Skip to main content

The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole

A preprint version of the article is available at arXiv.

Abstract

Bulk quantum fields are often said to contribute to the generalized entropy \( \frac{A}{4{G}_N}+{S}_{\mathrm{bulk}} \) only at O(1). Nonetheless, in the context of evaporating black holes, O(1/GN ) gradients in Sbulk can arise due to large boosts, introducing a quantum extremal surface far from any classical extremal surface. We examine the effect of such bulk quantum effects on quantum extremal surfaces (QESs) and the resulting entanglement wedge in a simple two-boundary 2d bulk system defined by Jackiw-Teitelboim gravity coupled to a 1+1 CFT. Turning on a coupling between one boundary and a further external auxiliary system which functions as a heat sink allows a two-sided otherwise-eternal black hole to evaporate on one side. We find the generalized entropy of the QES to behave as expected from general considerations of unitarity, and in particular that ingoing information disappears from the entanglement wedge after a scambling time \( \frac{\beta }{2\pi}\log \varDelta S+O(1) \) in accord with expectations for holographic implementations of the Hayden-Preskill protocol. We also find an interesting QES phase transition at what one might call the Page time for our process.

References

  1. 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].

    ADS  MathSciNet  Article  Google Scholar 

  2. 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].

    ADS  MathSciNet  Article  Google Scholar 

  3. M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP12 (2014) 162 [arXiv:1408.6300] [INSPIRE].

    ADS  Article  Google Scholar 

  4. V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP07 (2007) 062 [arXiv:0705.0016] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  5. M. Headrick and T. Takayanagi, A holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev.D 76 (2007) 106013 [arXiv:0704.3719] [INSPIRE].

  6. A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP04 (2015) 163 [arXiv:1411.7041] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  7. D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP06 (2016) 004 [arXiv:1512.06431] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  8. X. Dong, D. Harlow and A.C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Phys. Rev. Lett.117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  9. T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP07 (2017) 151 [arXiv:1704.05464] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  10. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  11. S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP08 (2006) 045 [hep-th/0605073] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  12. A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP08 (2013) 090 [arXiv:1304.4926] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  13. X. Dong, A. Lewkowycz and M. Rangamani, Deriving covariant holographic entanglement, JHEP11 (2016) 028 [arXiv:1607.07506] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  14. T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP11 (2013) 074 [arXiv:1307.2892] [INSPIRE].

    ADS  Article  Google Scholar 

  15. N. Engelhardt and A.C. Wall, Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime, JHEP01 (2015) 073 [arXiv:1408.3203] [INSPIRE].

    ADS  Article  Google Scholar 

  16. R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev.D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].

  17. V. Iyer and R.M. Wald, A comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev.D 52 (1995) 4430 [gr-qc/9503052] [INSPIRE].

  18. T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev.D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].

  19. X. Dong, Holographic entanglement entropy for general higher derivative gravity, JHEP01 (2014) 044 [arXiv:1310.5713] [INSPIRE].

  20. R.-X. Miao and W.-Z. Guo, Holographic entanglement entropy for the most general higher derivative gravity, JHEP08 (2015) 031 [arXiv:1411.5579] [INSPIRE].

  21. W.H. Zurek, Entropy evaporated by a black hole, Phys. Rev. Lett.49 (1982) 1683 [INSPIRE].

    ADS  Article  Google Scholar 

  22. D.N. Page, Comment on ‘entropy evaporated by a black hole’, Phys. Rev. Lett.50 (1983) 1013 [INSPIRE].

    ADS  Article  Google Scholar 

  23. P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP09 (2007) 120 [arXiv:0708.4025] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  24. A. Almheiri, Holographic quantum error correction and the projected black hole interior, arXiv:1810.02055 [INSPIRE].

  25. D. Harlow, Jerusalem lectures on black holes and quantum information, Rev. Mod. Phys.88 (2016) 015002 [arXiv:1409.1231] [INSPIRE].

    ADS  Article  Google Scholar 

  26. D. Marolf, The black hole information problem: past, present and future, Rept. Prog. Phys.80 (2017) 092001 [arXiv:1703.02143] [INSPIRE].

    ADS  Article  Google Scholar 

  27. G. Penington, Entanglement wedge reconstruction and the information paradox, arXiv:1905.08255 [INSPIRE].

  28. S. Leichenauer, Disrupting entanglement of black holes, Phys. Rev.D 90 (2014) 046009 [arXiv:1405.7365] [INSPIRE].

  29. P. Gao, D.L. Jafferis and A.C. Wall, Traversable wormholes via a double trace deformation, JHEP12 (2017) 151 [arXiv:1608.05687] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  30. J. Maldacena, D. Stanford and Z. Yang, Diving into traversable wormholes, Fortsch. Phys.65 (2017) 1700034 [arXiv:1704.05333] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  31. J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional nearly anti-de-Sitter space, PTEP2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].

