Time evolution of entanglement entropy from black hole interiors. (English) Zbl 1342.83170
Summary: 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.
MSC:
| 83C57 | Black holes |
| 83E30 | String and superstring theories in gravitational theory |
| 81P40 | Quantum coherence, entanglement, quantum correlations |
References:
| [1] | S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE]. · Zbl 1228.83110 · doi:10.1103/PhysRevLett.96.181602 |
| [2] | V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP07 (2007) 062 [arXiv:0705.0016] [INSPIRE]. · doi:10.1088/1126-6708/2007/07/062 |
| [3] | V.E. Hubeny, Extremal surfaces as bulk probes in AdS/CFT, JHEP07 (2012) 093 [arXiv:1203.1044] [INSPIRE]. · Zbl 1397.83155 · doi:10.1007/JHEP07(2012)093 |
| [4] | J. Abajo-Arrastia, J. Aparicio and E. Lopez, Holographic evolution of entanglement entropy, JHEP11 (2010) 149 [arXiv:1006.4090] [INSPIRE]. · Zbl 1294.81128 · doi:10.1007/JHEP11(2010)149 |
| [5] | J. Aparicio and E. Lopez, Evolution of two-point functions from holography, JHEP12 (2011) 082 [arXiv:1109.3571] [INSPIRE]. · Zbl 1306.81145 · doi:10.1007/JHEP12(2011)082 |
| [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]. · Zbl 1448.83015 · doi:10.1088/1367-2630/13/4/045017 |
| [7] | V. Balasubramanian et al., Thermalization of strongly coupled field theories, Phys. Rev. Lett.106 (2011) 191601 [arXiv:1012.4753] [INSPIRE]. · doi:10.1103/PhysRevLett.106.191601 |
| [8] | V. Balasubramanian et al., Holographic thermalization, Phys. Rev.D 84 (2011) 026010 [arXiv:1103.2683] [INSPIRE]. |
| [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]. · Zbl 0990.81564 · doi:10.1016/0550-3213(94)90402-2 |
| [10] | P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.0406 (2004) P06002 [hep-th/0405152] [INSPIRE]. · Zbl 1082.82002 · doi:10.1088/1742-5468/2004/06/P06002 |
| [11] | P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. (2005) P04010 [cond-mat/0503393]. · Zbl 1456.82578 |
| [12] | T. Takayanagi and T. Ugajin, Measuring black hole formations by entanglement entropy via coarse-graining, JHEP11 (2010) 054 [arXiv:1008.3439] [INSPIRE]. · Zbl 1294.81237 · doi:10.1007/JHEP11(2010)054 |
| [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]. |
| [14] | P. Basu and S.R. Das, Quantum quench across a holographic critical point, JHEP01 (2012) 103 [arXiv:1109.3909] [INSPIRE]. · Zbl 1306.81191 · doi:10.1007/JHEP01(2012)103 |
| [15] | P. Basu, D. Das, S.R. Das and T. Nishioka, Quantum quench across a zero temperature holographic superfluid transition, JHEP03 (2013) 146 [arXiv:1211.7076] [INSPIRE]. · Zbl 1342.83229 · doi:10.1007/JHEP03(2013)146 |
| [16] | A. Buchel, L. Lehner, R.C. Myers and A. van Niekerk, Quantum quenches of holographic plasmas, arXiv:1302.2924 [INSPIRE]. |
| [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]. |
| [18] | A. Allais and E. Tonni, Holographic evolution of the mutual information, JHEP01 (2012) 102 [arXiv:1110.1607] [INSPIRE]. · Zbl 1306.81144 · doi:10.1007/JHEP01(2012)102 |
| [19] | M. Nozaki, T. Numasawa and T. Takayanagi, Holographic local quenches and entanglement density, arXiv:1302.5703 [INSPIRE]. · Zbl 1342.83111 |
| [20] | J. Maldacena and G.L. Pimentel, Entanglement entropy in de Sitter space, JHEP02 (2013) 038 [arXiv:1210.7244] [INSPIRE]. · Zbl 1342.83253 · doi:10.1007/JHEP02(2013)038 |
| [21] | W. Israel, Thermo field dynamics of black holes, Phys. Lett.A 57 (1976) 107 [INSPIRE]. |
| [22] | J.M. Maldacena, Eternal black holes in Anti-de Sitter, JHEP04 (2003) 021 [hep-th/0106112] [INSPIRE]. · doi:10.1088/1126-6708/2003/04/021 |
| [23] | T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys.A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE]. · Zbl 1179.81138 |
| [24] | T. Azeyanagi, T. Nishioka and T. Takayanagi, Near extremal black hole entropy as entanglement entropy via AdS2/CF T1, Phys. Rev.D 77 (2008) 064005 [arXiv:0710.2956] [INSPIRE]. |
