Abstract
We consider the question of whether the leading contribution to the entanglement entropy in holographic CFTs is truly given by the expectation value of a linear operator as is suggested by the Ryu-Takayanagi formula. We investigate this property by computing the entanglement entropy, via the replica trick, in states dual to superpositions of macroscopically distinct geometries and find it consistent with evaluating the expectation value of the area operator within such states. However, we find that this fails once the number of semi-classical states in the superposition grows exponentially in the central charge of the CFT. Moreover, in certain such scenarios we find that the choice of surface on which to evaluate the area operator depends on the density matrix of the entire CFT. This nonlinearity is enforced in the bulk via the homology prescription of Ryu-Takayanagi. We thus conclude that the homology constraint is not a linear property in the CFT. We also discuss the existence of ‘entropy operators’ in general systems with a large number of degrees of freedom.
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References
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [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].
S.W. Hawking, Gravitational radiation from colliding black holes, Phys. Rev. Lett. 26 (1971) 1344 [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
N. Engelhardt and A.C. Wall, Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
D.L. Jafferis and S.J. Suh, The gravity duals of modular hamiltonians, JHEP 09 (2016) 068 [arXiv:1412.8465] [INSPIRE].
D.L. Jafferis et al., Relative entropy equals bulk relative entropy, arXiv:1512.06431 [INSPIRE].
K. Papadodimas and S. Raju, Remarks on the necessity and implications of state-dependence in the black hole interior, Phys. Rev. D 93 (2016) 084049 [arXiv:1503.08825] [INSPIRE].
N. Engelhardt and A.C. Wall, Extremal surface barriers, JHEP 03 (2014) 068 [arXiv:1312.3699] [INSPIRE].
B.S. DeWitt, Quantum theory of gravity. 1. The canonical theory, Phys. Rev. 160 (1967) 1113 [INSPIRE].
S.B. Giddings, D. Marolf and J.B. Hartle, Observables in effective gravity, Phys. Rev. D 74 (2006) 064018 [hep-th/0512200] [INSPIRE].
D. Marolf, Comments on microcausality, chaos, and gravitational observables, Class. Quant. Grav. 32 (2015) 245003 [arXiv:1508.00939] [INSPIRE].
W. Donnelly and S.B. Giddings, Diffeomorphism-invariant observables and their nonlocal algebra, Phys. Rev. D 93 (2016) 024030 [arXiv:1507.07921] [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].
I. Heemskerk, Construction of bulk fields with gauge redundancy, JHEP 09 (2012) 106 [arXiv:1201.3666] [INSPIRE].
D. Kabat, G. Lifschytz, S. Roy and D. Sarkar, Holographic representation of bulk fields with spin in AdS/CFT, Phys. Rev. D 86 (2012) 026004 [arXiv:1204.0126] [INSPIRE].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [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].
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].
I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, Bulk and Transhorizon Measurements in AdS/CFT, JHEP 10 (2012) 165 [arXiv:1201.3664] [INSPIRE].
J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.
M. Srednicki, Thermal fluctuations in quantized chaotic systems, J. Phys. A 29 (1996) L75 [chao-dyn/9511001].
S.W. Hawking and D.N. Page, Thermodynamics of black holes in Anti-de Sitter space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
J. de Boer, F. Denef, S. El-Showk, I. Messamah and D. Van den Bleeken, Black hole bound states in AdS 3 × S 2, JHEP 11 (2008) 050 [arXiv:0802.2257] [INSPIRE].
I. Bena, B.D. Chowdhury, J. de Boer, S. El-Showk and M. Shigemori, Moulting black holes, JHEP 03 (2012) 094 [arXiv:1108.0411] [INSPIRE].
C.A. Tracy and H. Widom, Level spacing distributions and the Airy kernel, Phys. Lett. B 305 (1993) 115 [hep-th/9210074] [INSPIRE].
C.A. Tracy and H. Widom, Level spacing distributions and the Airy kernel, Commun. Math. Phys. 159 (1994) 151 [hep-th/9211141] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 06 (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].
