Journal of High Energy Physics

, 2016:26 | Cite as

Rescuing complementarity with little drama

  • Ning Bao
  • Adam Bouland
  • Aidan Chatwin-DaviesEmail author
  • Jason Pollack
  • Henry Yuen
Open Access
Regular Article - Theoretical Physics


The AMPS paradox challenges black hole complementarity by apparently constructing a way for an observer to bring information from the outside of the black hole into its interior if there is no drama at its horizon, making manifest a violation of monogamy of entanglement. We propose a new resolution to the paradox: this violation cannot be explicitly checked by an infalling observer in the finite proper time they have to live after crossing the horizon. Our resolution depends on a weak relaxation of the no-drama condition (we call it “little-drama”) which is the “complementarity dual” of scrambling of information on the stretched horizon. When translated to the description of the black hole interior, this implies that the fine-grained quantum information of infalling matter is rapidly diffused across the entire interior while classical observables and coarse-grained geometry remain unaffected. Under the assumption that information has diffused throughout the interior, we consider the difficulty of the information-theoretic task that an observer must perform after crossing the event horizon of a Schwarzschild black hole in order to verify a violation of monogamy of entanglement. We find that the time required to complete a necessary subroutine of this task, namely the decoding of Bell pairs from the interior and the late radiation, takes longer than the maximum amount of time that an observer can spend inside the black hole before hitting the singularity. Therefore, an infalling observer cannot observe monogamy violation before encountering the singularity.


Black Holes Models of Quantum Gravity 


Open Access

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  1. [1]
    S.W. Hawking, Information loss in black holes, Phys. Rev. D 72 (2005) 084013 [hep-th/0507171] [INSPIRE].ADSMathSciNetGoogle 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].ADSMathSciNetCrossRefzbMATHGoogle 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]
    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
  5. [5]
    P. Chen, Y.C. Ong and D.-h. Yeom, Black Hole Remnants and the Information Loss Paradox, Phys. Rept. 603 (2015) 1 [arXiv:1412.8366] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    S. Lloyd and J. Preskill, Unitarity of black hole evaporation in final-state projection models, JHEP 08 (2014) 126 [arXiv:1308.4209] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    K. Papadodimas and S. Raju, An Infalling Observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    S.D. Mathur, The fuzzball proposal for black holes: An elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    S.B. Giddings, Nonviolent nonlocality, Phys. Rev. D 88 (2013) 064023 [arXiv:1211.7070] [INSPIRE].ADSGoogle Scholar
  11. [11]
    M. Hotta and A. Sugita, The Fall of Black Hole Firewall: Natural Nonmaximal Entanglement for Page Curve, PTEP 2015 (2015) 123B04 [arXiv:1505.05870] [INSPIRE].
  12. [12]
    Y. Nomura, F. Sanches and S.J. Weinberg, Black Hole Interior in Quantum Gravity, Phys. Rev. Lett. 114 (2015) 201301 [arXiv:1412.7539] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    Y. Nomura, F. Sanches and S.J. Weinberg, Relativeness in Quantum Gravity: Limitations and Frame Dependence of Semiclassical Descriptions, JHEP 04 (2015) 158 [arXiv:1412.7538] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    Y. Nomura and N. Salzetta, Why Firewalls Need Not Exist, Phys. Lett. B 761 (2016) 62 [arXiv:1602.07673] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    D. Harlow and P. Hayden, Quantum Computation vs. Firewalls, JHEP 06 (2013) 085 [arXiv:1301.4504] [INSPIRE].
  16. [16]
    C.H. Bennett, H.J. Bernstein, S. Popescu and B. Schumacher, Concentrating partial entanglement by local operations, Phys. Rev. A 53 (1996) 2046 [quant-ph/9511030] [INSPIRE].
  17. [17]
    J. Oppenheim and W.G. Unruh, Firewalls and flat mirrors: An alternative to the AMPS experiment which evades the Harlow-Hayden obstacle, JHEP 03 (2014) 120 [arXiv:1401.1523] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    S. Aaronson, Computational complexity underpinnings of the Harlow-Hayden argument, unpublished.Google Scholar
  19. [19]
    K.S. Thorne, R.H. Price and D.A. Macdonald, Black Holes: The Membrane Paradigm, Yale University Press, New Haven, U.S.A. (1986), pg. 367.Google Scholar
  20. [20]
    Y. Nomura, Physical Theories, Eternal Inflation and Quantum Universe, JHEP 11 (2011) 063 [arXiv:1104.2324] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  21. [21]
    P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the Fast Scrambling Conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    D.N. Page, Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole, Phys. Rev. D 13 (1976) 198 [INSPIRE].ADSGoogle Scholar
  28. [28]
    D.N. Page, Particle Emission Rates from a Black Hole. 2. Massless Particles from a Rotating Hole, Phys. Rev. D 14 (1976) 3260 [INSPIRE].
  29. [29]
    S.B. Giddings, Hawking radiation, the Stefan-Boltzmann law and unitarization, Phys. Lett. B 754 (2016) 39 [arXiv:1511.08221] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  30. [30]
    D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    D.N. Page, Time Dependence of Hawking Radiation Entropy, JCAP 09 (2013) 028 [arXiv:1301.4995] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    F.G.S.L. Brandao, A. Harrow and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, arXiv:1208.0692.
  33. [33]
    C. Dankert, R. Cleve, J. Emerson and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Phys. Rev. Lett. A 80 (2009) 012304.Google Scholar
  34. [34]
    J. Shore and R. Johnson, Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy, IEEE Trans. Inf. Theory 26 (1980) 26.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    B. Freivogel, R.A. Jefferson, L. Kabir and I.-S. Yang, Geometry of the Infalling Causal Patch, Phys. Rev. D 91 (2015) 044036 [arXiv:1406.6043] [INSPIRE].ADSMathSciNetGoogle Scholar
  36. [36]
    S. Raju, Smooth Causal Patches for AdS Black Holes, arXiv:1604.03095 [INSPIRE].
  37. [37]
    S.D. Mathur and D. Turton, The flaw in the firewall argument, Nucl. Phys. B 884 (2014) 566 [arXiv:1306.5488] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  38. [38]
    S.D. Mathur, A model with no firewall, arXiv:1506.04342 [INSPIRE].
  39. [39]
    G. Dotti and R.J. Gleiser, Gravitational instability of the inner static region of a Reissner-Nordstrom black hole, Class. Quant. Grav. 27 (2010) 185007 [arXiv:1001.0152] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    N. Engelhardt and G.T. Horowitz, Holographic Consequences of a No Transmission Principle, Phys. Rev. D 93 (2016) 026005 [arXiv:1509.07509] [INSPIRE].ADSMathSciNetGoogle Scholar

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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Walter Burke Institute for Theoretical Physics, California Institute of TechnologyPasadenaU.S.A.
  2. 2.Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of TechnologyCambridgeU.S.A.
  3. 3.Computer Science DivisionUniversity of California, BerkeleyBerkeleyU.S.A.

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