  32. J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS2 backreaction and holography, JHEP07 (2016) 139 [arXiv:1606.03438] [INSPIRE].

  33. A. Almheiri and B. Kang, Conformal symmetry breaking and thermodynamics of near-extremal black holes, JHEP10 (2016) 052 [arXiv:1606.04108] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  34. S. Sachdev, Universal low temperature theory of charged black holes with AdS2 horizons, J. Math. Phys.60 (2019) 052303 [arXiv:1902.04078] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  35. A. Almheiri and J. Polchinski, Models of AdS2 backreaction and holography, JHEP11 (2015) 014 [arXiv:1402.6334] [INSPIRE].

    ADS  Article  Google Scholar 

  36. K. Jensen, Chaos in AdS2 holography, Phys. Rev. Lett.117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].

    ADS  Article  Google Scholar 

  37. J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv:1804.00491 [INSPIRE].

  38. G.J. Galloway and M. Graf, Rigidity of asymptotically AdS2 × S 2spacetimes, arXiv:1803.10529 [INSPIRE].

  39. T. Faulkner, R.G. Leigh, O. Parrikar and H. Wang, Modular Hamiltonians for deformed half-spaces and the averaged null energy condition, JHEP09 (2016) 038 [arXiv:1605.08072] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  40. P. Calabrese and J. Cardy, Entanglement and correlation functions following a local quench: a conformal field theory approach, J. Stat. Mech.0710 (2007) P10004 [arXiv:0708.3750] [INSPIRE].

    Article  Google Scholar 

  41. P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys.A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].

  42. C.T. Asplund and A. Bernamonti, Mutual information after a local quench in conformal field theory, Phys. Rev.D 89 (2014) 066015 [arXiv:1311.4173] [INSPIRE].

  43. I. Affleck and A.W.W. Ludwig, The Fermi edge singularity and boundary condition changing operators, J. Phys.A 27 (1994) 5375 [cond-mat/9405057].

  44. R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev.D 93 (2016) 064044 [arXiv:1506.02669] [INSPIRE].

  45. D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett.71 (1993) 1291 [gr-qc/9305007] [INSPIRE].

  46. D. Harlow, The Ryu-Takayanagi formula from quantum error correction, Commun. Math. Phys.354 (2017) 865 [arXiv:1607.03901] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  47. R.P. Geroch, The domain of dependence, J. Math. Phys.11 (1970) 437 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  48. E.H. Lieb and M.B. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys.14 (1973) 1938 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  49. P. Hayden, R. Jozsa, D. Petz and A. Winter, Structure of states which satisfy strong subadditivity of quantum entropy with equality, Commun. Math. Phys.246 (2004) 359 [quant-ph/0304007].

  50. O. Fawzi and R. Renner, Quantum conditional mutual information and approximate Markov chains, Commun. Math. Phys.340 (2015) 575 [arXiv:1410.0664].

    ADS  MathSciNet  Article  Google Scholar 

  51. D. Sutter, O. Fawzi and R. Renner, Universal recovery map for approximate Markov chains, Proc. Roy. Soc. Lond.A 472 (2016) 20150623 [arXiv:1504.07251] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  52. M. Berta and M. Tomamichel, The fidelity of recovery is multiplicative, arXiv:1502.07973.

  53. K.P. Seshadreesan and M.M. Wilde, Fidelity of recovery, squashed entanglement, and measurement recoverability, Phys. Rev.A 92 (2015) 042321 [arXiv:1410.1441].

  54. A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, University of California, Santa Barbara, CA, U.S.A., 7 April 2015.

  55. A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, University of California, Santa Barbara, CA, U.S.A., 27 May 2015.

  56. J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev.D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].

  57. S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett.70 (1993) 3339 [cond-mat/9212030] [INSPIRE].

  58. A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP02 (2013) 062 [arXiv:1207.3123] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  59. A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An apologia for firewalls, JHEP09 (2013) 018 [arXiv:1304.6483] [INSPIRE].

    ADS  Article  Google Scholar 

  60. D. Marolf and J. Polchinski, Gauge/gravity duality and the black hole interior, Phys. Rev. Lett.111 (2013) 171301 [arXiv:1307.4706] [INSPIRE].

    ADS  Article  Google Scholar 

  61. E. Verlinde and H. Verlinde, Behind the horizon in AdS/CFT, arXiv:1311.1137 [INSPIRE].

  62. J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys.61 (2013) 781 [arXiv:1306.0533] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

Download references

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Netta Engelhardt.

Additional information

ArXiv ePrint: 1905.08762

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Almheiri, A., Engelhardt, N., Marolf, D. et al. The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole. J. High Energ. Phys. 2019, 63 (2019). https://doi.org/10.1007/JHEP12(2019)063

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP12(2019)063

Keywords

  • 2D Gravity
  • AdS-CFT Correspondence
  • Black Holes in String Theory
  • Gauge- gravity correspondence