Advertisement

Time evolution of entanglement entropy from black hole interiors

  • Thomas Hartman
  • Juan Maldacena
Article

Abstract

We compute the time-dependent entanglement entropy of a CFT which starts in relatively simple initial states. The initial states are the thermofield double for thermal states, dual to eternal black holes, and a particular pure state, dual to a black hole formed by gravitational collapse. The entanglement entropy grows linearly in time. This linear growth is directly related to the growth of the black hole interior measured along “nice” spatial slices. These nice slices probe the spacelike direction in the interior, at a fixed special value of the interior time. In the case of a two-dimensional CFT, we match the bulk and boundary computations of the entanglement entropy. We briefly discuss the long time behavior of various correlators, computed via classical geodesics or surfaces, and point out that their exponential decay comes about for similar reasons. We also present the time evolution of the wavefunction in the tensor network description.

Keywords

AdS-CFT Correspondence Black Holes 

References

  1. [1]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    V.E. Hubeny, Extremal surfaces as bulk probes in AdS/CFT, JHEP 07 (2012) 093 [arXiv:1203.1044] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    J. Abajo-Arrastia, J. Aparicio and E. Lopez, Holographic evolution of entanglement entropy, JHEP 11 (2010) 149 [arXiv:1006.4090] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    J. Aparicio and E. Lopez, Evolution of two-point functions from holography, JHEP 12 (2011) 082 [arXiv:1109.3571] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    T. Albash and C.V. Johnson, Evolution of holographic entanglement entropy after thermal and electromagnetic quenches, New J. Phys. 13 (2011) 045017 [arXiv:1008.3027] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    V. Balasubramanian et al., Thermalization of strongly coupled field theories, Phys. Rev. Lett. 106 (2011) 191601 [arXiv:1012.4753] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    V. Balasubramanian et al., Holographic thermalization, Phys. Rev. D 84 (2011) 026010 [arXiv:1103.2683] [INSPIRE].ADSGoogle Scholar
  9. [9]
    C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].CrossRefGoogle Scholar
  11. [11]
    P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. (2005) P04010 [cond-mat/0503393].
  12. [12]
    T. Takayanagi and T. Ugajin, Measuring black hole formations by entanglement entropy via coarse-graining, JHEP 11 (2010) 054 [arXiv:1008.3439] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    C.T. Asplund and S.G. Avery, Evolution of entanglement entropy in the D1-D5 brane system, Phys. Rev. D 84 (2011) 124053 [arXiv:1108.2510] [INSPIRE].ADSGoogle Scholar
  14. [14]
    P. Basu and S.R. Das, Quantum quench across a holographic critical point, JHEP 01 (2012) 103 [arXiv:1109.3909] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    P. Basu, D. Das, S.R. Das and T. Nishioka, Quantum quench across a zero temperature holographic superfluid transition, JHEP 03 (2013) 146 [arXiv:1211.7076] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    A. Buchel, L. Lehner, R.C. Myers and A. van Niekerk, Quantum quenches of holographic plasmas, arXiv:1302.2924 [INSPIRE].
  17. [17]
    V. Balasubramanian, A. Bernamonti, N. Copland, B. Craps and F. Galli, Thermalization of mutual and tripartite information in strongly coupled two dimensional conformal field theories, Phys. Rev. D 84 (2011) 105017 [arXiv:1110.0488] [INSPIRE].ADSGoogle Scholar
  18. [18]
    A. Allais and E. Tonni, Holographic evolution of the mutual information, JHEP 01 (2012) 102 [arXiv:1110.1607] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    M. Nozaki, T. Numasawa and T. Takayanagi, Holographic local quenches and entanglement density, arXiv:1302.5703 [INSPIRE].
  20. [20]
