Abstract
The modular Hamiltonian of reduced states, given essentially by the logarithm of the reduced density matrix, plays an important role within the AdS/CFT correspondence in view of its relation to quantum information. In particular, it is an essential ingredient for quantum information measures of distances between states, such as the relative entropy and the Fisher information metric. However, the modular Hamiltonian is known explicitly only for a few examples. For a family of states ρλ that is parametrized by a scalar λ, the first order contribution in \( \overset{\sim }{\lambda } = \lambda -{\lambda}_0 \) of the modular Hamiltonian to the relative entropy between ρλ and a reference state ρλ0 is completely determined by the entanglement entropy, via the first law of entanglement. For several examples, e.g. for ball-shaped regions in the ground state of CFTs, higher order contributions are known to vanish. In these cases the modular Hamiltonian contributes to the Fisher information metric in a trivial way. We investigate under which conditions the modular Hamiltonian provides a non-trivial contribution to the Fisher information metric, i.e. when the contribution of the modular Hamiltonian to the relative entropy is of higher order in \( \overset{\sim }{\lambda } \). We consider one-parameter families of reduced states on two entangling regions that form an entanglement plateau, i.e. the entanglement entropies of the two regions saturate the Araki-Lieb inequality. We show that in general, at least one of the relative entropies of the two entangling regions is expected to involve \( \overset{\sim }{\lambda } \) contributions of higher order from the modular Hamiltonian. Furthermore, we consider the implications of this observation for prominent AdS/CFT examples that form entanglement plateaux in the large N limit.
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References
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149 [arXiv:1503.06237] [INSPIRE].
N. Lashkari and M. Van Raamsdonk, Canonical Energy is Quantum Fisher Information, JHEP 04 (2016) 153 [arXiv:1508.00897] [INSPIRE].
S. Banerjee, J. Erdmenger and D. Sarkar, Connecting Fisher information to bulk entanglement in holography, JHEP 08 (2018) 001 [arXiv:1701.02319] [INSPIRE].
A. Bhattacharya and S. Roy, Holographic Entanglement Entropy, Subregion Complexity and Fisher Information metric of ‘black’ non-SUSY D3 Brane, arXiv:1807.06361 [INSPIRE].
L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [INSPIRE].
D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].
M. Alishahiha, Holographic Complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].
R. Abt et al., Topological Complexity in AdS 3 /CFT 2, Fortsch. Phys. 66 (2018) 1800034 [arXiv:1710.01327] [INSPIRE].
R. Abt, J. Erdmenger, M. Gerbershagen, C.M. Melby-Thompson and C. Northe, Holographic Subregion Complexity from Kinematic Space, arXiv:1805.10298 [INSPIRE].
V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys. 74 (2002) 197 [INSPIRE].
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
G. Sárosi and T. Ugajin, Relative entropy of excited states in two dimensional conformal field theories, JHEP 07 (2016) 114 [arXiv:1603.03057] [INSPIRE].
G. Sárosi and T. Ugajin, Relative entropy of excited states in conformal field theories of arbitrary dimensions, JHEP 02 (2017) 060 [arXiv:1611.02959] [INSPIRE].
G. Wong, I. Klich, L.A. Pando Zayas and D. Vaman, Entanglement Temperature and Entanglement Entropy of Excited States, JHEP 12 (2013) 020 [arXiv:1305.3291] [INSPIRE].
D.L. Jafferis and S.J. Suh, The Gravity Duals of Modular Hamiltonians, JHEP 09 (2016) 068 [arXiv:1412.8465] [INSPIRE].
N. Lashkari, Modular Hamiltonian for Excited States in Conformal Field Theory, Phys. Rev. Lett. 117 (2016) 041601 [arXiv:1508.03506] [INSPIRE].
T. Faulkner, R.G. Leigh, O. Parrikar and H. Wang, Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition, JHEP 09 (2016) 038 [arXiv:1605.08072] [INSPIRE].
T. Ugajin, Mutual information of excited states and relative entropy of two disjoint subsystems in CFT, JHEP 10 (2017) 184 [arXiv:1611.03163] [INSPIRE].
R. Arias, D. Blanco, H. Casini and M. Huerta, Local temperatures and local terms in modular Hamiltonians, Phys. Rev. D 95 (2017) 065005 [arXiv:1611.08517] [INSPIRE].
J. Koeller, S. Leichenauer, A. Levine and A. Shahbazi-Moghaddam, Local Modular Hamiltonians from the Quantum Null Energy Condition, Phys. Rev. D 97 (2018) 065011 [arXiv:1702.00412] [INSPIRE].
H. Casini, E. Teste and G. Torroba, Modular Hamiltonians on the null plane and the Markov property of the vacuum state, J. Phys. A 50 (2017) 364001 [arXiv:1703.10656] [INSPIRE].
G. Sárosi and T. Ugajin, Modular Hamiltonians of excited states, OPE blocks and emergent bulk fields, JHEP 01 (2018) 012 [arXiv:1705.01486] [INSPIRE].
R. Arias, H. Casini, M. Huerta and D. Pontello, Anisotropic Unruh temperatures, Phys. Rev. D 96 (2017) 105019 [arXiv:1707.05375] [INSPIRE].
H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].
D.D. Blanco and H. Casini, Localization of Negative Energy and the Bekenstein Bound, Phys. Rev. Lett. 111 (2013) 221601 [arXiv:1309.1121] [INSPIRE].
J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. 1612 (2016) 123103 [arXiv:1608.01283] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [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].
D. Blanco, Quantum information measures and their applications in quantum field theory, Ph.D. thesis, Balseiro Inst., San Carlos de Bariloche, Argentina, 2016, arXiv:1702.07384 [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].
D. Blanco, H. Casini, M. Leston and F. Rosso, Modular energy inequalities from relative entropy, JHEP 01 (2018) 154 [arXiv:1711.04816] [INSPIRE].
V.E. Hubeny, H. Maxfield, M. Rangamani and E. Tonni, Holographic entanglement plateaux, JHEP 08 (2013) 092 [arXiv:1306.4004] [INSPIRE].
H. Araki and E.H. Lieb, Entropy inequalities, Commun. Math. Phys. 18 (1970) 160 [INSPIRE].
A. Uhlmann, Relative Entropy and the Wigner-Yanase-Dyson-Lieb Concavity in an Interpolation Theory, Commun. Math. Phys. 54 (1977) 21 [INSPIRE].
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].
M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [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].
R.E. Arias, H. Casini, M. Huerta and D. Pontello, Entropy and modular Hamiltonian for a free chiral scalar in two intervals, arXiv:1809.00026 [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches, JHEP 02 (2015) 171 [arXiv:1410.1392] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
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Abt, R., Erdmenger, J. Properties of modular Hamiltonians on entanglement plateaux. J. High Energ. Phys. 2018, 2 (2018). https://doi.org/10.1007/JHEP11(2018)002
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DOI: https://doi.org/10.1007/JHEP11(2018)002