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Entanglement entropy: Holography and renormalization group

Tatsuma Nishioka

Tatsuma Nishioka

  • Department of Physics, Faculty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan

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Rev. Mod. Phys. 90, 035007 – Published 17 September, 2018

DOI: https://doi.org/10.1103/RevModPhys.90.035007

Abstract

Entanglement entropy plays a variety of roles in quantum field theory, including the connections between quantum states and gravitation through the holographic principle. This article provides a review of entanglement entropy from a mixed viewpoint of field theory and holography. A set of basic methods for the computation is developed and illustrated with simple examples such as free theories and conformal field theories. The structures of the ultraviolet divergences and the universal parts are determined and compared with the holographic descriptions of entanglement entropy. The utility of quantum inequalities of entanglement are discussed and shown to derive the C theorem that constrains renormalization group flows of quantum field theories in diverse dimensions.

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References (247)

  1. Adesso, Gerardo, Davide Girolami, and Alessio Serafini, 2012, “Measuring Gaussian quantum information and correlations using the Rényi entropy of order 2,” Phys. Rev. Lett. 109, 190502.
  2. Aharony, Ofer, Steven S. Gubser, Juan Martin Maldacena, Hirosi Ooguri, and Yaron Oz, 2000, “Large N Field Theories, String Theory and Gravity,” Phys. Rep. 323, 183–386.
  3. Alday, Luis F., Paul Richmond, and James Sparks, 2015, “The holographic supersymmetric Rényi entropy in five dimensions,” J. High Energy Phys. 02, 102.
  4. Anselmi, D., D. Z. Freedman, Marcus T. Grisaru, and A. A. Johansen, 1998, “Nonperturbative formulas for central functions of supersymmetric gauge theories,” Nucl. Phys. B 526, 543–571.
  5. Aoki, Sinya, Takumi Iritani, Masahiro Nozaki, Tokiro Numasawa, Noburo Shiba, and Hal Tasaki, 2015, “On the Definition of Entanglement Entropy in Lattice Gauge Theories,” J. High Energy Phys. 06, 187.
  6. Appelquist, Thomas, Andrew G. Cohen, and Martin Schmaltz, 1999, “A New constraint on strongly coupled gauge theories,” Phys. Rev. D 60, 045003.
  7. Araki, H., and E. H. Lieb, 1970, “Entropy inequalities,” Commun. Math. Phys. 18, 160–170.
  8. Araki, Huzihiro, 1975, “Relative entropy of states of von neumann algebras,” Publ. RIMS 11, 809–833.
  9. Azeyanagi, Tatsuo, R. Loganayagam, and Gim Seng Ng, 2017, “Holographic Entanglement for Chern-Simons Terms,” J. High Energy Phys. 02, 001.
  10. Bakhmatov, I., N. S. Deger, J. Gutowski, E. Ó Colgáin, and H. Yavartanoo, 2017, “Calibrated Entanglement Entropy,” J. High Energy Phys. 07, 117.
  11. Balakrishnan, Srivatsan, Thomas Faulkner, Zuhair U. Khandker, and Huajia Wang, 2017, “A General Proof of the Quantum Null Energy Condition,” arXiv:1706.09432.
  12. Bao, Ning, Sepehr Nezami, Hirosi Ooguri, Bogdan Stoica, James Sully, and Michael Walter, 2015, “The Holographic Entropy Cone,” J. High Energy Phys. 09, 130.
  13. Barnes, Edwin, Kenneth A. Intriligator, Brian Wecht, and Jason Wright, 2004, “Evidence for the strongest version of the 4d a-theorem, via a-maximization along RG flows,” Nucl. Phys. B 702, 131–162.
  14. Beigi, Salman, 2013, “Sandwiched Rényi divergence satisfies data processing inequality,” J. Math. Phys. (N.Y.) 54, 122202.
  15. Benini, Francesco, Po-Shen Hsin, and Nathan Seiberg, 2017, “Comments on global symmetries, anomalies, and duality in (2+1)d,” J. High Energy Phys. 04, 135.
  16. Bhattacharya, Jyotirmoy, Veronika E. Hubeny, Mukund Rangamani, and Tadashi Takayanagi, 2015, “Entanglement density and gravitational thermodynamics,” Phys. Rev. D 91, 106009.
  17. Birrell, N. D., and P. C. W. Davies, 1982, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, England).
  18. Bisognano, J. J., and E. H. Wichmann, 1975, “On the Duality Condition for a Hermitian Scalar Field,” J. Math. Phys. (N.Y.) 16, 985–1007.
  19. Bisognano, J. J., and E. H. Wichmann, 1976, “On the Duality Condition for Quantum Fields,” J. Math. Phys. (N.Y.) 17, 303–321.
  20. Bobev, Nikolay, and P. Marcos Crichigno, 2017, “Universal RG Flows Across Dimensions and Holography,” J. High Energy Phys. 12, 065.
  21. Bombelli, Luca, Rabinder K. Koul, Joohan Lee, and Rafael D. Sorkin, 1986, “A Quantum Source of Entropy for Black Holes,” Phys. Rev. D 34, 373–383.
  22. Bonora, L., P. Pasti, and M. Bregola, 1986, “Weyl Cocycles,” Classical Quantum Gravity 3, 635.
  23. Bousso, Raphael, Horacio Casini, Zachary Fisher, and Juan Maldacena, 2014, “Proof of a Quantum Bousso Bound,” Phys. Rev. D 90, 044002.
  24. Bousso, Raphael, Horacio Casini, Zachary Fisher, and Juan Maldacena, 2015, “Entropy on a null surface for interacting quantum field theories and the Bousso bound,” Phys. Rev. D 91, 084030.