    J. Maldacena and G.L. Pimentel, Entanglement entropy in de Sitter space, JHEP 02 (2013) 038 [arXiv:1210.7244] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    W. Israel, Thermo field dynamics of black holes, Phys. Lett. A 57 (1976) 107 [INSPIRE].MathSciNetADSGoogle Scholar
  22. [22]
    J.M. Maldacena, Eternal black holes in Anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].MathSciNetGoogle Scholar
  24. [24]
    T. Azeyanagi, T. Nishioka and T. Takayanagi, Near extremal black hole entropy as entanglement entropy via AdS 2 /CF T 1, Phys. Rev. D 77 (2008) 064005 [arXiv:0710.2956] [INSPIRE].MathSciNetADSGoogle Scholar
  25. [25]
    J. Louko and D. Marolf, Single exterior black holes and the AdS/CFT conjecture, Phys. Rev. D 59 (1999) 066002 [hep-th/9808081] [INSPIRE].MathSciNetADSGoogle Scholar
  26. [26]
    J. Polchinski, String theory and black hole complementarity, hep-th/9507094 [INSPIRE].
  27. [27]
    C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [INSPIRE].MathSciNetADSGoogle Scholar
  28. [28]
    P. Calabrese and J. Cardy, Quantum quenches in extended systems, J. Stat. Mech. 0706 (2007) P06008 [arXiv:0704.1880] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  29. [29]
    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].MathSciNetGoogle Scholar
  30. [30]
    I.A. Morrison and M.M. Roberts, Mutual information between thermo-field doubles and disconnected holographic boundaries, arXiv:1211.2887 [INSPIRE].
  31. [31]
    M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].ADSGoogle Scholar
  32. [32]
    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].MathSciNetADSGoogle Scholar
  33. [33]
    V.E. Hubeny and M. Rangamani, Holographic entanglement entropy for disconnected regions, JHEP 03 (2008) 006 [arXiv:0711.4118] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  35. [35]
    J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    M. Parikh and P. Samantray, Rindler-AdS/CFT, arXiv:1211.7370 [INSPIRE].
  37. [37]
    G. Festuccia and H. Liu, A Bohr-Sommerfeld quantization formula for quasinormal frequencies of AdS black holes, Adv. Sci. Lett. 2 (2009) 221 [arXiv:0811.1033] [INSPIRE].CrossRefGoogle Scholar
  38. [38]
    L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The black hole singularity in AdS/CFT, JHEP 02 (2004) 014 [hep-th/0306170] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    G. Festuccia and H. Liu, Excursions beyond the horizon: black hole singularities in Yang-Mills theories. I, JHEP 04 (2006) 044 [hep-th/0506202] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    G. Vidal, Entanglement Renormalization: an introduction, in Understanding quantum phase transitions, L.D. Carr ed., Taylor & Francis, Boca Raton U.S.A. (2010), arXiv:0912.1651.
  41. [41]
    S. Ostlund and S. Rommer, Thermodynamic limit of density matrix renormalization for the spin-1 Heisenberg chain, Phys. Rev. Lett. 75 (1995) 3537 [cond-mat/9503107] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    S.R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69 (1992) 2863 [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326 (2011) 96 [arXiv:1008.3477].MathSciNetADSMATHCrossRefGoogle Scholar
  44. [44]
    F. Verstraete and J.I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, cond-mat/0407066.
  45. [45]
    G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101 (2008) 110501 [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].ADSGoogle Scholar
  47. [47]
    G. Evenbly and G. Vidal, Tensor network states and geometry, J. Stat. Phys. 145 (2011) 891 [arXiv:1106.1082].MathSciNetADSMATHCrossRefGoogle Scholar
  48. [48]
    B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].
  49. [49]
    M. Nozaki, S. Ryu and T. Takayanagi, Holographic geometry of entanglement renormalization in quantum field theories, arXiv:1208.3469 [INSPIRE].
  50. [50]
    J.M. Maldacena and L. Susskind, D-branes and fat black holes, Nucl. Phys. B 475 (1996) 679 [hep-th/9604042] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    D. Harlow and P. Hayden, Quantum computation vs. firewalls, arXiv:1301.4504 [INSPIRE].

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.

Personalised recommendations