Towards a derivation of holographic entanglement entropy

We provide a derivation of holographic entanglement entropy for spherical entangling surfaces. Our construction relies on conformally mapping the boundary CFT to a hyperbolic geometry and observing that the vacuum state is mapped to a thermal state in the latter geometry. Hence the conformal transformation maps the entanglement entropy to the thermodynamic entropy of this thermal state. The AdS/CFT dictionary allows us to calculate this thermodynamic entropy as the horizon entropy of a certain topological black hole. In even dimensions, we also demonstrate that the universal contribution to the entanglement entropy is given by A-type trace anomaly for any CFT, without reference to holography.

[1]  Tadashi Takayanagi,et al.  Holographic entanglement entropy: an overview , 2009, 0905.0932.

[2]  J.S.Dowker.  Entanglement entropy for odd spheres , 2010, 1012.1548.

[3]  J. Boer,et al.  AdS7/CFT6, Gauss-Bonnet gravity, and viscosity bound , 2009, 0910.5347.

[4]  I. Klebanov,et al.  Towards the F-theorem: $ \mathcal{N} = 2 $ field theories on the three-sphere , 2011, 1103.1181.

[5]  Geometry and thermofields , 1989 .

[6]  W. Unruh.  Notes on black-hole evaporation , 1976 .

[7]  Xiao-Gang Wen,et al.  Detecting topological order in a ground state wave function. , 2005, Physical review letters.

[8]  On black hole entropy. , 1993, Physical review. D, Particles and fields.

[9]  S. C. Bariloche,et al.  Entanglement entropy in free quantum field theory , 2009, 0905.2562.

[10]  D. Lovelock.  Divergence-free tensorial concomitants , 1969 .

[11]  L. Susskind,et al.  The Holographic bound in anti-de Sitter space , 1998, hep-th/9805114.

[12]  Iyer,et al.  Some properties of the Noether charge and a proposal for dynamical black hole entropy. , 1994, Physical review. D, Particles and fields.

[13]  S. Theisen,et al.  Entanglement entropy, trace anomalies and holography ✩ , 2008, 0802.1017.

[14]  Rudolf Haag,et al.  Local quantum physics : fields, particles, algebras , 1993 .

[15]  M. Duff.  Observations on Conformal Anomalies , 1977 .

[16]  S. C. Bariloche,et al.  Mutual information challenges entropy bounds , 2006, gr-qc/0609126.

[17]  Miele,et al.  Finite-temperature scalar field theory in static de Sitter space. , 1994, Physical review. D, Particles and fields.

[18]  D. Vassilevich,et al.  Heat kernel expansion: user's manual , 2003, hep-th/0306138.

[19]  J. Edelstein,et al.  Causality in AdS/CFT and Lovelock theory , 2009, 0912.1944.

[20]  M. Grisaru,et al.  Four-Loop β-Function for the N = 1 And N = 2 Supersymmetric Non-Linear Sigma Model In Two Dimensions , 1986 .

[21]  M. Paulos,et al.  Holographic GB gravity in arbitrary dimensions , 2009, 0911.4257.

[22]  R. Longo,et al.  Modular structure of the local algebras associated with the free massless scalar field theory , 1982 .

[23]  L. Fidkowski.  Entanglement spectrum of topological insulators and superconductors. , 2009, Physical review letters.

[24]  J. Boer,et al.  Holographic Lovelock gravities and black holes , 2009, 0912.1877.

[25]  AdS membranes wrapped on surfaces of arbitrary genus , 1998, hep-th/9804031.

[26]  R. Myers,et al.  Holographic calculations of Rényi entropy , 2011, 1110.1084.

[27]  D. Gross,et al.  Superstring Modifications of Einstein's Equations , 1986 .

[28]  T. Takayanagi,et al.  Aspects of Holographic Entanglement Entropy , 2006, hep-th/0605073.

[29]  D. Birmingham.  Topological black holes in anti-de Sitter space , 1998, hep-th/9808032.

[30]  The exact superconformal R-symmetry maximizes a , 2003, hep-th/0304128.

[31]  Matthew Headrick,et al.  Entanglement Renyi entropies in holographic theories , 2010, 1006.0047.

[32]  R. Myers,et al.  Seeing a c-theorem with holography , 2010, 1006.1263.

[33]  R. Myers,et al.  Beyond η/s = 1/4π , 2009 .

[34]  D. Fursaev.  Proof of the holographic formula for entanglement entropy , 2006, hep-th/0606184.

[35]  John Preskill,et al.  Topological entanglement entropy. , 2005, Physical Review Letters.

[36]  L. Brown,et al.  Stress tensors and their trace anomalies in conformally flat space-time , 1977 .

[37]  R. Myers,et al.  Black holes in quasi-topological gravity , 2010, 1003.5357.

[38]  Frank Pollmann,et al.  Topological Phases of One-Dimensional Fermions: An Entanglement Point of View , 2010, 1008.4346.

[39]  M. Dehghani.  Rotating topological black branes in various dimensions and AdS/CFT correspondence , 2002, hep-th/0203049.

[40]  S. Theisen,et al.  Numerical determination of entanglement entropy for a sphere , 2009, 0911.4283.

[41]  T. Takayanagi,et al.  Holographic Derivation of Entanglement Entropy from AdS/CFT , 2006, hep-th/0603001.

[42]  Making anti-de Sitter black holes , 1996, gr-qc/9604005.

[43]  J. S. Dowker,et al.  FIELD THEORIES ON CONFORMALLY RELATED SPACE-TIMES: SOME GLOBAL CONSIDERATIONS , 1979 .

[44]  James T. Liu,et al.  Holographic c-theorems and higher derivative gravity , 2010, 1012.3382.

