Abstract
Relative entropy between two states in the same Hilbert space is a fundamental statistical measure of the distance between these states. Relative entropy is always positive and increasing with the system size. Interestingly, for two states which are infinitesimally different to each other, vanishing of relative entropy gives a powerful equation ΔS = ΔH for the first order variation of the entanglement entropy ΔS and the expectation value of the modular Hamiltonian ΔH. We evaluate relative entropy between the vacuum and other states for spherical regions in the AdS/CFT framework. We check that the relevant equations and inequalities hold for a large class of states, giving a strong support to the holographic entropy formula. We elaborate on potential uses of the equation ΔS = ΔH for vacuum state tomography and obtain modified versions of the Bekenstein bound.
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M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613].
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].
A. Hamma, R. Ionicioiu and P. Zanardi, Ground state entanglement and geometric entropy in the Kitaev model [rapid communication], Phys. Lett. A 337 (2005) 22 [quant-ph/0406202].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory: A Non-technical introduction, Int. J. Quant. Inf. 4 (2006) 429 [quant-ph/0505193] [INSPIRE].
I.R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of confinement, Nucl. Phys. B 796 (2008) 274 [arXiv:0709.2140] [INSPIRE].
T. Nishioka and T. Takayanagi, AdS Bubbles, Entropy and Closed String Tachyons, JHEP 01 (2007) 090 [hep-th/0611035] [INSPIRE].
P. Buividovich and M. Polikarpov, Numerical study of entanglement entropy in SU(2) lattice gauge theory, Nucl. Phys. B 802 (2008) 458 [arXiv:0802.4247] [INSPIRE].
Y. Nakagawa, A. Nakamura, S. Motoki and V. Zakharov, Quantum entanglement in SU(3) lattice Yang-Mills theory at zero and finite temperatures, PoS(Lattice 2010)281 [arXiv:1104.1011] [INSPIRE].
H. Casini and M. Huerta, A Finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].
V. Balasubramanian, M.B. McDermott and M. Van Raamsdonk, Momentum-space entanglement and renormalization in quantum field theory, Phys. Rev. D 86 (2012) 045014 [arXiv:1108.3568] [INSPIRE].
H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].
R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].
R.D. Sorkin, On the Entropy of the Vacuum Outside a Horizon, in proceedings of 10th Int. Conf. on General Relativity and Gravitation, Padova, Italy, 4-9 July 1983, General Relativity and Gravitation, Vol. 1, Classical Relativity, B. Bertotti, F. de Felice and A. Pascolini eds., Consiglio Nazionale delle Ricerche, Rome, Italy (1983).
L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A Quantum Source of Entropy for Black Holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
V.P. Frolov and I. Novikov, Dynamical origin of the entropy of a black hole, Phys. Rev. D 48 (1993) 4545 [gr-qc/9309001] [INSPIRE].
L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50 (1994) 2700 [hep-th/9401070] [INSPIRE].
S.N. Solodukhin, Entanglement entropy of black holes, Living Rev. Rel. 14 (2011) 8 [arXiv:1104.3712] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
S.D. Mathur, The Information paradox: A Pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
S.L. Braunstein, S. Pirandola and K. yczkowski, Entangled black holes as ciphers of hidden information, Physical Review Letters 110 (2013) 101301 [arXiv:0907.1190] [INSPIRE].
M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
V.E. Hubeny and M. Rangamani, Causal Holographic Information, JHEP 06 (2012) 114 [arXiv:1204.1698] [INSPIRE].
B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The Gravity Dual of a Density Matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].
E. Bianchi and R.C. Myers, On the Architecture of Spacetime Geometry, arXiv:1212.5183 [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
T. Nishioka, S. Ryu and T. Takayanagi, Holographic Entanglement Entropy: An Overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].
T. Takayanagi, Entanglement Entropy from a Holographic Viewpoint, Class. Quant. Grav. 29 (2012) 153001 [arXiv:1204.2450] [INSPIRE].
M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, On Holographic Entanglement Entropy and Higher Curvature Gravity, JHEP 04 (2011) 025 [arXiv:1101.5813] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, arXiv:1304.4926 [INSPIRE].
A. Rényi, On measures of information and entropy, in proceedings of the 4 th Berkeley Symposium on Mathematics, Statistics and Probability, 1 (1961) 547, Uiversity of California Press, Berkeley, CA, U.S.A. [http://digitalassets.lib.berkeley.edu/math/ucb/text/math s4 v1 article-27.pdf].