  25. Bousso, Raphael, Zachary Fisher, Jason Koeller, Stefan Leichenauer, and Aron C. Wall, 2016, “Proof of the Quantum Null Energy Condition,” Phys. Rev. D 93, 024017.
  26. Brown, Lowell S., 1977, “Stress Tensor Trace Anomaly in a Gravitational Metric: Scalar Fields,” Phys. Rev. D 15, 1469.
  27. Bueno, Pablo, and Robert C. Myers, 2015, “Corner contributions to holographic entanglement entropy,” J. High Energy Phys. 08, 068.
  28. Bueno, Pablo, Robert C. Myers, and William Witczak-Krempa, 2015, “Universality of corner entanglement in conformal field theories,” Phys. Rev. Lett. 115, 021602.
  29. Calabrese, Pasquale, and John Cardy, 2007, “Quantum Quenches in Extended Systems,” J. Stat. Mech. 06, P06008.
  30. Calabrese, Pasquale, and John Cardy, 2009, “Entanglement Entropy and Conformal Field Theory,” J. Phys. A 42, 504005.
  31. Calabrese, Pasquale, and John Cardy, 2016, “Quantum quenches in 1+1 dimensional conformal field theories,” J. Stat. Mech. 06, 064003.
  32. Calabrese, Pasquale, and John L. Cardy, 2004, “Entanglement Entropy and Quantum Field Theory,” J. Stat. Mech. 06, P06002.
  33. Calabrese, Pasquale, and John L. Cardy, 2005, “Evolution of entanglement entropy in one-dimensional systems,” J. Stat. Mech. 04, P04010.
  34. Camporesi, Roberto, and Atsushi Higuchi, 1996, “On the Eigen Functions of the Dirac Operator on Spheres and Real Hyperbolic Spaces,” J. Geom. Phys. 20, 1–18.
  35. Camps, Joan, 2014, “Generalized Entropy and Higher Derivative Gravity,” J. High Energy Phys. 03, 070.
  36. Cappelli, Andrea, Daniel Friedan, and Jose I. Latorre, 1991, “C theorem and spectral representation,” Nucl. Phys. B 352, 616–670.
  37. Cardy, John, and Christopher P. Herzog, 2014, “Universal Thermal Corrections to Single Interval Entanglement Entropy for Two Dimensional Conformal Field Theories,” Phys. Rev. Lett. 112, 171603.
  38. Cardy, John L., 1988, “Is There a c-Theorem in Four-Dimensions?,” Phys. Lett. B 215, 749–752.
  39. Casini, H., and M. Huerta, 2004, “A Finite Entanglement Entropy and the C-Theorem,” Phys. Lett. B 600, 142–150.
  40. Casini, H., and M. Huerta, 2005, “Entanglement and Alpha Entropies for a Massive Scalar Field in Two Dimensions,” J. Stat. Mech. 12, P12012.
  41. Casini, H., and M. Huerta, 2007, “Universal terms for the entanglement entropy in 2+1 dimensions,” Nucl. Phys. B 764, 183–201.
  42. Casini, H., and M. Huerta, 2009, “Entanglement Entropy in Free Quantum Field Theory,” J. Phys. A 42, 504007.
  43. Casini, H., M. Huerta, and L. Leitao, 2009, “Entanglement entropy for a Dirac fermion in three dimensions: Vertex contribution,” Nucl. Phys. B 814, 594–609.
  44. Casini, H., and Marina Huerta, 2012, “On the RG Running of the Entanglement Entropy of a Circle,” Phys. Rev. D 85, 125016.
  45. Casini, Horacio, Marina Huerta, and Robert C. Myers, 2011, “Towards a Derivation of Holographic Entanglement Entropy,” J. High Energy Phys. 05, 036.
  46. Casini, Horacio, Marina Huerta, Robert C. Myers, and Alexandre Yale, 2015, “Mutual information and the F-theorem,” J. High Energy Phys. 10, 003.
  47. Casini, Horacio, Marina Huerta, and Jose Alejandro Rosabal, 2014, “Remarks on Entanglement Entropy for Gauge Fields,” Phys. Rev. D 89, 085012.
  48. Casini, Horacio, F. D. Mazzitelli, and Eduardo Testé, 2015, “Area terms in entanglement entropy,” Phys. Rev. D 91, 104035.
  49. Casini, Horacio, Eduardo Testé, and Gonzalo Torroba, 2017a, “Markov Property of the Conformal Field Theory Vacuum and the a-Theorem,” Phys. Rev. Lett. 118, 261602.
  50. Casini, Horacio, Eduardo Testé, and Gonzalo Torroba, 2017b, “Modular Hamiltonians on the null plane and the Markov property of the vacuum state,” J. Phys. A 50, 364001.
  51. Casini, Horacio, Eduardo Testé, and Gonzalo Torroba, 2018, “All the entropies on the light-cone,” arXiv:1802.04278.
  52. Castro, Alejandra, Stephane Detournay, Nabil Iqbal, and Eric Perlmutter, 2014, “Holographic Entanglement Entropy and Gravitational Anomalies,” J. High Energy Phys. 07, 114.
  53. Chen, Jiunn-Wei, Shou-Huang Dai, and Jin-Yi Pang, 2015, “Strong Coupling Expansion of the Entanglement Entropy of Yang-Mills Gauge Theories,” arXiv:1503.01766.
  54. Chung, Ming-Chiang, and Ingo Peschel, 2000, “Density-matrix spectra for two-dimensional quantum systems,” Phys. Rev. B 62, 4191.
