8. [8]
    W.-C. Gan and F.-W. Shu, Holographic complexity: A tool to probe the property of reduced fidelity susceptibility, Phys. Rev. D 96 (2017) 026008 [arXiv:1702.07471] [INSPIRE].
  9. [9]
    M. Flory, A complexity/fidelity susceptibility g-theorem for AdS 3 /BCFT 2, JHEP 06 (2017) 131 [arXiv:1702.06386] [INSPIRE].
  10. [10]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    F. Pastawski and J. Preskill, Code properties from holographic geometries, Phys. Rev. X 7 (2017) 021022 [arXiv:1612.00017] [INSPIRE].
  12. [12]
    N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement ‘thermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    C.M. Papadimitriou, Computational complexity, Addison-Wesley, Reading Massachusetts U.S.A. (1994).Google Scholar
  14. [14]
    M.A. Nielsen, A geometric approach to quantum circuit lower bounds, Quant. Inf. Comput. 6 (2006) 213 [quant-ph/0502070].
  15. [15]
    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].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    L. Hackl and R.C. Myers, Circuit complexity for free fermions, JHEP 07 (2018) 139 [arXiv:1803.10638] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    R. Khan, C. Krishnan and S. Sharma, Circuit Complexity in Fermionic Field Theory, arXiv:1801.07620 [INSPIRE].
  19. [19]
    L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    A.R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE].
  21. [21]
    M. Alishahiha, Holographic Complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].
  22. [22]
    O. Ben-Ami and D. Carmi, On Volumes of Subregions in Holography and Complexity, JHEP 11 (2016) 129 [arXiv:1609.02514] [INSPIRE].
  23. [23]
    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
  24. [24]
    D. Carmi, More on Holographic Volumes, Entanglement and Complexity, arXiv:1709.10463 [INSPIRE].
  25. [25]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    C.A. Agón, M. Headrick and B. Swingle, Subsystem Complexity and Holography, arXiv:1804.01561 [INSPIRE].
  27. [27]
    R. Abt et al., Topological Complexity in AdS 3 /CFT 2, Fortsch. Phys. 66 (2018) 1800034 [arXiv:1710.01327] [INSPIRE].
  28. [28]
    B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral Geometry and Holography, JHEP 10 (2015) 175 [arXiv:1505.05515] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, A Stereoscopic Look into the Bulk, JHEP 07 (2016) 129 [arXiv:1604.03110] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    J. de Boer, F.M. Haehl, M.P. Heller and R.C. Myers, Entanglement, holography and causal diamonds, JHEP 08 (2016) 162 [arXiv:1606.03307] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    B. Czech and L. Lamprou, Holographic definition of points and distances, Phys. Rev. D 90 (2014) 106005 [arXiv:1409.4473] [INSPIRE].
  32. [32]
    B. Czech, L. Lamprou, S. McCandlish and J. Sully, Tensor Networks from Kinematic Space, JHEP 07 (2016) 100 [arXiv:1512.01548] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    B. Czech, L. Lamprou, S. Mccandlish and J. Sully, Modular Berry Connection for Entangled Subregions in AdS/CFT, Phys. Rev. Lett. 120 (2018) 091601 [arXiv:1712.07123] [INSPIRE].
  34. [34]
    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].
  35. [35]
    L.A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley Publishing Company, Reading Massachusetts U.S.A. (1976).Google Scholar
  36. [36]
    Q. Wen, Fine structure in holographic entanglement and entanglement contour, Phys. Rev. D 98 (2018) 106004 [arXiv:1803.05552] [INSPIRE].
  37. [37]
    R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].
  38. [38]
    R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    V. Balasubramanian, B.D. Chowdhury, B. Czech and J. de Boer, Entwinement and the emergence of spacetime, JHEP 01 (2015) 048 [arXiv:1406.5859] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  40. [40]
    V. Balasubramanian, A. Bernamonti, B. Craps, T. De Jonckheere and F. Galli, Entwinement in discretely gauged theories, JHEP 12 (2016) 094 [arXiv:1609.03991] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    J.C. Cresswell and A.W. Peet, Kinematic space for conical defects, JHEP 11 (2017) 155 [arXiv:1708.09838] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    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
  43. [43]
    J.-d. Zhang and B. Chen, Kinematic Space and Wormholes, JHEP 01 (2017) 092 [arXiv:1610.07134] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    C.T. Asplund, N. Callebaut and C. Zukowski, Equivalence of Emergent de Sitter Spaces from Conformal Field Theory, JHEP 09 (2016) 154 [arXiv:1604.02687] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    V.E. Hubeny, H. Maxfield, M. Rangamani and E. Tonni, Holographic entanglement plateaux, JHEP 08 (2013) 092 [arXiv:1306.4004] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
  47. [47]
    L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    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].
  49. [49]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    A.P. Reynolds and S.F. Ross, Complexity of the AdS Soliton, Class. Quant. Grav. 35 (2018) 095006 [arXiv:1712.03732] [INSPIRE].
  51. [51]
    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].
  52. [52]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    B. Czech, Einstein Equations from Varying Complexity, Phys. Rev. Lett. 120 (2018) 031601 [arXiv:1706.00965] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Raimond Abt
    • 1
  • Johanna Erdmenger
    • 1
    Email author
  • Marius Gerbershagen
    • 1
  • Charles M. Melby-Thompson
    • 1
  • Christian Northe
    • 1
  1. 1.Institut für Theoretische Physik und AstrophysikJulius-Maximilians-Universität WürzburgWürzburgGermany

Personalised recommendations