A. Rényi, On the foundations of information theory, Rev. Int. Stat. Inst. 33 (1965) 1.
K. Zyczkowski, Renyi extrapolation of Shannon entropy, Open Syst. Inf. Dyn. 10 (2003) 297 [quant-ph/0305062].
C. Beck and F. Schlögl, Thermodynamics of chaotic systems, Cambridge University Press, Cambridge, U.K. (1993).
P. Calabrese and A. Lefevre, Entanglement spectrum in one-dimensional systems, Phys. Rev. A 78 (2008) 032329 [arXiv:0806.3059].
T. Faulkner, The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955 [INSPIRE].
L.-Y. Hung, R.C. Myers, M. Smolkin and A. Yale, Holographic Calculations of Renyi Entropy, JHEP 12 (2011) 047 [arXiv:1110.1084] [INSPIRE].
A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50 (1978) 221 [INSPIRE].
V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys. 74 (2002) 197 [INSPIRE].
R. Haag, Local quantum physics: Fields, particles, algebras, Texts and monographs in physics, Springer, Berlin, Germany (1992) [INSPIRE].
H. Li and F.D.M. Haldane, Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States, Phys. Rev. Lett. 101 (2008) 010504 [arXiv:0805.0332].
A.M. Turner, F. Pollmann and E. Berg, Topological phases of one-dimensional fermions: An entanglement point of view, Phys. Rev. B 83 (2011) 075102 [arXiv:1008.4346].
L. Fidkowski, Entanglement Spectrum of Topological Insulators and Superconductors, Phys. Rev. Lett. 104 (2010) 130502 [arXiv:0909.2654].
H. Yao and X.-L. Qi, Entanglement Entropy and Entanglement Spectrum of the Kitaev Model, Phys. Rev. Lett. 105 (2010) 080501 [arXiv:1001.1165].
J. Bisognano and E. Wichmann, On the Duality Condition for Quantum Fields, J. Math. Phys. 17 (1976) 303 [INSPIRE].
J. Bisognano and E. Wichmann, On the Duality Condition for a Hermitian Scalar Field, J. Math. Phys. 16 (1975) 985 [INSPIRE].
W. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [INSPIRE].
P.D. Hislop and R. Longo, Modular structure of the local algebras associated with the free massless scalar field theory, Commun. Math. Phys. 84 (1982) 71 [INSPIRE].
H. Borchers and J. Yngvason, Modular groups of quantum fields in thermal states, J. Math. Phys. 40 (1999) 601 [math-ph/9805013] [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
R.C. Myers, Stress tensors and Casimir energies in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 046002 [hep-th/9903203] [INSPIRE].
W. Fischler, A. Kundu and S. Kundu, Holographic Mutual Information at Finite Temperature, Phys. Rev. D 87 (2013) 126012 [arXiv:1212.4764] [INSPIRE].
J. Bhattacharya, M. Nozaki, T. Takayanagi and T. Ugajin, Thermodynamical Property of Entanglement Entropy for Excited States, Phys. Rev. Lett. 110 (2013) 091602 [arXiv:1212.1164] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
R.C. Myers, M.F. Paulos and A. Sinha, Holographic Hydrodynamics with a Chemical Potential, JHEP 06 (2009) 006 [arXiv:0903.2834] [INSPIRE].
C. Fefferman and C. R. Graham, Conformal Invariants, in lie Cartan et les Mathématiques d’aujourd hui, Astérisque (1985), pg. 95.
C. Fefferman and C.R. Graham, The ambient metric, arXiv:0710.0919 [INSPIRE].
S. Nojiri and S.D. Odintsov, On the conformal anomaly from higher derivative gravity in AdS/CFT correspondence, Int. J. Mod. Phys. A 15 (2000) 413 [hep-th/9903033] [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
A. Schwimmer and S. Theisen, Entanglement Entropy, Trace Anomalies and Holography, Nucl. Phys. B 801 (2008) 1 [arXiv:0802.1017] [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, Some Calculable Contributions to Holographic Entanglement Entropy, JHEP 08 (2011) 039 [arXiv:1105.6055] [INSPIRE].
M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
S. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
V.E. Korepin, Universality of Entropy Scaling in One Dimensional Gapless Models, Phys. Rev. Lett. 92 (2004) 096402 [cond-mat/0311056].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, New York, U.S.A. (1997).
J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].
S.S. Gubser, Curvature singularities: The Good, the bad and the naked, Adv. Theor. Math. Phys. 4 (2000) 679 [hep-th/0002160] [INSPIRE].
R.C. Myers and O. Tafjord, Superstars and giant gravitons, JHEP 11 (2001) 009 [hep-th/0109127] [INSPIRE].
M.A. Nielsen and I.L. Chuang, Quantum Computation and quantum Information, Cambridge University Press, Cambridge, U.K. (2000).