  55. Closset, Cyril, Thomas T. Dumitrescu, Guido Festuccia, Zohar Komargodski, and Nathan Seiberg, 2012, “Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories,” J. High Energy Phys. 10, 053.
  56. Colgáin, Eoin Ó., 2018, “Holographic Entanglement Entropy, SUSY & Calibrations,” Eur. Phys. J. Web Conf. 168, 03003.
  57. Cordova, Clay, Thomas T. Dumitrescu, and Kenneth Intriligator, 2016, “Anomalies, renormalization group flows, and the a-theorem in six-dimensional (1,0) theories,” J. High Energy Phys. 10, 080.
  58. Cordova, Clay, Thomas T. Dumitrescu, and Xi Yin, 2015, “Higher Derivative Terms, Toroidal Compactification, and Weyl Anomalies in Six-Dimensional (2,0) Theories,” arXiv:1505.03850.
  59. de Boer, Jan, Manuela Kulaxizi, and Andrei Parnachev, 2011, “Holographic Entanglement Entropy in Lovelock Gravities,” J. High Energy Phys. 07, 109.
  60. Deser, Stanley, and A. Schwimmer, 1993, “Geometric Classification of Conformal Anomalies in Arbitrary Dimensions,” Phys. Lett. B 309, 279–284.
  61. Di Francesco, P., P. Mathieu, and D. Senechal, 1997, Conformal Field Theory, Graduate Texts in Contemporary Physics (Springer-Verlag, New York).
  62. Di Pietro, Lorenzo, Zohar Komargodski, Itamar Shamir, and Emmanuel Stamou, 2016, “Quantum Electrodynamics in d=3 from the ε Expansion,” Phys. Rev. Lett. 116, 131601.
  63. Di Pietro, Lorenzo, and Emmanuel Stamou, 2017, “Scaling dimensions in QED3 from the ε-expansion,” J. High Energy Phys. 12, 054.
  64. Dong, Xi, 2014, “Holographic Entanglement Entropy for General Higher Derivative Gravity,” J. High Energy Phys. 01, 044.
  65. Dong, Xi, 2016, “The Gravity Dual of Renyi Entropy,” Nat. Commun. 7, 12472.
  66. Dong, Xi, Aitor Lewkowycz, and Mukund Rangamani, 2016, “Deriving covariant holographic entanglement,” J. High Energy Phys. 11, 028.
  67. Donnelly, William, 2014, “Entanglement Entropy and Nonabelian Gauge Symmetry,” Classical Quantum Gravity 31, 214003.
  68. Donnelly, William, and Aron C. Wall, 2015, “Entanglement Entropy of Electromagnetic Edge Modes,” Phys. Rev. Lett. 114, 111603.
  69. Donnelly, William, and Aron C. Wall, 2016, “Geometric entropy and edge modes of the electromagnetic field,” Phys. Rev. D 94, 104053.
  70. Dowker, J. S., 2010, “Entanglement Entropy for Odd Spheres,” arXiv:1012.1548.
  71. Duff, M. J., 1994, “Twenty Years of the Weyl Anomaly,” Classical Quantum Gravity 11, 1387–1404.
  72. Dumitrescu, Thomas T., 2017, “An introduction to supersymmetric field theories in curved space,” J. Phys. A 50, 443005.
  73. Dur, W., G. Vidal, and J. I. Cirac, 2000, “Three Qubits Can Be Entangled in Two Inequivalent Ways,” Phys. Rev. A 62, 062314.
  74. Eisert, J., 2006, “Entanglement in quantum information theory,” arXiv:quant-ph/0610253v1.
  75. Eisert, Jens, Marcus Cramer, and Martin B. Plenio, 2010, “Colloquium: Area laws for the entanglement entropy,” Rev. Mod. Phys. 82, 277.
  76. Eisler, Viktor, and Ingo Peschel, 2007, “Evolution of entanglement after a local quench,” J. Stat. Mech. 06, P06005.
  77. Elvang, Henriette, and Marios Hadjiantonis, 2015, “Exact results for corner contributions to the entanglement entropy and Rényi entropies of free bosons and fermions in 3d,” Phys. Lett. B 749, 383–388.
  78. Erdmenger, J., and H. Osborn, 1997, “Conserved Currents and the Energy Momentum Tensor in Conformally Invariant Theories for General Dimensions,” Nucl. Phys. B 483, 431–474.
  79. Faulkner, Thomas, 2015, “Bulk Emergence and the RG Flow of Entanglement Entropy,” J. High Energy Phys. 05, 033.
  80. Faulkner, Thomas, Robert G. Leigh, Onkar Parrikar, and Huajia Wang, 2016, “Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition,” J. High Energy Phys. 09, 038.
  81. Fei, Lin, Simone Giombi, and Igor R. Klebanov, 2014, “Critical O(N) models in 6−ε dimensions,” Phys. Rev. D 90, 025018.
  82. Fradkin, Eduardo, and Joel E. Moore, 2006, “Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum,” Phys. Rev. Lett. 97, 050404.
  83. Frank, Rupert L., and Elliott H. Lieb, 2013, “Monotonicity of a relative Rényi entropy,” J. Math. Phys. (N.Y.) 54, 122201.
  84. Freedman, D. Z., S. S. Gubser, K. Pilch, and N. P. Warner, 1999, “Renormalization Group Flows from Holography Supersymmetry and a C-Theorem,” Adv. Theor. Math. Phys. 3, 363–417.
  85. Friedan, Daniel, and Anatoly Konechny, 2010, “Gradient formula for the beta-function of 2d quantum field theory,” J. Phys. A 43, 215401.
