Abstract
We consider the gravity dual of the modular Hamiltonian associated to a general subregion of a boundary theory. We use it to argue that the relative entropy of nearby states is given by the relative entropy in the bulk, to leading order in the bulk gravitational coupling. We also argue that the boundary modular flow is dual to the bulk modular flow in the entanglement wedge, with implications for entanglement wedge reconstruction.
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References
M. Ohya and D. Petz, Quantum entropy and its use, corrected 2nd printing, Text and Monographs in Physics, Springer Study Edition, Springer (2004).
H. Araki, Relative entropy of states of von Neumann algebras, Publ. Res. Inst. Math. Sci. Kyoto 1976 (1976) 809 [INSPIRE].
H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].
A.C. Wall, A Proof of the generalized second law for rapidly-evolving Rindler horizons, Phys. Rev. D 82 (2010) 124019 [arXiv:1007.1493] [INSPIRE].
A.C. Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices, Phys. Rev. D 85 (2012) 104049 [arXiv:1105.3445] [INSPIRE].
D.L. Jafferis and S.J. Suh, The gravity duals of modular hamiltonians, arXiv:1412.8465 [INSPIRE].
D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative entropy and holography, JHEP 08 (2013) 060 [arXiv:1305.3182] [INSPIRE].
N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement ‘thermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].
T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from entanglement in holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
J. Lin, M. Marcolli, H. Ooguri and B. Stoica, Locality of gravitational systems from entanglement of conformal field theories, Phys. Rev. Lett. 114 (2015) 221601 [arXiv:1412.1879] [INSPIRE].
N. Lashkari, C. Rabideau, P. Sabella-Garnier and M. Van Raamsdonk, Inviolable energy conditions from entanglement inequalities, JHEP 06 (2015) 067 [arXiv:1412.3514] [INSPIRE].
N. Lashkari and M. Van Raamsdonk, Canonical energy is quantum Fisher information, JHEP 04 (2016) 153 [arXiv:1508.00897] [INSPIRE].
N. Lashkari, J. Lin, H. Ooguri, M. van Raamsdonk, B. Stoica, to appear.
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].
H. Casini, F.D. Mazzitelli and E. Testé, Area terms in entanglement entropy, Phys. Rev. D 91 (2015) 104035 [arXiv:1412.6522] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].
W. Donnelly and A.C. Wall, Geometric entropy and edge modes of the electromagnetic field, arXiv:1506.05792 [INSPIRE].
D.N. Kabat, Black hole entropy and entropy of entanglement, Nucl. Phys. B 453 (1995) 281 [hep-th/9503016] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].
J.M. Bardeen, B. Carter and S.W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE].
V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
B. Swingle and M. Van Raamsdonk, Universality of Gravity from Entanglement, arXiv:1405.2933 [INSPIRE].
J. Lee, A. Lewkowycz, E. Perlmutter and B.R. Safdi, Rényi entropy, stationarity and entanglement of the conformal scalar, JHEP 03 (2015) 075 [arXiv:1407.7816] [INSPIRE].
S. Hollands and R.M. Wald, Stability of black holes and black branes, Commun. Math. Phys. 321 (2013) 629 [arXiv:1201.0463] [INSPIRE].
H. Casini and M. Huerta, Reduced density matrix and internal dynamics for multicomponent regions, Class. Quant. Grav. 26 (2009) 185005 [arXiv:0903.5284] [INSPIRE].
A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
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].
R. Bousso, S. Leichenauer and V. Rosenhaus, Light-sheets and AdS/CFT, Phys. Rev. D 86 (2012) 046009 [arXiv:1203.6619] [INSPIRE].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
J.M. Maldacena, Eternal black holes in Anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
S. Carlip and C. Teitelboim, The off-shell black hole, Class. Quant. Grav. 12 (1995) 1699 [gr-qc/9312002] [INSPIRE].
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].
V.E. Hubeny and M. Rangamani, Causal holographic information, JHEP 06 (2012) 114 [arXiv:1204.1698] [INSPIRE].
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].
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].
K. Papadodimas and S. Raju, An infalling observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
W.R. Kelly and A.C. Wall, Coarse-grained entropy and causal holographic information in AdS/CFT, JHEP 03 (2014) 118 [arXiv:1309.3610] [INSPIRE].
T. Jacobson, Entanglement equilibrium and the Einstein equation, Phys. Rev. Lett. 116 (2016) 201101 [arXiv:1505.04753] [INSPIRE].
R.M. Soni and S.P. Trivedi, Aspects of entanglement entropy for gauge theories, JHEP 01 (2016) 136 [arXiv:1510.07455] [INSPIRE].
K. Van Acoleyen, N. Bultinck, J. Haegeman, M. Marien, V.B. Scholz and F. Verstraete, The entanglement of distillation for gauge theories, arXiv:1511.04369 [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An apologia for firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
D. Marolf and J. Polchinski, Gauge/gravity duality and the black hole interior, Phys. Rev. Lett. 111 (2013) 171301 [arXiv:1307.4706] [INSPIRE].
K.-W. Huang, Central charge and entangled gauge fields, Phys. Rev. D 92 (2015) 025010 [arXiv:1412.2730] [INSPIRE].
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Jafferis, D.L., Lewkowycz, A., Maldacena, J. et al. Relative entropy equals bulk relative entropy. J. High Energ. Phys. 2016, 4 (2016). https://doi.org/10.1007/JHEP06(2016)004
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DOI: https://doi.org/10.1007/JHEP06(2016)004