H. Casini, M. Huerta and R. C. Myers, Mutual information and a c-theorem for d = 3, in preparation.
H. Halvorson, Reeh-Schlieder defeats Newton-Wigner: On alternative localization schemes in relativistic quantum field theory, Phil. Sci. 68 (2001) 111 [quant-ph/0007060] [INSPIRE].
L.Y. Hung, R.C. Myers and M. Smolkin, Twist operators in higher dimensions, in preparation.
H. Liu and A.A. Tseytlin, On four point functions in the CFT/AdS correspondence, Phys. Rev. D 59 (1999) 086002 [hep-th/9807097] [INSPIRE].
E. D’Hoker and D.Z. Freedman, Supersymmetric gauge theories and the AdS/CFT correspondence, hep-th/0201253 [INSPIRE].
J.D. Bekenstein, A Universal Upper Bound on the Entropy to Energy Ratio for Bounded Systems, Phys. Rev. D 23 (1981) 287 [INSPIRE].
J.D. Bekenstein, Generalized second law of thermodynamics in black hole physics, Phys. Rev. D 9 (1974) 3292 [INSPIRE].
H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].
D. Marolf, D. Minic and S.F. Ross, Notes on space-time thermodynamics and the observer dependence of entropy, Phys. Rev. D 69 (2004) 064006 [hep-th/0310022] [INSPIRE].
D. Marolf, A Few words on entropy, thermodynamics and horizons, hep-th/0410168 [INSPIRE].
R. Bousso, Light sheets and Bekenstein’s bound, Phys. Rev. Lett. 90 (2003) 121302 [hep-th/0210295] [INSPIRE].
E. Bianchi, Horizon entanglement entropy and universality of the graviton coupling, arXiv:1211.0522 [INSPIRE].
M. Nozaki, T. Numasawa, A. Prudenziati and T. Takayanagi, Dynamics of Entanglement Entropy from Einstein Equation, arXiv:1304.7100 [INSPIRE].
M. Nozaki, T. Numasawa and T. Takayanagi, Holographic Local Quenches and Entanglement Density, JHEP 05 (2013) 080 [arXiv:1302.5703] [INSPIRE].
D. Allahbakhshi, M. Alishahiha and A. Naseh, Entanglement Thermodynamics, arXiv:1305.2728 [INSPIRE].
G. Wong, I. Klich, L.A. Pando Zayas and D. Vaman, Entanglement Temperature and Entanglement Entropy of Excited States, arXiv:1305.3291 [INSPIRE].
V. Vedral, Introduction to quantum information science, Oxford University Press, New York, U.S.A. (2006).
P. Martinetti and C. Rovelli, Diamonds’s temperature: Unruh effect for bounded trajectories and thermal time hypothesis, Class. Quant. Grav. 20 (2003) 4919 [gr-qc/0212074] [INSPIRE].
V. Balasubramanian, P. Kraus and A.E. Lawrence, Bulk versus boundary dynamics in anti-de Sitter space-time, Phys. Rev. D 59 (1999) 046003 [hep-th/9805171] [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].
A.C. Wall, Maximin Surfaces and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, arXiv:1211.3494 [INSPIRE].
T. Sagawa, Second Law-Like Inequalities with Quantum Relative Entropy: An Introduction, arXiv:1202.0983.
R.D. Sorkin, Toward a Proof of Entropy Increase in the Presence of Quantum Black Holes, Phys. Rev. Lett. 56 (1986) 1885.
R.D. Sorkin, The statistical mechanics of black hole thermodynamics, gr-qc/9705006 [INSPIRE].
A.C. Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices, Phys. Rev. D 85 (2012) 104049 [arXiv:1105.3445] [INSPIRE].
A.C. Wall, A Proof of the generalized second law for rapidly-evolving Rindler horizons, Phys. Rev. D 82 (2010) 124019 [arXiv:1007.1493] [INSPIRE].
M. Pelath and R.M. Wald, Comment on entropy bounds and the generalized second law, Phys. Rev. D 60 (1999) 104009 [gr-qc/9901032] [INSPIRE].
D. Marolf and R.D. Sorkin, On the status of highly entropic objects, Phys. Rev. D 69 (2004) 024014 [hep-th/0309218] [INSPIRE].
D.N. Page, Comment on a universal upper bound on the entropy to energy ratio for bounded systems, Phys. Rev. D 26 (1982) 947 [INSPIRE].
W. Unruh and R.M. Wald, Acceleration Radiation and Generalized Second Law of Thermodynamics, Phys. Rev. D 25 (1982) 942 [INSPIRE].
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Blanco, D.D., Casini, H., Hung, LY. et al. Relative entropy and holography. J. High Energ. Phys. 2013, 60 (2013). https://doi.org/10.1007/JHEP08(2013)060
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DOI: https://doi.org/10.1007/JHEP08(2013)060