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Journal of High Energy Physics

, 2018:72 | Cite as

Holographic complexity is nonlocal

  • Zicao Fu
  • Alexander Maloney
  • Donald Marolf
  • Henry Maxfield
  • Zhencheng Wang
Open Access
Regular Article - Theoretical Physics

Abstract

We study the “complexity equals volume” (CV) and “complexity equals action” (CA) conjectures by examining moments of of time symmetry for AdS3 wormholes having n asymptotic regions and arbitrary (orientable) internal topology. For either prescription, the complexity relative to n copies of the M = 0 BTZ black hole takes the form ΔC = αcχ, where c is the central charge and χ is the Euler character of the bulk time-symmetric surface. The coefficients α V = −4π/3, α A = 1/6 defined by CV and CA are independent of both temperature and any moduli controlling the geometry inside the black hole. Comparing with the known structure of dual CFT states in the hot wormhole limit, the temperature and moduli independence of α V , α A implies that any CFT gate set defining either complexity cannot be local. In particular, the complexity of an efficient quantum circuit building local thermofield-double-like entanglement of thermal-sized patches does not depend on the separation of the patches so entangled. We also comment on implications of the (positive) sign found for α A , which requires the associated complexity to decrease when handles are added to our wormhole.

Keywords

AdS-CFT Correspondence Black Holes 

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]
    L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle 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]
    D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    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
  5. [5]
    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].ADSMathSciNetGoogle Scholar
  6. [6]
    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].ADSGoogle Scholar
  7. [7]
    R.A. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    D. Marolf, H. Maxfield, A. Peach and S.F. Ross, Hot multiboundary wormholes from bipartite entanglement, Class. Quant. Grav. 32 (2015) 215006 [arXiv:1506.04128] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    D.R. Brill, Multi-black hole geometries in (2 + 1)-dimensional gravity, Phys. Rev. D 53 (1996) 4133 [gr-qc/9511022] [INSPIRE].
  10. [10]
    S. Aminneborg, I. Bengtsson, D. Brill, S. Holst and P. Peldan, Black holes and wormholes in (2 + 1)-dimensions, Class. Quant. Grav. 15 (1998) 627 [gr-qc/9707036] [INSPIRE].
  11. [11]
    D. Brill, Black holes and wormholes in (2 + 1)-dimensions, gr-qc/9904083 [INSPIRE].
  12. [12]
    K. Krasnov, Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 (2000) 929 [hep-th/0005106] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    K. Skenderis and B.C. van Rees, Holography and wormholes in 2 + 1 dimensions, Commun. Math. Phys. 301 (2011) 583 [arXiv:0912.2090] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    V. Balasubramanian, P. Hayden, A. Maloney, D. Marolf and S.F. Ross, Multiboundary Wormholes and Holographic Entanglement, Class. Quant. Grav. 31 (2014) 185015 [arXiv:1406.2663] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  16. [16]
    S. Chapman, H. Marrochio and R.C. Myers, Complexity of formation in holography, JHEP 01 (2017) 062 [arXiv:1610.08063] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    R. Abt, J. Erdmenger, H. Hinrichsen, C.M. Melby-Thompson, R. Meyer, C. Northe et al., Topological Complexity in AdS3/CFT2, arXiv:1710.01327 [INSPIRE].
  18. [18]
    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].ADSGoogle Scholar
  19. [19]
    A.P. Reynolds and S.F. Ross, Complexity of the AdS Soliton, arXiv:1712.03732 [INSPIRE].
  20. [20]
    A. Maloney, Geometric Microstates for the Three Dimensional Black Hole?, arXiv:1508.04079 [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of CaliforniaSanta BarbaraU.S.A.
  2. 2.Department of PhysicsMcGill UniversityMontealCanada
  3. 3.School of PhysicsNankai UniversityTianjinChina

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