| [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]. |
| [26] | J. Polchinski, String theory and black hole complementarity, hep-th/9507094 [INSPIRE]. · Zbl 0850.53020 |
| [27] | C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett.B 333 (1994) 55 [hep-th/9401072] [INSPIRE]. |
| [28] | P. Calabrese and J. Cardy, Quantum quenches in extended systems, J. Stat. Mech.0706 (2007) P06008 [arXiv:0704.1880] [INSPIRE]. · Zbl 1456.81358 · doi:10.1088/1742-5468/2007/06/P06008 |
| [29] | P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys.A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE]. · Zbl 1179.81026 |
| [30] | I.A. Morrison and M.M. Roberts, Mutual information between thermo-field doubles and disconnected holographic boundaries, arXiv:1211.2887 [INSPIRE]. · Zbl 1342.83195 |
| [31] | M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev.D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE]. |
| [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]. |
| [33] | V.E. Hubeny and M. Rangamani, Holographic entanglement entropy for disconnected regions, JHEP03 (2008) 006 [arXiv:0711.4118] [INSPIRE]. · doi:10.1088/1126-6708/2008/03/006 |
| [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]. · Zbl 0968.83514 · doi:10.1103/PhysRevLett.69.1849 |
| [35] | J.M. Maldacena and A. Strominger, AdS3black holes and a stringy exclusion principle, JHEP12 (1998) 005 [hep-th/9804085] [INSPIRE]. · Zbl 0951.83019 · doi:10.1088/1126-6708/1998/12/005 |
| [36] | M. Parikh and P. Samantray, Rindler-AdS/CFT, arXiv:1211.7370 [INSPIRE]. · Zbl 1402.83060 |
| [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]. · doi:10.1166/asl.2009.1029 |
| [38] | L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The black hole singularity in AdS/CFT, JHEP02 (2004) 014 [hep-th/0306170] [INSPIRE]. · doi:10.1088/1126-6708/2004/02/014 |
| [39] | G. Festuccia and H. Liu, Excursions beyond the horizon: black hole singularities in Yang-Mills theories. I, JHEP04 (2006) 044 [hep-th/0506202] [INSPIRE]. · doi:10.1088/1126-6708/2006/04/044 |
| [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] | 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]. · doi:10.1103/PhysRevLett.75.3537 |
| [42] | S.R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett.69 (1992) 2863 [INSPIRE]. · doi:10.1103/PhysRevLett.69.2863 |
| [43] | U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics326 (2011) 96 [arXiv:1008.3477]. · Zbl 1213.81178 · doi:10.1016/j.aop.2010.09.012 |
| [44] | F. Verstraete and J.I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, cond-mat/0407066. · Zbl 1181.82010 |
| [45] | G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett.101 (2008) 110501 [INSPIRE]. · doi:10.1103/PhysRevLett.101.110501 |
| [46] | B. Swingle, Entanglement renormalization and holography, Phys. Rev.D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE]. |
| [47] | G. Evenbly and G. Vidal, Tensor network states and geometry, J. Stat. Phys.145 (2011) 891 [arXiv:1106.1082]. · Zbl 1231.82021 · doi:10.1007/s10955-011-0237-4 |
| [48] | B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE]. |
| [49] | M. Nozaki, S. Ryu and T. Takayanagi, Holographic geometry of entanglement renormalization in quantum field theories, arXiv:1208.3469 [INSPIRE]. · Zbl 1397.81046 |
| [50] | J.M. Maldacena and L. Susskind, D-branes and fat black holes, Nucl. Phys.B 475 (1996) 679 [hep-th/9604042] [INSPIRE]. · Zbl 0925.83076 · doi:10.1016/0550-3213(96)00323-9 |
| [51] | A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP02 (2013) 062 [arXiv:1207.3123] [INSPIRE]. · Zbl 1342.83121 · doi:10.1007/JHEP02(2013)062 |
| [52] | D. Harlow and P. Hayden, Quantum computation vs. firewalls, arXiv:1301.4504 [INSPIRE]. · Zbl 1342.83169 |
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