Abstract
The relative entropy is a measure of the distinguishability of two quantum states. A great deal of progress has been made in the study of the relative entropy between an excited state and the vacuum state of a conformal field theory (CFT) reduced to a spherical region. For example, when the excited state is a small perturbation of the vacuum state, the relative entropy is known to have a universal expression for all CFT’s [1]. Specifically, the perturbative relative entropy can be written as the symplectic flux of a certain scalar field in an auxiliary AdS-Rindler spacetime [1]. Moreover, if the CFT has a semi-classical holographic dual, the relative entropy is known to be related to conserved charges in the bulk dual spacetime [2]. In this paper, we introduce a one-parameter generalization of the relative entropy which we call refined Rényi relative entropy. We study this quantity in CFT’s and find a one-parameter generalization of the aforementioned known results about the relative entropy. We also discuss a new family of positive energy theorems in asymptotically locally AdS spacetimes that arises from the holographic dual of the refined Rényi relative entropy.
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References
T. Faulkner et al., Nonlinear gravity from entanglement in conformal field theories, JHEP08 (2017) 057 [arXiv:1705.03026] [INSPIRE].
N. Lashkari et al., Gravitational positive energy theorems from information inequalities, PTEP12 (2016) 12C109 [arXiv:1605.01075].
H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav.25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].
R. Longo and F. Xu, Comment on the Bekenstein bound, J. Geom. Phys.130 (2018) 113 [arXiv:1802.07184] [INSPIRE].
A.C. Wall, A proof of the generalized second law for rapidly-evolving Rindler horizons, Phys. Rev.D82 (2010) 124019 [arXiv:1007.1493] [INSPIRE].
A.C. Wall, Proof of the generalized second law for rapidly changing fields and arbitrary horizon slices, Phys. Rev.D 85 (2012) 104049 [Erratum ibid. D 87 (2013) 069904] [arXiv:1105.3445].
R. Bousso, H. Casini, Z. Fisher and J. Maldacena, Proof of a quantum bousso bound, Phys. Rev.D 90 (2014) 044002 [arXiv:1404.5635] [INSPIRE].
R. Bousso, H. Casini, Z. Fisher and J. Maldacena, Entropy on a null surface for interacting quantum field theories and the Bousso bound, Phys. Rev.D 91 (2015) 084030 [arXiv:1406.4545] [INSPIRE].
T. Faulkner, R.G. Leigh, O. Parrikar and H. Wang, Modular Hamiltonians for deformed half-spaces and the averaged null energy condition, JHEP09 (2016) 038.
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].
S. Leichenauer, A. Levine and A. Shahbazi-Moghaddam, Energy density from second shape variations of the von Neumann entropy, Phys. Rev.D 98 (2018) 086013 [arXiv:1802.02584] [INSPIRE].
F. Ceyhan and T. Faulkner, Recovering the QNEC from the ANEC, arXiv:1812.04683 [INSPIRE].
H. Casini, E. Teste and G. Torroba, Relative entropy and the RG flow, JHEP03 (2017) 089 [arXiv:1611.00016] [INSPIRE].
N. Bao and H. Ooguri, Distinguishability of black hole microstates, Phys. Rev.D 96 (2017) 066017 [arXiv:1705.07943] [INSPIRE].
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
X. Dong, D. Harlow and A.C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Phys. Rev. Lett.117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP07 (2017) 151 [arXiv:1704.05464] [INSPIRE].
J. Cotler et al., Entanglement wedge reconstruction via universal recovery channels, Phys. Rev.X 9 (2019) 031011 [arXiv:1704.05839] [INSPIRE].
N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement ‘thermodynamics’, JHEP04 (2014) 195 [arXiv:1308.3716] [INSPIRE].