  86. Fursaev, Dmitri V., 1994, “Spectral Geometry and One Loop Divergences on Manifolds with Conical Singularities,” Phys. Lett. B 334, 53–60.
  87. Fursaev, Dmitri V., 2006, “Proof of the Holographic Formula for Entanglement Entropy,” J. High Energy Phys. 09, 018.
  88. Fursaev, Dmitri V., Alexander Patrushev, and Sergey N. Solodukhin, 2013, “Distributional Geometry of Squashed Cones,” Phys. Rev. D 88, 044054.
  89. Fursaev, Dmitri V., and Sergey N. Solodukhin, 1995, “On the Description of the Riemannian Geometry in the Presence of Conical Defects,” Phys. Rev. D 52, 2133–2143.
  90. Ghasemi, Mostafa, and Shahrokh Parvizi, 2018, “Entanglement entropy of singular surfaces under relevant deformations in holography,” J. High Energy Phys. 02, 009.
  91. Ghosh, Sudip, Ronak M. Soni, and Sandip P. Trivedi, 2015, “On the Entanglement Entropy for Gauge Theories,” J. High Energy Phys. 09, 069.
  92. Gioev, Dimitri, and Israel Klich, 2006, “Entanglement Entropy of Fermions in Any Dimension and the Widom Conjecture,” Phys. Rev. Lett. 96, 100503.
  93. Giombi, Simone, and Igor R. Klebanov, 2015, “Interpolating between a and F,” J. High Energy Phys. 03, 117.
  94. Giombi, Simone, Igor R. Klebanov, and Grigory Tarnopolsky, 2016, “Conformal QEDd, F-Theorem and the ε Expansion,” J. Phys. A 49, 135403.
  95. Girardello, L., M. Petrini, M. Porrati, and A. Zaffaroni, 1998, “Novel Local CFT and Exact Results on Perturbations of N=4 Super Yang-Mills from AdS Dynamics,” J. High Energy Phys. 12, 022.
  96. Giveon, Amit, and David Kutasov, 2016, “Supersymmetric Rényi entropy in CFT2 and AdS3,” J. High Energy Phys. 01, 042.
  97. Graham, C. Robin, and Edward Witten, 1999, “Conformal anomaly of submanifold observables in AdS/CFT correspondence,” Nucl. Phys. B 546, 52–64.
  98. Greenberger, Daniel M., Michael A. Horne, Abner Shimony, and Anton Zeilinger, 1990, “Bells theorem without inequalities,” Am. J. Physiol. 58, 1131–1143.
  99. Greenberger, Daniel M., Michael A. Horne, and Anton Zeilinger, 1989, “Going beyond Bells theorem,” in Bells theorem, quantum theory and conceptions of the universe (Springer, New York), pp. 69–72.
  100. Grover, Tarun, 2014, “Entanglement Monotonicity and the Stability of Gauge Theories in Three Spacetime Dimensions,” Phys. Rev. Lett. 112, 151601.
  101. Grover, Tarun, Ari M. Turner, and Ashvin Vishwanath, 2011, “Entanglement Entropy of Gapped Phases and Topological Order in Three Dimensions,” Phys. Rev. B 84, 195120.
  102. Gubser, S. S., Igor R. Klebanov, and Alexander M. Polyakov, 1998, “Gauge Theory Correlators from Non-Critical String Theory,” Phys. Lett. B 428, 105–114.
  103. Gukov, Sergei, 2016, “Counting RG flows,” J. High Energy Phys. 01, 020.
  104. Gukov, Sergei, 2017, “RG Flows and Bifurcations,” Nucl. Phys. B 919, 583–638.
  105. Gusynin, V. P., and P. K. Pyatkovskiy, 2016, “Critical number of fermions in three-dimensional QED,” Phys. Rev. D 94, 125009.
  106. Hama, Naofumi, Tatsuma Nishioka, and Tomonori Ugajin, 2014, “Supersymmetric Rényi entropy in five dimensions,” J. High Energy Phys. 12, 048.
  107. Harlow, Daniel, 2016, “Jerusalem Lectures on Black Holes and Quantum Information,” Rev. Mod. Phys. 88, 015002.
  108. Hawking, S. W., and G. F. R. Ellis, 2011, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England).
  109. Hayden, Patrick, Matthew Headrick, and Alexander Maloney, 2013, “Holographic Mutual Information is Monogamous,” Phys. Rev. D 87, 046003.
  110. Headrick, Matthew, 2014, “General Properties of Holographic Entanglement Entropy,” J. High Energy Phys. 03, 085.
  111. Headrick, Matthew, and Tadashi Takayanagi, 2007, “A Holographic Proof of the Strong Subadditivity of Entanglement Entropy,” Phys. Rev. D 76, 106013.
  112. Herbut, Igor F., 2016, “Chiral symmetry breaking in three-dimensional quantum electrodynamics as fixed point annihilation,” Phys. Rev. D 94, 025036.
  113. Hertzberg, Mark P., 2013, “Entanglement Entropy in Scalar Field Theory,” J. Phys. A 46, 015402.
  114. Hertzberg, Mark P., and Frank Wilczek, 2011, “Some Calculable Contributions to Entanglement Entropy,” Phys. Rev. Lett. 106, 050404.
  115. Herzog, Christopher, and Kuo-Wei Huang, 2017, “Boundary Fluctuations and A Reduction Entropy,” Phys. Rev. D 95, 021901.
  116. Herzog, Christopher P., 2014, “Universal Thermal Corrections to Entanglement Entropy for Conformal Field Theories on Spheres,” J. High Energy Phys. 10, 28.