[45]  R. Wald,et al.  Black hole entropy is Noether charge. , 1993, Physical review. D, Particles and fields.

[46]  B. Swingle,et al.  Mutual information and the structure of entanglement in quantum field theory , 2010, 1010.4038.

[47]  S. Solodukhin.  Entanglement entropy of round spheres , 2010, 1008.4314.

[48]  J. Edelstein,et al.  Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity , 2009, 0911.3160.

[49]  A. Lefevre,et al.  Entanglement spectrum in one-dimensional systems , 2008, 0806.3059.

[50]  J. S. Dowker.  Hyperspherical entanglement entropy , 2010 .

[51]  L. Vanzo,et al.  Rotating topological black holes , 1997, Physical Review D.

[52]  Aninda Sinha.  On higher derivative gravity, c-theorems and cosmology , 2010, 1008.4315.

[53]  D. Das,et al.  Geometric Entropy , 2010, 1007.4085.

[54]  T. Takayanagi,et al.  Holographic derivation of entanglement entropy from the anti-de Sitter space/conformal field theory correspondence. , 2006, Physical review letters.

[55]  S. Solodukhin.  Entanglement entropy, conformal invariance and extrinsic geometry , 2008, 0802.3117.

[56]  Thermodynamics and stability of hyperbolic charged black holes , 2004, hep-th/0406057.

[57]  J. Boer,et al.  Holographic entanglement entropy in Lovelock gravities , 2011, 1101.5781.

[58]  D. Gross,et al.  The Quartic Effective Action for the Heterotic String , 1987 .

[59]  J. Cardy,et al.  Entanglement entropy and quantum field theory , 2004, hep-th/0405152.

[60]  Effects of D instantons , 1997, hep-th/9701093.

[61]  Twenty years of the Weyl anomaly , 1993, hep-th/9308075.

[62]  M. Paulos,et al.  Lovelock theories, holography and the fate of the viscosity bound , 2010, 1010.1682.

[64]  Strings in background fields , 1985 .

[65]  H. Casini,et al.  Entanglement entropy in free quantum field theory , 2009 .

[66]  Thermodynamics of (3+1)-dimensional black holes with toroidal or higher genus horizons , 1997, gr-qc/9705012.

[67]  M. Raamsdonk,et al.  Building up spacetime with quantum entanglement , 2010, 1005.3035.

[68]  X. Qi,et al.  Entanglement entropy and entanglement spectrum of the Kitaev model. , 2010, Physical review letters.

[69]  P. Davies,et al.  Quantum field theory in de Sitter space: renormalization by point-splitting , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[70]  S. C. Bariloche,et al.  Entanglement entropy for the n-sphere , 2010, 1007.1813.

[71]  D. Lovelock.  The Einstein Tensor and Its Generalizations , 1971 .

[72]  M. Raamsdonk,et al.  Comments on quantum gravity and entanglement , 2009, 0907.2939.

[73]  ENTANGLEMENT ENTROPY AND QUANTUM FIELD THEORY: A NON-TECHNICAL INTRODUCTION , 2005, quant-ph/0505193.

[74]  M. Paulos,et al.  Universal holographic hydrodynamics at finite coupling , 2008, 0808.1837.

[75]  R. Myers,et al.  On holographic entanglement entropy and higher curvature gravity , 2011, 1101.5813.

[76]  R. Myers,et al.  Holographic c-theorems in arbitrary dimensions , 2010, 1011.5819.

[77]  Curvature terms in D-brane actions and their M theory origin , 1999, hep-th/9903210.

[78]  The holographic Weyl anomaly , 1998, hep-th/9806087.

[79]  E. Fradkin,et al.  Universal entanglement entropy in 2D conformal quantum critical points , 2008 .

[80]  I. Klebanov,et al.  Entanglement as a probe of confinement , 2007, 0709.2140.

[81]  Holography and the Weyl anomaly , 1998, hep-th/9812032.

[82]  H. Casini,et al.  Remarks on the entanglement entropy for disconnected regions , 2008, 0812.1773.

[83]  D. Jafferis.  The exact superconformal R-symmetry extremizes Z , 2010, 1012.3210.

[84]  M. Paulos,et al.  Quantum corrections to ?/s , 2008, 0806.2156.

[85]  M. Paulos.  Holographic phase space: c-functions and black holes as renormalization group flows , 2011, 1101.5993.

[86]  J. Maldacena,et al.  Eternal black holes in anti-de Sitter , 2001, hep-th/0106112.

[87]  F. Wilczek,et al.  Geometric and renormalized entropy in conformal field theory , 1994, hep-th/9403108.

[88]  Y. Kats,et al.  Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory , 2007, 0712.0743.

[89]  Hui Li,et al.  Entanglement spectrum as a generalization of entanglement entropy: identification of topological order in non-Abelian fractional quantum Hall effect states. , 2008, Physical review letters.

[90]  N. D. Birrell,et al.  Quantum fields in curved space , 2007 .

[91]  E. Wichmann,et al.  ON THE DUALITY CONDITION FOR QUANTUM FIELDS , 1976 .

[92]  E. Bergshoeff,et al.  The quartic effective action of the heterotic string and supersymmetry , 1989 .

[93]  Eyvind H. Wichmann,et al.  On the duality condition for a Hermitian scalar field , 1975 .

[94]  M. Paulos,et al.  Holographic studies of quasi-topological gravity , 2010, 1004.2055.

[95]  Geometric classification of conformal anomalies in arbitrary dimensions , 1993, hep-th/9302047.

[96]  J. S. Dowker.  Entanglement entropy for even spheres , 2010 .