T. Faulkner et al., Gravitation from entanglement in holographic CFTs, JHEP03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
S. Banerjee et al., Constraining gravity using entanglement in AdS/CFT, JHEP05 (2014) 029 [arXiv:1401.5089] [INSPIRE].
B. Swingle and M. Van Raamsdonk, Universality of gravity from entanglement, arXiv:1405.2933 [INSPIRE].
S. Banerjee, A. Kaviraj and A. Sinha, Nonlinear constraints on gravity from entanglement, Class. Quant. Grav.32 (2015) 065006 [arXiv:1405.3743] [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, JHEP06 (2015) 067.
N. Lashkari and M. Van Raamsdonk, Canonical energy is quantum Fisher information, JHEP04 (2016) 153 [arXiv:1508.00897] [INSPIRE].
T. Faulkner, Bulk emergence and the RG flow of entanglement entropy, JHEP05 (2015) 033 [arXiv:1412.5648] [INSPIRE].
N. Lashkari, Relative entropies in conformal field theory, Phys. Rev. Lett.113 (2014) 051602 [arXiv:1404.3216] [INSPIRE].
D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative entropy and holography, JHEP08 (2013) 060 [arXiv:1305.3182] [INSPIRE].
V. Rosenhaus and M. Smolkin, Entanglement entropy: a perturbative calculation, JHEP12 (2014) 179 [arXiv:1403.3733] [INSPIRE].
V. Rosenhaus and M. Smolkin, Entanglement entropy, planar surfaces and spectral functions, JHEP09 (2014) 119 [arXiv:1407.2891] [INSPIRE].
A. Allais and M. Mezei, Some results on the shape dependence of entanglement and Rényi entropies, Phys. Rev.D 91 (2015) 046002 [arXiv:1407.7249] [INSPIRE].
A. Lewkowycz and E. Perlmutter, Universality in the geometric dependence of Renyi entropy, JHEP01 (2015) 080 [arXiv:1407.8171] [INSPIRE].
V. Rosenhaus and M. Smolkin, Entanglement entropy for relevant and geometric perturbations, JHEP02 (2015) 015 [arXiv:1410.6530] [INSPIRE].
M. Mezei, Entanglement entropy across a deformed sphere, Phys. Rev.D 91 (2015) 045038 [arXiv:1411.7011] [INSPIRE].
D. Carmi, On the shape dependence of entanglement entropy, JHEP12 (2015) 043 [arXiv:1506.07528] [INSPIRE].
T. Faulkner, R.G. Leigh and O. Parrikar, Shape dependence of entanglement entropy in conformal field theories, JHEP04 (2016) 088 [arXiv:1511.05179] [INSPIRE].
S. Leichenauer, M. Moosa and M. Smolkin, Dynamics of the area law of entanglement entropy, JHEP09 (2016) 035 [arXiv:1604.00388] [INSPIRE].
X. Dong, The gravity dual of Renyi entropy, Nature Commun. 7 (2016) 12472 [arXiv:1601.06788] [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, JHEP07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
X. Dong, A. Lewkowycz and M. Rangamani, Deriving covariant holographic entanglement, JHEP11 (2016) 028 [arXiv:1607.07506] [INSPIRE].
M.M. Wilde, A. Winter and D. Yang, Strong converse for the classical capacity of entanglement-breaking and hadamard channels via a sandwiched renyi relative entropy, Commun. Math. Phys.331 (2014) 593 [arXiv:1306.1586] [INSPIRE].
M. Müller-Lennert et al., On quantum Rényi entropies: a new generalization and some properties, J. Math. Phys.54 (2013) 122203 [arXiv:1306.3142].
R.L. Frank and E.H. Lieb, Monotonicity of a relative Rényi entropy, J. Math. Phys.54 (2013) 122201 [arXiv:1306.5358].
S. Beigi, Sandwiched Rényi divergence satisfies data processing inequality, J. Math. Phys.54 (2013) 122202 [arXiv:1306.5920].
M. Mosonyi and T. Ogawa, Quantum hypothesis testing and the operational interpretation of the quantum Rényi relative entropies, Commun. Math. Phys.334 (2015) 1617 [arXiv:1309.3228].