  117. Herzog, Christopher P., Kuo-Wei Huang, and Kristan Jensen, 2016, “Universal Entanglement and Boundary Geometry in Conformal Field Theory,” J. High Energy Phys. 01, 162.
  118. Herzog, Christopher P., and Tatsuma Nishioka, 2013, “Entanglement Entropy of a Massive Fermion on a Torus,” J. High Energy Phys. 03, 077.
  119. Herzog, Christopher P., and Tatsuma Nishioka, 2016, “The Edge of Entanglement: Getting the Boundary Right for Non-Minimally Coupled Scalar Fields,” J. High Energy Phys. 12, 138.
  120. Hirata, Tomoyoshi, and Tadashi Takayanagi, 2007, “AdS/CFT and strong subadditivity of entanglement entropy,” J. High Energy Phys. 02, 042.
  121. Holzhey, Christoph, Finn Larsen, and Frank Wilczek, 1994, “Geometric and Renormalized Entropy in Conformal Field Theory,” Nucl. Phys. B 424, 443–467.
  122. Horodecki, Ryszard, Pawel Horodecki, Michal Horodecki, and Karol Horodecki, 2009, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942.
  123. Huang, Kuo-Wei, 2015, “Central Charge and Entangled Gauge Fields,” Phys. Rev. D 92, 025010.
  124. Huang, Xing, Soo-Jong Rey, and Yang Zhou, 2014, “Three-dimensional SCFT on conic space as hologram of charged topological black hole,” J. High Energy Phys. 03, 127.
  125. Huang, Xing, and Yang Zhou, 2015, “N=4 Super-Yang-Mills on conic space as hologram of STU topological black hole,” J. High Energy Phys. 02, 068.
  126. Hubeny, Veronika E., Mukund Rangamani, and Tadashi Takayanagi, 2007, “A Covariant Holographic Entanglement Entropy Proposal,” J. High Energy Phys. 07, 062.
  127. Huerta, Marina, 2012, “Numerical Determination of the Entanglement Entropy for Free Fields in the Cylinder,” Phys. Lett. B 710, 691–696.
  128. Hughes, Taylor L., Robert G. Leigh, Onkar Parrikar, and Srinidhi T. Ramamurthy, 2016, “Entanglement entropy and anomaly inflow,” Phys. Rev. D 93, 065059.
  129. Huijse, Liza, Subir Sachdev, and Brian Swingle, 2012, “Hidden Fermi Surfaces in Compressible States of Gauge-Gravity Duality,” Phys. Rev. B 85, 035121.
  130. Hung, Ling-Yan, Robert C. Myers, and Michael Smolkin, 2011, “On Holographic Entanglement Entropy and Higher Curvature Gravity,” J. High Energy Phys. 04, 025.
  131. Hung, Ling-Yan, Robert C. Myers, Michael Smolkin, and Alexandre Yale, 2011, “Holographic Calculations of Rényi Entropy,” J. High Energy Phys. 12, 047.
  132. Hung, Ling-Yan, and Yidun Wan, 2015, “Revisiting Entanglement Entropy of Lattice Gauge Theories,” J. High Energy Phys. 04, 122.
  133. Intriligator, Kenneth A., and Brian Wecht, 2003, “The Exact superconformal R symmetry maximizes a,” Nucl. Phys. B 667, 183–200.
  134. Iqbal, Nabil, and Aron C. Wall, 2016, “Anomalies of the Entanglement Entropy in Chiral Theories,” J. High Energy Phys. 10, 111.
  135. Israel, W., 1976, “Thermo field dynamics of black holes,” Phys. Lett. A 57, 107–110.
  136. Jack, I., and H. Osborn, 1990, “Analogs for the c Theorem for Four-dimensional Renormalizable Field Theories,” Nucl. Phys. B 343, 647–688.
  137. Jacobson, Ted, and Robert C. Myers, 1993, “Black Hole Entropy and Higher Curvature Interactions,” Phys. Rev. Lett. 70, 3684–3687.
  138. Jafferis, Daniel L., 2010, “The Exact Superconformal R-Symmetry Extremizes Z,” arXiv:1012.3210.
  139. Jafferis, Daniel L., Igor R. Klebanov, Silviu S. Pufu, and Benjamin R. Safdi, 2011, “Towards the F-Theorem: N=2 Field Theories on the Three-Sphere,” J. High Energy Phys. 06, 102.
  140. Jafferis, Daniel L., and Silviu S. Pufu, 2014, “Exact results for five-dimensional superconformal field theories with gravity duals,” J. High Energy Phys. 05, 032.
  141. Jin, B-Q, and Vladimir E. Korepin, 2004, “Quantum spin chain, Toeplitz determinants and the Fisher-Hartwig conjecture,” J. Stat. Phys. 116, 79–95.
  142. Kabat, Daniel N., 1995, “Black Hole Entropy and Entropy of Entanglement,” Nucl. Phys. B 453, 281–302.
  143. Karthik, Nikhil, and Rajamani Narayanan, 2016a, “No evidence for bilinear condensate in parity-invariant three-dimensional QED with massless fermions,” Phys. Rev. D 93, 045020.
  144. Karthik, Nikhil, and Rajamani Narayanan, 2016b, “Scale-invariance of parity-invariant three-dimensional QED,” Phys. Rev. D 94, 065026.
  145. Kawano, Teruhiko, Yuki Nakaguchi, and Tatsuma Nishioka, 2014, “Holographic Interpolation between a and F,” J. High Energy Phys. 12, 161.
