A toy model of black hole complementarity

  • Souvik Banerjee
  • Jan-Willem Bryan
  • Kyriakos Papadodimas
  • Suvrat Raju
Open Access
Regular Article - Theoretical Physics


We consider the algebra of simple operators defined in a time band in a CFT with a holographic dual. When the band is smaller than the light crossing time of AdS, an entire causal diamond in the center of AdS is separated from the band by a horizon. We show that this algebra obeys a version of the Reeh-Schlieder theorem: the action of the algebra on the CFT vacuum can approximate any low energy state in the CFT arbitrarily well, but no operator within the algebra can exactly annihilate the vacuum. We show how to relate local excitations in the complement of the central diamond to simple operators in the band. Local excitations within the diamond are invisible to the algebra of simple operators in the band by causality, but can be related to complicated operators called “precursors”. We use the Reeh-Schlieder theorem to write down a simple and explicit formula for these precursors on the boundary. We comment on the implications of our results for black hole complementarity and the emergence of bulk locality from the boundary.


AdS-CFT Correspondence Black Holes 1/N Expansion 


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.D. Mathur, The information paradox: A pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An Apologia for Firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    D. Marolf and J. Polchinski, Gauge/Gravity Duality and the Black Hole Interior, Phys. Rev. Lett. 111 (2013) 171301 [arXiv:1307.4706] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    S.L. Braunstein, S. Pirandola and K. Życzkowski, Better Late than Never: Information Retrieval from Black Holes, Phys. Rev. Lett. 110 (2013) 101301 [arXiv:0907.1190] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    G. ’t Hooft, On the Quantum Structure of a Black Hole, Nucl. Phys. B 256 (1985) 727 [INSPIRE].
  7. [7]
    L. Susskind, L. Thorlacius and J. Uglum, The stretched horizon and black hole complementarity, Phys. Rev. D 48 (1993) 3743 [hep-th/9306069] [INSPIRE].ADSMathSciNetGoogle Scholar
  8. [8]
    K. Papadodimas and S. Raju, State-Dependent Bulk-Boundary Maps and Black Hole Complementarity, Phys. Rev. D 89 (2014) 086010 [arXiv:1310.6335] [INSPIRE].ADSGoogle Scholar
  9. [9]
    K. Papadodimas and S. Raju, Black Hole Interior in the Holographic Correspondence and the Information Paradox, Phys. Rev. Lett. 112 (2014) 051301 [arXiv:1310.6334] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    K. Papadodimas and S. Raju, Comments on the Necessity and Implications of State-Dependence in the Black Hole Interior, arXiv:1503.08825 [INSPIRE].
  11. [11]
    K. Papadodimas and S. Raju, Local Operators in the Eternal Black Hole, Phys. Rev. Lett. 115 (2015) 211601 [arXiv:1502.06692] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    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].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    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].ADSGoogle Scholar
  16. [16]
    R.C. Myers, J. Rao and S. Sugishita, Holographic Holes in Higher Dimensions, JHEP 06 (2014) 044 [arXiv:1403.3416] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    M. Headrick, R.C. Myers and J. Wien, Holographic Holes and Differential Entropy, JHEP 10 (2014) 149 [arXiv:1408.4770] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    D. Marolf, Holographic Thought Experiments, Phys. Rev. D 79 (2009) 024029 [arXiv:0808.2845] [INSPIRE].ADSMathSciNetGoogle Scholar
  19. [19]
    D. Marolf, Unitarity and Holography in Gravitational Physics, Phys. Rev. D 79 (2009) 044010 [arXiv:0808.2842] [INSPIRE].ADSMathSciNetGoogle Scholar
  20. [20]
    E. Mintun, J. Polchinski and V. Rosenhaus, Bulk-Boundary Duality, Gauge Invariance and Quantum Error Corrections, Phys. Rev. Lett. 115 (2015) 151601 [arXiv:1501.06577] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    S.B. Giddings, Hilbert space structure in quantum gravity: an algebraic perspective, JHEP 12 (2015) 099 [arXiv:1503.08207] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    K. Papadodimas and S. Raju, An Infalling Observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, arXiv:1512.06431 [INSPIRE].
  26. [26]
    X. Dong, D. Harlow and A.C. Wall, Bulk Reconstruction in the Entanglement Wedge in AdS/CFT, arXiv:1601.05416 [INSPIRE].
  27. [27]
    B. Freivogel, R.A. Jefferson and L. Kabir, Precursors, Gauge Invariance and Quantum Error Correction in AdS/CFT, arXiv:1602.04811 [INSPIRE].
  28. [28]
    I.A. Morrison, Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography, JHEP 05 (2014) 053 [arXiv:1403.3426] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    R. Bousso, B. Freivogel, S. Leichenauer, V. Rosenhaus and C. Zukowski, Null Geodesics, Local CFT Operators and AdS/CFT for Subregions, Phys. Rev. D 88 (2013) 064057 [arXiv:1209.4641] [INSPIRE].ADSGoogle Scholar
  30. [30]
    R. Haag, Local quantum physics: Fields, particles, algebras, 2nd edition, Springer (1992).Google Scholar
  31. [31]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    A.L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Duffin, Effective Conformal Theory and the Flat-Space Limit of AdS, JHEP 07 (2011) 023 [arXiv:1007.2412] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    S. El-Showk and K. Papadodimas, Emergent Spacetime and Holographic CFTs, JHEP 10 (2012) 106 [arXiv:1101.4163] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    J. Polchinski, L. Susskind and N. Toumbas, Negative energy, superluminosity and holography, Phys. Rev. D 60 (1999) 084006 [hep-th/9903228] [INSPIRE].ADSMathSciNetGoogle Scholar
  35. [35]
    T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
  36. [36]
    I. Bena, On the construction of local fields in the bulk of AdS 5 and other spaces, Phys. Rev. D 62 (2000) 066007 [hep-th/9905186] [INSPIRE].ADSMathSciNetGoogle Scholar
  37. [37]
    A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].ADSMathSciNetGoogle Scholar
  38. [38]
    A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: A boundary view of horizons and locality, Phys. Rev. D 73 (2006) 086003 [hep-th/0506118] [INSPIRE].ADSMathSciNetGoogle Scholar
  39. [39]
    A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT and the fate of the BTZ singularity, AMS/IP Stud. Adv. Math. 44 (2008) 85 [arXiv:0710.4334] [INSPIRE].MathSciNetMATHGoogle Scholar
  40. [40]
    L. Susskind and N. Toumbas, Wilson loops as precursors, Phys. Rev. D 61 (2000) 044001 [hep-th/9909013] [INSPIRE].ADSMathSciNetGoogle Scholar
  41. [41]
    S.B. Giddings and M. Lippert, Precursors, black holes and a locality bound, Phys. Rev. D 65 (2002) 024006 [hep-th/0103231] [INSPIRE].ADSMathSciNetGoogle Scholar
  42. [42]
    D.L. Jafferis and S.J. Suh, The Gravity Duals of Modular Hamiltonians, arXiv:1412.8465 [INSPIRE].
  43. [43]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Souvik Banerjee
    • 1
  • Jan-Willem Bryan
    • 1
  • Kyriakos Papadodimas
    • 1
    • 2
  • Suvrat Raju
    • 3
  1. 1.Van Swinderen Institute for Particle Physics and GravityUniversity of GroningenGroningenThe Netherlands
  2. 2.Theoretical Physics DepartmentCERNGeneva 23Switzerland
  3. 3.International Centre for Theoretical SciencesTata Institute of Fundamental ResearchBengaluruIndia

Personalised recommendations