H. Araki, Type of von Neumann Algebra Associated with Free Field, Prog. Theor. Phys.32 (1964) 956.
R. Longo, Algebraic and modular structure of von Neumann algebras of physics, Commun. Math. Phys.38 (1982) 551 [INSPIRE].
K. Fredenhagen, On the modular structure of local algebras of observables, Comm. Math. Phys.97 (1985) 79.
H. Araki, Relative entropy of states of von Neumann algebras, Publ. Res. Inst. Math. Sci.11 (1976) 809.
H. Araki, Inequalities in von Neumann algebras, Les rencontres Physiciens-Mathématiciens de Strasbourg (RCP25)22 (1975) 1.
E. Witten, APS medal for exceptional achievement in research: invited article on entanglement properties of quantum field theory, Rev. Mod. Phys.90 (2018) 045003 [arXiv:1803.04993] [INSPIRE].
N. Lashkari, Constraining quantum fields using modular theory, JHEP01 (2019) 059 [arXiv:1810.09306] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
T. Ugajin, Perturbative expansions of Rényi relative divergences and holography, arXiv:1812.01135 [INSPIRE].
A. Bernamonti, F. Galli, R.C. Myers and J. Oppenheim, Holographic second laws of black hole thermodynamics, JHEP07 (2018) 111 [arXiv:1803.03633] [INSPIRE].
A. May and E. Hijano, The holographic entropy zoo, JHEP10 (2018) 036 [arXiv:1806.06077] [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev.D 48 (1993) R3427 [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].
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys.31 (1990) 725 [INSPIRE].
R.M. Wald and A. Zoupas, A general definition of ‘conserved quantities’ in general relativity and other theories of gravity, Phys. Rev.D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
S. Hollands and R.M. Wald, Stability of black holes and black branes, Commun. Math. Phys.321 (2013) 629 [arXiv:1201.0463] [INSPIRE].
G. Lindblad, Expectations and entropy inequalities for finite quantum systems, Comm. Math. Phys.39 (1974) 111.
R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Japan12 (1957) 570.
P.C. Martin and J.S. Schwinger, Theory of many particle systems. 1., Phys. Rev.115 (1959) 1342 [INSPIRE].
R. Haag, N.M. Hugenholtz and M. Winnink, On the equilibrium states in quantum statistical mechanics, Commun. Math. Phys.5 (1967) 215 [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150] [INSPIRE].
D. Marolf et al., From Euclidean sources to lorentzian spacetimes in holographic conformal field theories, JHEP06 (2018) 077 [arXiv:1709.10101] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett.B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
C. Akers and P. Rath, Holographic Renyi entropy from quantum error correction, JHEP05 (2019) 052 [arXiv:1811.05171] [INSPIRE].
X. Dong, D. Harlow and D. Marolf, Flat entanglement spectra in fixed-area states of quantum gravity, arXiv:1811.05382 [INSPIRE].
L.F. Abbott and S. Deser, Stability of gravity with a cosmological constant, Nucl. Phys. B 195 (1982) 76 [INSPIRE].
E. Woolgar, The positivity of energy for asymptotically Anti-de Sitter space-times, Class. Quant. Grav.11 (1994) 1881 [gr-qc/9404019] [INSPIRE].
R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev.D 93 (2016) 064044 [arXiv:1506.02669] [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].
H. Casini, R. Medina, I. Salazar Landea and G. Torroba, Renyi relative entropies and renormalization group flows, JHEP09 (2018) 166 [arXiv:1807.03305] [INSPIRE].
P. Hayden, M. Headrick and A. Maloney, Holographic mutual information is monogamous, Phys. Rev.D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].
N. Bao et al., The holographic entropy cone, JHEP09 (2015) 130 [arXiv:1505.07839] [INSPIRE].
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Bao, N., Moosa, M. & Shehzad, I. The holographic dual of Rényi relative entropy. J. High Energ. Phys. 2019, 99 (2019). https://doi.org/10.1007/JHEP08(2019)099
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DOI: https://doi.org/10.1007/JHEP08(2019)099