  146. Kitaev, Alexei, and John Preskill, 2006, “Topological Entanglement Entropy,” Phys. Rev. Lett. 96, 110404.
  147. Klebanov, Igor R., 2000, “TASI Lectures: Introduction to the AdS/CFT Correspondence,” arXiv:hep-th/0009139.
  148. Klebanov, Igor R., David Kutasov, and Arvind Murugan, 2008, “Entanglement as a Probe of Confinement,” Nucl. Phys. B 796, 274–293.
  149. Klebanov, Igor R., Tatsuma Nishioka, Silviu S. Pufu, and Benjamin R. Safdi, 2012a, “Is Renormalized Entanglement Entropy Stationary at RG Fixed Points?,” J. High Energy Phys. 10, 058.
  150. Klebanov, Igor R., Tatsuma Nishioka, Silviu S. Pufu, and Benjamin R. Safdi, 2012b, “On Shape Dependence and RG Flow of Entanglement Entropy,” J. High Energy Phys. 07, 001.
  151. Klebanov, Igor R., Silviu S. Pufu, Subir Sachdev, and Benjamin R. Safdi, 2012c, “Entanglement Entropy of 3−d Conformal Gauge Theories with Many Flavors,” J. High Energy Phys. 05, 036.
  152. Klebanov, Igor R., Silviu S. Pufu, Subir Sachdev, and Benjamin R. Safdi, 2012d, “Rényi Entropies for Free Field Theories,” J. High Energy Phys. 04, 074.
  153. Klebanov, Igor R., Silviu S. Pufu, and Benjamin R. Safdi, 2011, “F-Theorem without Supersymmetry,” J. High Energy Phys. 10, 038.
  154. Komargodski, Zohar, 2012, “The Constraints of Conformal Symmetry on RG Flows,” J. High Energy Phys. 07, 069.
  155. Komargodski, Zohar, and Adam Schwimmer, 2011, “On Renormalization Group Flows in Four Dimensions,” arXiv:1107.3987.
  156. Laflamme, R., 1989, “Geometry and Thermofields,” Nucl. Phys. B 324, 233–252.
  157. Laflorencie, Nicolas, 2016, “Quantum entanglement in condensed matter systems,” Phys. Rep. 646, 1–59.
  158. Lashkari, Nima, 2017, “Entanglement at a Scale and Renormalization Monotones,” arXiv:1704.05077.
  159. Lee, Jeongseog, Aitor Lewkowycz, Eric Perlmutter, and Benjamin R. Safdi, 2015, “Rényi entropy, stationarity, and entanglement of the conformal scalar,” J. High Energy Phys. 03, 075.
  160. Levin, Michael, and Xiao-Gang Wen, 2006, “Detecting Topological Order in a Ground State Wave Function,” Phys. Rev. Lett. 96, 110405.
  161. Lewkowycz, Aitor, and Juan Maldacena, 2013, “Generalized Gravitational Entropy,” J. High Energy Phys. 08, 090.
  162. Lieb, E. H., and M. B. Ruskai, 1973, “Proof of the Strong Subadditivity of Quantum-Mechanical Entropy,” J. Math. Phys. (N.Y.) 14, 1938–1941.
  163. Lieb, Elliott H., and Jakob Yngvason, 1999, “The Physics and Mathematics of the Second Law of Thermodynamics,” Phys. Rep. 310, 1–96.
  164. Liu, Hong, and Mark Mezei, 2013a, “A Refinement of Entanglement Entropy and the Number of Degrees of Freedom,” J. High Energy Phys. 04, 162.
  165. Liu, Hong, and Mark Mezei, 2013b, “Probing Renormalization Group Flows Using Entanglement Entropy,” arXiv:1309.6935.
  166. Lohmayer, R., H. Neuberger, A. Schwimmer, and S. Theisen, 2010, “Numerical Determination of Entanglement Entropy for a Sphere,” Phys. Lett. B 685, 222–227.
  167. Ma, Chen-Te, 2016, “Entanglement with Centers,” J. High Energy Phys. 01, 070.
  168. Maldacena, Juan, and Leonard Susskind, 2013, “Cool horizons for entangled black holes,” Fortschr. Phys. 61, 781–811.
  169. Maldacena, Juan Martin, 1998, “The Large N Limit of Superconformal Field Theories and Supergravity,” Adv. Theor. Math. Phys. 2, 231–252.
  170. McGreevy, John, 2010, “Holographic Duality with a View Toward Many-Body Physics,” Adv. High Energy Phys. 2010, 1.
  171. Miao, Rong-Xin, and Wu-zhong Guo, 2015, “Holographic Entanglement Entropy for the Most General Higher Derivative Gravity,” J. High Energy Phys. 08, 031.
  172. Mori, Hironori, 2016, “Supersymmetric Rényi entropy in two dimensions,” J. High Energy Phys. 03, 058.
  173. Müller-Lennert, Martin, Frédéric Dupuis, Oleg Szehr, Serge Fehr, and Marco Tomamichel, 2013, “On quantum Rényi entropies: A new generalization and some properties,” J. Math. Phys. (N.Y.) 54, 122203.
  174. Myers, Robert C., and Ajay Singh, 2012a, “Comments on Holographic Entanglement Entropy and RG Flows,” arXiv:1202.2068.
  175. Myers, Robert C., and Ajay Singh, 2012b, “Entanglement Entropy for Singular Surfaces,” J. High Energy Phys. 09, 013.
  176. Myers, Robert C., and Aninda Sinha, 2010, “Seeing a C-Theorem with Holography,” Phys. Rev. D 82, 046006.
  177. Myers, Robert C., and Aninda Sinha, 2011, “Holographic c-theorems in arbitrary dimensions,” J. High Energy Phys. 01, 125.
  178. Nakaguchi, Yuki, and Tatsuma Nishioka, 2016, “A holographic proof of Rényi entropic inequalities,” J. High Energy Phys. 12, 129.
  179. Nakayama, Yu, 2015, “Scale invariance vs conformal invariance,” Phys. Rep. 569, 1–93.
  180. Narnhofer, H., and Walter E. Thirring, 1985, “From Relative Entropy to Entropy,” Fizika (Zegreb) 17, 257–265.
  181. Nian, Jun, and Yang Zhou, 2016, “Rényi entropy of a free (2,0) tensor multiplet and its supersymmetric counterpart,” Phys. Rev. D 93, 125010.
  182. Nielsen, Michael A., and Isaac L. Chuang, 2010, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England).
  183. Nishioka, Tatsuma, 2014a, “Relevant Perturbation of Entanglement Entropy and Stationarity,” Phys. Rev. D 90, 045006.
  184. Nishioka, Tatsuma, 2014b, “The Gravity Dual of Supersymmetric Renyi Entropy,” J. High Energy Phys. 07, 061.
  185. Nishioka, Tatsuma, Shinsei Ryu, and Tadashi Takayanagi, 2009, “Holographic Entanglement Entropy: an Overview,” J. Phys. A 42, 504008.
  186. Nishioka, Tatsuma, and Tadashi Takayanagi, 2007, “AdS Bubbles, Entropy and Closed String Tachyons,” J. High Energy Phys. 01, 090.
  187. Nishioka, Tatsuma, and Itamar Yaakov, 2013, “Supersymmetric Rényi Entropy,” J. High Energy Phys. 10, 155.
  188. Nishioka, Tatsuma, and Itamar Yaakov, 2017, “Supersymmetric Rényi entropy and defect operators,” J. High Energy Phys. 11, 071.
  189. Nishioka, Tatsuma, and Amos Yarom, 2016, “Anomalies and Entanglement Entropy,” J. High Energy Phys. 03, 077.
  190. Nishioka, Tatsuma, and Kazuya Yonekura, 2013, “On RG Flow of τRR for Supersymmetric Field Theories in Three-Dimensions,” J. High Energy Phys. 05, 165.
  191. Ogawa, Noriaki, Tadashi Takayanagi, and Tomonori Ugajin, 2012, “Holographic Fermi Surfaces and Entanglement Entropy,” J. High Energy Phys. 01, 125.
  192. Ohmori, Kantaro, and Yuji Tachikawa, 2015, “Physics at the entangling surface,” J. Stat. Mech. 04, P04010.
  193. Ohya, Masanori, 2004, Quantum entropy and its use (Springer, New York).
  194. Osborn, H., 1989, “Derivation of a Four-dimensional c Theorem,” Phys. Lett. B 222, 97–102.
  195. Osborn, H., and A. C. Petkou, 1994, “Implications of Conformal Invariance in Field Theories for General Dimensions,” Ann. Phys. (N.Y.) 231, 311–362.
  196. Pakman, Ari, and Andrei Parnachev, 2008, “Topological Entanglement Entropy and Holography,” J. High Energy Phys. 07, 097.
  197. Perlmutter, Eric, 2014, “A universal feature of CFT Rényi entropy,” J. High Energy Phys. 03, 117.
  198. Peschel, I., 2003, “Calculation of reduced density matrices from correlation functions,” J. Phys. A 36, L205–L208.
  199. Peschel, Ingo, and Ming-Chiang Chung, 1999, “Density matrices for a chain of oscillators,” J. Phys. A 32, 8419.
  200. Peschel, Ingo, and Viktor Eisler, 2009, “Reduced density matrices and entanglement entropy in free lattice models,” J. Phys. A 42, 504003.
  201. Petkou, Anastasios C., 1995, “CT and CJ up to next-to-leading order in 1/N in the conformally invariant O(N) vector model for 2<d<4,” Phys. Lett. B 359, 101–107.
  202. Polchinski, Joseph, 1984, “Renormalization and Effective Lagrangians,” Nucl. Phys. B 231, 269–295.
  203. Preskill, J., 1997, “Lecture notes for ph219/cs219,” http://www.theory.caltech.edu/people/preskill/ph229/.
  204. Pretko, Michael, 2018, “On the Entanglement Entropy of Maxwell Theory: A Condensed Matter Perspective,” arXiv:1801.01158.
  205. Pretko, Michael, and T. Senthil, 2016, “Entanglement entropy of U(1) quantum spin liquids,” Phys. Rev. B 94, 125112.
  206. Pufu, Silviu S., 2016, “The F-Theorem and F-Maximization,” arXiv:1608.02960.
  207. Radicevic, Djordje, 2014, “Notes on Entanglement in Abelian Gauge Theories,” arXiv:1404.1391.
  208. Radicevic, Djordje, 2016, “Entanglement in Weakly Coupled Lattice Gauge Theories,” J. High Energy Phys. 04, 163.
  209. Rangamani, Mukund, and Tadashi Takayanagi, 2017, “Holographic Entanglement Entropy,” Lect. Notes Phys. 931, 1–246.
  210. Rényi, Alfred, 1961, “On measures of entropy and information,” in Fourth Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley, CA), pp. 547–561.
  211. Rosenhaus, Vladimir, and Michael Smolkin, 2014, “Entanglement Entropy: A Perturbative Calculation,” J. High Energy Phys. 12, 179.
  212. Rosenhaus, Vladimir, and Michael Smolkin, 2015, “Entanglement Entropy for Relevant and Geometric Perturbations,” J. High Energy Phys. 02, 015.
  213. Ryu, Shinsei, and Tadashi Takayanagi, 2006a, “Aspects of Holographic Entanglement Entropy,” J. High Energy Phys. 08, 045.
  214. Ryu, Shinsei, and Tadashi Takayanagi, 2006b, “Holographic Derivation of Entanglement Entropy from AdS/CFT,” Phys. Rev. Lett. 96, 181602.
  215. Sachdev, Subir, 1993, “Polylogarithm identities in a conformal field theory in three-dimensions,” Phys. Lett. B 309, 285–288.
  216. Safdi, Benjamin R., 2012, “Exact and Numerical Results on Entanglement Entropy in (5+1)-Dimensional CFT,” J. High Energy Phys. 12, 005.
  217. Solodukhin, Sergey N., 2008, “Entanglement Entropy, Conformal Invariance and Extrinsic Geometry,” Phys. Lett. B 665, 305–309.
  218. Solodukhin, Sergey N., 2011, “Entanglement Entropy of Black Holes,” Living Rev. Relativity 14, 8.
  219. Solodukhin, Sergey N., 2013, “The a-theorem and entanglement entropy,” arXiv:1304.4411.
  220. Soni, Ronak M., and Sandip P. Trivedi, 2016, “Aspects of Entanglement Entropy for Gauge Theories,” J. High Energy Phys. 01, 136.
  221. Srednicki, Mark, 1993, “Entropy and Area,” Phys. Rev. Lett. 71, 666–669.
  222. Swingle, Brian, 2012, “Entanglement Renormalization and Holography,” Phys. Rev. D 86, 065007.
  223. Takayanagi, Tadashi, 2012, “Entanglement Entropy from a Holographic Viewpoint,” Classical Quantum Gravity 29, 153001.
  224. Taylor, Marika, and William Woodhead, 2016a, “Renormalized entanglement entropy,” J. High Energy Phys. 08, 165.
  225. Taylor, Marika, and William Woodhead, 2016b, “The holographic F theorem,” arXiv:1604.06809.
  226. Umegaki, Hisaharu, et al., 1962, “Conditional expectation in an operator algebra. IV. Entropy and information,” in Kodai Mathematical Seminar Reports, Vol. 14 (Tokyo Institute of Technology, Department of Mathematics), pp. 59–85.
  227. Van Acoleyen, Karel, Nick Bultinck, Jutho Haegeman, Michael Marien, Volkher B. Scholz, and Frank Verstraete, 2016, “The entanglement of distillation for gauge theories,” Phys. Rev. Lett. 117, 131602.
  228. Van Raamsdonk, Mark, 2009, “Comments on quantum gravity and entanglement,” arXiv:0907.2939.
  229. Van Raamsdonk, Mark, 2017, “Lectures on Gravity and Entanglement,” in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015) (World Scientific, Singapore), pp. 297–351.
  230. Vassilevich, D. V., 2003, “Heat Kernel Expansion: User’s Manual,” Phys. Rep. 388, 279–360.
  231. Vedral, V., 2002, “The role of relative entropy in quantum information theory,” Rev. Mod. Phys. 74, 197–234.
  232. Vidal, G., J. I. Latorre, E Rico, and A. Kitaev, 2003, “Entanglement in quantum critical phenomena,” Phys. Rev. Lett. 90, 227902.
  233. Wall, Aron C., 2014, “Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy,” Classical Quantum Gravity 31, 225007.
  234. Wilde, Mark M., Andreas Winter, and Dong Yang, 2013, “Strong converse for the classical capacity of entanglement-breaking channels,” arXiv:1306.1586.
  235. Willett, Brian, 2017, “Localization on three-dimensional manifolds,” J. Phys. A 50, 443006.
  236. Wilson, K. G., and John B. Kogut, 1974, “The Renormalization group and the epsilon expansion,” Phys. Rep. 12, 75–200.
  237. Witten, Edward, 1998, “Anti-de Sitter Space and Holography,” Adv. Theor. Math. Phys. 2, 253–291.
  238. Witten, Edward, 2018, “Notes on Some Entanglement Properties of Quantum Field Theory,” arXiv:1803.04993.
  239. Wolf, Michael M., 2006, “Violation of the Entropic Area Law for Fermions,” Phys. Rev. Lett. 96, 010404.
  240. Wolf, Michael M., Frank Verstraete, Matthew B. Hastings, and J Ignacio Cirac, 2008, “Area laws in quantum systems: mutual information and correlations,” Phys. Rev. Lett. 100, 070502.
  241. Yankielowicz, Shimon, and Yang Zhou, 2017, “Supersymmetric Rényi entropy and Anomalies in 6d (1,0) SCFTs,” J. High Energy Phys. 04, 128.
  242. Yao, Hong, and Xiao-Liang Qi, 2010, “Entanglement entropy and entanglement spectrum of the Kitaev model,” Phys. Rev. Lett. 105, 080501.
  243. Yonekura, Kazuya, 2013, “Perturbative c-theorem in d-dimensions,” J. High Energy Phys. 04, 011.
  244. Zamolodchikov, A. B., 1986, “Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,” JETP Lett. 43, 730–732 [http://www.jetpletters.ac.ru/ps/1413/article_21504.shtml].
  245. Zhou, Yang, 2015, “Universal Features of Four-Dimensional Superconformal Field Theory on Conic Space,” J. High Energy Phys. 08, 052.
  246. Zhou, Yang, 2016, “Supersymmetric Rényi entropy and Weyl anomalies in six-dimensional (2,0) theories,” J. High Energy Phys. 06, 064.
  247. Życzkowski, Karol, 2003, “Rényi extrapolation of Shannon entropy,” Open Syst. Inf. Dyn. 10, 297–310.

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Rev. Mod. Phys. 90, 035007– Published 17 September, 2018
  • Received 27 January 2018
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