Abstract
The Hubeny-Rangamani causal holographic information (CHI) defined by a region R of a holographic quantum field theory (QFT) is a modern version of the idea that the area of event horizons might be related to an entropy. Here the event horizon lives in a dual gravitational bulk theory with Newton’s constant Gbulk, and the relation involves a factor of 4Gbulk. The fact that CHI is bounded below by the von Neumann entropy S suggests that CHI is coarse-grained. Its properties could thus differ markedly from those of S. In particular, recent results imply that when d ≤ 4 holographic QFTs are perturbatively coupled to d-dimensional gravity, the combined system satisfies the so-called quantum focusing condition (QFC) at leading order in the new gravitational coupling Gd when the QFT entropy is taken to be that of von Neumann. However, by studying states dual to spherical bulk (anti-de Sitter) Schwarschild black holes in the conformal frame for which the boundary is a (2 + 1)-dimensional de Sitter space, we find the QFC defined by CHI is violated even when perturbing about a Killing horizon and using a single null congruence. Since it is known that a generalized second law (GSL) holds in this context, our work demonstrates that the QFC is not required in order for an entropy, or an entropy-like quantity, to satisfy such a GSL.
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References
V.E. Hubeny and M. Rangamani, Causal holographic information, JHEP 06 (2012) 114 [arXiv:1204.1698] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [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].
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].
N. Engelhardt and A.C. Wall, No simple dual to the causal holographic information?, JHEP 04 (2017) 134 [arXiv:1702.01748] [INSPIRE].
B. Freivogel and B. Mosk, Properties of causal holographic information, JHEP 09 (2013) 100 [arXiv:1304.7229] [INSPIRE].
W. Bunting, Z. Fu and D. Marolf, A coarse-grained generalized second law for holographic conformal field theories, Class. Quant. Grav. 33 (2016) 055008 [arXiv:1509.00074] [INSPIRE].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
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].
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 and D. Marolf, Does horizon entropy satisfy a quantum null energy conjecture?, Class. Quant. Grav. 33 (2016) 245011 [arXiv:1606.04713] [INSPIRE].
R.C. Myers, Stress tensors and Casimir energies in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 046002 [hep-th/9903203] [INSPIRE].
B. Grado-White and D. Marolf, Marginally trapped surfaces and AdS/CFT, JHEP 02 (2018) 049 [arXiv:1708.00957] [INSPIRE].
N. Engelhardt and A.C. Wall, Decoding the apparent horizon: a coarse-grained holographic entropy, arXiv:1706.02038 [INSPIRE].
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Fu, Z., Marolf, D. & Qi, M. Causal holographic information does not satisfy the linearized quantum focusing condition. J. High Energ. Phys. 2018, 86 (2018). https://doi.org/10.1007/JHEP04(2018)086
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DOI: https://doi.org/10.1007/JHEP04(2018)086