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Generalized entanglement entropy

  • Marika Taylor
Open Access
Regular Article - Theoretical Physics

Abstract

We discuss two measures of entanglement in quantum field theory and their holographic realizations. For field theories admitting a global symmetry, we introduce a global symmetry entanglement entropy, associated with the partitioning of the symmetry group. This quantity is proposed to be related to the generalized holographic entanglement entropy defined via the partitioning of the internal space of the bulk geometry. Thesecond measure of quantum field theory entanglement is the field space entanglement entropy, obtained by integrating out a subset of the quantum fields. We argue that field space entanglement entropy cannot be precisely realised geometrically in a holographic dual. However, for holographic geometries with interior decoupling regions, the differential entropy provides a close analogue to the field space entanglement entropy. We derive generic descriptions of such inner throat regions in terms of gravity coupled to massive scalars and show how the differential entropy in the throat captures features of the field space entanglement entropy.

Keywords

AdS-CFT Correspondence Gauge-gravity correspondence 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    T. Takayanagi, Entanglement Entropy from a Holographic Viewpoint, Class. Quant. Grav. 29 (2012) 153001 [arXiv:1204.2450] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].ADSGoogle Scholar
  5. [5]
    M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
  6. [6]
    M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [Int. J. Mod. Phys. D 19 (2010) 2429] [arXiv:1005.3035] [INSPIRE].
  7. [7]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    E. Bianchi and R.C. Myers, On the Architecture of Spacetime Geometry, Class. Quant. Grav. 31 (2014) 214002 [arXiv:1212.5183] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    V. Balasubramanian, B. Czech, B.D. Chowdhury and J. de Boer, The entropy of a hole in spacetime, JHEP 10 (2013) 220 [arXiv:1305.0856] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    V. Balasubramanian, B.D. Chowdhury, B. Czech, J. de Boer and M.P. Heller, Bulk curves from boundary data in holography, Phys. Rev. D 89 (2014) 086004 [arXiv:1310.4204] [INSPIRE].ADSGoogle Scholar
  11. [11]
    R.C. Myers, J. Rao and S. Sugishita, Holographic Holes in Higher Dimensions, JHEP 06 (2014) 044 [arXiv:1403.3416] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    B. Czech, X. Dong and J. Sully, Holographic Reconstruction of General Bulk Surfaces, JHEP 11 (2014) 015 [arXiv:1406.4889] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    V. Balasubramanian, B.D. Chowdhury, B. Czech and J. de Boer, Entwinement and the emergence of spacetime, JHEP 01 (2015) 048 [arXiv:1406.5859] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    M. Headrick, R.C. Myers and J. Wien, Holographic Holes and Differential Entropy, JHEP 10 (2014) 149 [arXiv:1408.4770] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    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].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral Geometry and Holography, JHEP 10 (2015) 175 [arXiv:1505.05515] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    B. Freivogel, R.A. Jefferson, L. Kabir, B. Mosk and I.-S. Yang, Casting Shadows on Holographic Reconstruction, Phys. Rev. D 91 (2015) 086013 [arXiv:1412.5175] [INSPIRE].ADSGoogle Scholar
  18. [18]
    N. Engelhardt and S. Fischetti, Covariant Constraints on Hole-ography, Class. Quant. Grav. 32 (2015) 195021 [arXiv:1507.00354] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    N. Engelhardt and A.C. Wall, Extremal Surface Barriers, JHEP 03 (2014) 068 [arXiv:1312.3699] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    M. Rangamani and M. Rota, Comments on Entanglement Negativity in Holographic Field Theories, JHEP 10 (2014) 060 [arXiv:1406.6989] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    M. Kulaxizi, A. Parnachev and G. Policastro, Conformal Blocks and Negativity at Large Central Charge, JHEP 09 (2014) 010 [arXiv:1407.0324] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    E. Perlmutter, M. Rangamani and M. Rota, Central Charges and the Sign of Entanglement in 4D Conformal Field Theories, Phys. Rev. Lett. 115 (2015) 171601 [arXiv:1506.01679] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    S. Furukawa and Y.B. Kim, Entanglement entropy between two coupled Tomonaga-Luttinger liquids, Phys. Rev. B 83 (2011) 085112 [Erratum ibid. B 87 (2013) 119901] [arXiv:1009.3016] [INSPIRE].
  24. [24]
    C. Xu, Entanglement Entropy of Coupled Conformal Field Theories and Fermi Liquids, Phys. Rev. B 84 (2011) 125119 [arXiv:1102.5345] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    X. Chen and E. Fradkin, Quantum Entanglement and Thermal Reduced Density Matrices in Fermion and Spin Systems on Ladders, J. Stat. Mech. (2013) P08013 [arXiv:1305.6538] [INSPIRE].
  26. [26]
    R. Lundgren, Y. Fuji, S. Furukawa and M. Oshikawa, Entanglement spectra between coupled Tomonaga-Luttinger liquids: Applications to ladder systems and topological phases, Phys. Rev. B 88 (2013) 245137 [Erratum ibid. B 92 (2015) 039903] [INSPIRE].
  27. [27]
    A. