Abstract
We investigate the quantum null energy condition (QNEC) in holographic CFTs, focusing on half-spaces and particular classes of states. We present direct, and in certain cases nonperturbative, calculations for both the diagonal and off-diagonal variational derivatives of entanglement entropy. In d ≥ 3, we find that the QNEC is saturated. We compute relations between the off-diagonal variation of entanglement, boundary relative entropy, and the bulk stress tensor. Strong subadditivity then leads to energy conditions in the bulk. In d = 2, we find that the QNEC is in general not saturated when the Ryu-Takayanagi surface intersects bulk matter. Moreover, when bulk matter is present the QNEC can imply new bulk energy conditions. For a simple class of states, we derive an example that is stronger than the bulk averaged null energy condition and reduces to it in certain limits.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [INSPIRE].
R. Bousso, Z. Fisher, J. Koeller, S. Leichenauer and A.C. Wall, Proof of the Quantum Null Energy Condition, Phys. Rev. D 93 (2016) 024017 [arXiv:1509.02542] [INSPIRE].
J. Koeller and S. Leichenauer, Holographic Proof of the Quantum Null Energy Condition, Phys. Rev. D 94 (2016) 024026 [arXiv:1512.06109] [INSPIRE].
S. Balakrishnan, T. Faulkner, Z.U. Khandker and H. Wang, A General Proof of the Quantum Null Energy Condition, arXiv:1706.09432 [INSPIRE].
Z. Fu, J. Koeller and D. Marolf, The Quantum Null Energy Condition in Curved Space, Class. Quant. Grav. 34 (2017) 225012 [Erratum ibid. 35 (2018) 049501] [arXiv:1706.01572] [INSPIRE].
C. Akers, V. Chandrasekaran, S. Leichenauer, A. Levine and A. Shahbazi Moghaddam, The Quantum Null Energy Condition, Entanglement Wedge Nesting and Quantum Focusing, arXiv:1706.04183 [INSPIRE].
Z. Fu, J. Koeller and D. Marolf, Violating the quantum focusing conjecture and quantum covariant entropy bound in d ≥ 5 dimensions, Class. Quant. Grav. 34 (2017) 175006 [arXiv:1705.03161] [INSPIRE].
Z. Fu and D. Marolf, Bare Quantum Null Energy Condition, Phys. Rev. Lett. 120 (2018) 071601 [arXiv:1711.02330] [INSPIRE].
C. Ecker, D. Grumiller, W. van der Schee and P. Stanzer, Saturation of the Quantum Null Energy Condition in Far-From-Equilibrium Systems, Phys. Rev. D 97 (2018) 126016 [arXiv:1710.09837] [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, C. Rabideau, P. Sabella-Garnier and M. Van Raamsdonk, Inviolable energy conditions from entanglement inequalities, JHEP 06 (2015) 067 [arXiv:1412.3514] [INSPIRE].
J. Bhattacharya, V.E. Hubeny, M. Rangamani and T. Takayanagi, Entanglement density and gravitational thermodynamics, Phys. Rev. D 91 (2015) 106009 [arXiv:1412.5472] [INSPIRE].
S. Leichenauer, A. Levine and A. Shahbazi-Moghaddam, Energy is Entanglement, arXiv:1802.02584 [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].
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].
N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Shockwaves from the Operator Product Expansion, arXiv:1709.03597 [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, J. Lin, H. Ooguri, B. Stoica and M. Van Raamsdonk, Gravitational positive energy theorems from information inequalities, PTEP 2016 (2016) 12C109 [arXiv:1605.01075] [INSPIRE].
D. Neuenfeld, K. Saraswat and M. Van Raamsdonk, Positive gravitational subsystem energies from CFT cone relative entropies, JHEP 06 (2018) 050 [arXiv:1802.01585] [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].
T. Faulkner, R.G. Leigh and O. Parrikar, Shape Dependence of Entanglement Entropy in Conformal Field Theories, JHEP 04 (2016) 088 [arXiv:1511.05179] [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].
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].
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].
T. Faulkner, F.M. Haehl, E. Hijano, O. Parrikar, C. Rabideau and M. Van Raamsdonk, Nonlinear Gravity from Entanglement in Conformal Field Theories, JHEP 08 (2017) 057 [arXiv:1705.03026] [INSPIRE].
A.C. Wall, Testing the Generalized Second Law in 1+1 dimensional Conformal Vacua: An Argument for the Causal Horizon, Phys. Rev. D 85 (2012) 024015 [arXiv:1105.3520] [INSPIRE].
J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
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
Corresponding author
Additional information
ArXiv ePrint: 1803.03997
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/.
About this article
Cite this article
Khandker, Z.U., Kundu, S. & Li, D. Bulk matter and the boundary quantum null energy condition. J. High Energ. Phys. 2018, 162 (2018). https://doi.org/10.1007/JHEP08(2018)162
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2018)162