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Bulk locality and quantum error correction in AdS/CFT

  • Ahmed Almheiri
  • Xi Dong
  • Daniel Harlow
Open Access
Regular Article - Theoretical Physics

Abstract

We point out a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction. Bulk notions such as Bogoliubov transformations, location in the radial direction, and the holographic entropy bound all have natural CFT interpretations in the language of quantum error correction. We also show that the question of whether bulk operator reconstruction works only in the causal wedge or all the way to the extremal surface is related to the question of whether or not the quantum error correcting code realized by AdS/CFT is also a “quantum secret sharing scheme”, and suggest a tensor network calculation that may settle the issue. Interestingly, the version of quantum error correction which is best suited to our analysis is the somewhat nonstandard “operator algebra quantum error correction” of Beny, Kempf, and Kribs. Our proposal gives a precise formulation of the idea of “subregion-subregion” duality in AdS/CFT, and clarifies the limits of its validity.

Keywords

AdS-CFT Correspondence 1/N Expansion Black Holes in String Theory 

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]
    T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
  2. [2]
    D. Harlow and D. Stanford, Operator dictionaries and wave functions in AdS/CFT and dS/CFT, arXiv:1104.2621 [INSPIRE].
  3. [3]
    R. Streater and A. Wightman, PCT, spin and statistics, and all that, Benjamin Cummings (1964).Google Scholar
  4. [4]
    R. Haag, Local quantum physics: fields, particles, algebras, Springer (1996).Google Scholar
  5. [5]
    B. Chilian and K. Fredenhagen, The time slice axiom in perturbative quantum field theory on globally hyperbolic spacetimes, Commun. Math. Phys. 287 (2009) 513 [arXiv:0802.1642] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    M. Benini, C. Dappiaggi, T.-P. Hack and A. Schenkel, A C -algebra for quantized principal U(1)-connections on globally hyperbolic Lorentzian manifolds, Commun. Math. Phys. 332 (2014) 477 [arXiv:1307.3052] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    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
  8. [8]
    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
  9. [9]
    P.W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52 (1995) R2493.ADSCrossRefGoogle Scholar
  10. [10]
    D. Gottesman, Stabilizer codes and quantum error correction, quant-ph/9705052 [INSPIRE].
  11. [11]
    R. Bousso, S. Leichenauer and V. Rosenhaus, Light-sheets and AdS/CFT, Phys. Rev. D 86 (2012) 046009 [arXiv:1203.6619] [INSPIRE].ADSGoogle Scholar
  12. [12]
    B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a density matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    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
  14. [14]
    V.E. Hubeny and M. Rangamani, Causal holographic information, JHEP 06 (2012) 114 [arXiv:1204.1698] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    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
  16. [16]
    D. Harlow, Aspects of the Papadodimas-Raju proposal for the black hole interior, JHEP 11 (2014) 055 [arXiv:1405.1995] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    E. Verlinde and H. Verlinde, Black hole entanglement and quantum error correction, JHEP 10 (2013) 107 [arXiv:1211.6913] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, Bulk and transhorizon measurements in AdS/CFT, JHEP 10 (2012) 165 [arXiv:1201.3664] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    D. Kabat, G. Lifschytz and D.A. Lowe, Constructing local bulk observables in interacting AdS/CFT, Phys. Rev. D 83 (2011) 106009 [arXiv:1102.2910] [INSPIRE].ADSGoogle Scholar
  20. [20]
    R. Bousso and L. Randall, Holographic domains of Anti-de Sitter space, JHEP 04 (2002) 057 [hep-th/0112080] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    K. Papadodimas and S. Raju, An infalling observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    J. Preskill, Lecture notes on quantum computation, available online (1998).
  23. [23]
    M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge U.K. (2010).CrossRefMATHGoogle Scholar
  24. [24]
    D. Harlow and P. Hayden, Quantum computation vs. firewalls, JHEP 06 (2013) 085 [arXiv:1301.4504] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    R. Cleve, D. Gottesman and H.-K. Lo, How to share a quantum secret, Phys. Rev. Lett. 83 (1999) 648 [quant-ph/9901025] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    B. Schumacher and M.A. Nielsen, Quantum data processing and error correction, Phys. Rev. A 54 (1996) 2629 [quant-ph/9604022] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    D. Harlow, Jerusalem lectures on black holes and quantum information, arXiv:1409.1231 [INSPIRE].
  28. [28]
    M. Grassl, T. Beth, and T. Pellizzari, Codes for the quantum erasure channel, Phys. Rev. A 56 (1997) 33 [quant-ph/9610042] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    B. Schumacher and M.D. Westmoreland, Approximate quantum error correction, Quant. Informat. Process. 1 (2002) 5.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    M. M. Wolf, F. Verstraete, M. B. Hastings and J.I. Cirac, Area laws in quantum systems: mutual information and correlations, Phys. Rev. Lett. 100 (2008) 070502 [arXiv:0704.3906].ADSMathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    C. Bény, A. Kempf and D.W. Kribs, Generalization of quantum error correction via the heisenberg picture, Phys. Rev. Lett. 98 (2007) 100502 [quant-ph/0608071].CrossRefGoogle Scholar
  33. [33]
    C. Bény, A. Kempf and D.W. Kribs, Quantum error correction of observables, Phys. Rev. A 76 (2007) 042303 [arXiv:0705.1574].ADSCrossRefGoogle Scholar
  34. [34]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    D.L. Jafferis and S.J. Suh, The gravity duals of modular hamiltonians, arXiv:1412.8465 [INSPIRE].
  37. [37]
    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101 (2008) 110501 [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].ADSGoogle Scholar
  42. [42]
    G. Evenbly and G. Vidal, Tensor network states and geometry, J. Stat. Phys. 145 (2011) 891 [arXiv:1106.1082].ADSMathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    I. Heemskerk, Construction of bulk fields with gauge redundancy, JHEP 09 (2012) 106 [arXiv:1201.3666] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    D. Kabat and G. Lifschytz, Decoding the hologram: scalar fields interacting with gravity, Phys. Rev. D 89 (2014) 066010 [arXiv:1311.3020] [INSPIRE].ADSGoogle Scholar
  45. [45]
    H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].ADSGoogle Scholar
  46. [46]
    V.E. Hubeny, Extremal surfaces as bulk probes in AdS/CFT, JHEP 07 (2012) 093 [arXiv:1203.1044] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative entropy and holography, JHEP 08 (2013) 060 [arXiv:1305.3182] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    G. Gibbons, S. Hawking, G. T. Horowitz and M.J. Perry, Positive mass theorems for black holes, Commun. Math. Phys. 88 (1983) 295.ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    L. Susskind and E. Witten, The holographic bound in Anti-de Sitter space, hep-th/9805114 [INSPIRE].
  50. [50]
    L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    S. Leichenauer and V. Rosenhaus, AdS black holes, the bulk-boundary dictionary and smearing functions, Phys. Rev. D 88 (2013) 026003 [arXiv:1304.6821] [INSPIRE].ADSGoogle Scholar
  52. [52]
    S.-J. Rey and V. Rosenhaus, Scanning tunneling macroscopy, black holes and AdS/CFT bulk locality, JHEP 07 (2014) 050 [arXiv:1403.3943] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    B.D. Chowdhury and M.K. Parikh, When UV and IR collide: inequivalent CFTs from different foliations of AdS, arXiv:1407.4467 [INSPIRE].

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Stanford Institute for Theoretical Physics, Department of PhysicsStanford UniversityStanfordUnited States
  2. 2.Princeton Center for Theoretical SciencePrinceton UniversityPrincetonUnited States

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