S. Giusto and R. Russo, Entanglement entropy and D1-D5 geometries, Phys. Rev. D 90 (2014) 066004 [arXiv:1405.6185] [INSPIRE].
S. Giusto, E. Moscato and R. Russo, AdS 3 holography for 1/4 and 1/8 BPS geometries, JHEP 11 (2015) 004 [arXiv:1507.00945].
T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [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].
N. Linden, S. Popescu and J.A. Smolin, Entanglement of superpositions, Phys. Rev. Lett. 97 (2006) 100502 [quant-ph/0507049].
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. Berenstein and A. Miller, Topology and geometry cannot be measured by an operator measurement in quantum gravity, arXiv:1605.06166 [INSPIRE].
K. Papadodimas and S. Raju, Local operators in the eternal black hole, Phys. Rev. Lett. 115 (2015) 211601 [arXiv:1502.06692] [INSPIRE]
D. Marolf and A.C. Wall, Eternal black holes and superselection in AdS/CFT, Class. Quant. Grav. 30 (2013) 025001 [arXiv:1210.3590] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
S.H. Shenker and D. Stanford, Multiple shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
M. Keyl and R.F. Werner, Estimating the spectrum of a density operator, Phys. Rev. A 64 (2001) 052311 [quant-ph/0102027].
R. Alicki, S. Rudnicki and S. Sadowski, Symmetry properties of product states for the system of N n-level atoms, J. Math. Phys. 29 (1988) 1158.
M. Hayashi and K. Matsumoto, Quantum universal variable-length source coding, Phys. Rev. A 66 (2002) 022311 [quant-ph/0202001].
M. Christandl and G. Mitchison, The spectra of quantum states and the Kronecker coefficients of the symmetric group, Commun. Math. Phys. 261 (2006) 789 [quant-ph/0409016].
J. Eisert, T. Tyc, T. Rudolph and B.C. Sanders, Gaussian quantum marginal problem, Commun. Math. Phys. 280 (2008) 263 [quant-ph/0703225].
D. Gross and M. Walter, Stabilizer information inequalities from phase space distributions, J. Math. Phys. 54 (2013) 082201 [arXiv:1302.6990].
A. Klyachko, Quantum marginal problem and representations of the symmetric group, quant-ph/0409113.
S. Daftuar and P. Hayden, Quantum state transformations and the Schubert calculus, Ann. Phys. 315 (2005) 80 [quant-ph/0410052].
M. Altunbulak and A. Klyachko, The Pauli principle revisited Commun. Math. Phys. 282 (2008) 287 [arXiv:0802.0918].
M. Christandl, M. Burak ¸ahinoğlu, and M. Walter, Recoupling coefficients and quantum entropies, arXiv:1210.0463.
K. Papadodimas and S. Raju, Remarks on the necessity and implications of state-dependence in the black hole interior, Phys. Rev. D 93 (2016) 084049 [arXiv:1503.08825] [INSPIRE].
N. Bao et al., The holographic entropy cone, JHEP 09 (2015) 130 [arXiv:1505.07839] [INSPIRE].
L. Susskind, ER=EPR, GHZ and the consistency of quantum measurements, Fortsch. Phys. 64 (2016)72 [arXiv:1412.8483] [INSPIRE].
D. Garfinkle and A. Strominger, Semiclassical Wheeler wormhole production, Phys. Lett. B 256 (1991) 146 [INSPIRE].
D. Garfinkle, S.B. Giddings and A. Strominger, Entropy in black hole pair production, Phys. Rev. D 49 (1994) 958 [gr-qc/9306023] [INSPIRE].
B. Czech, P. Hayden, N. Lashkari and B. Swingle, The information theoretic interpretation of the length of a curve, JHEP 06 (2015) 157 [arXiv:1410.1540] [INSPIRE].
B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].
P. Hayden et al., Holographic duality from random tensor networks, arXiv:1601.01694.
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Almheiri, A., Dong, X. & Swingle, B. Linearity of holographic entanglement entropy. J. High Energ. Phys. 2017, 74 (2017). https://doi.org/10.1007/JHEP02(2017)074
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DOI: https://doi.org/10.1007/JHEP02(2017)074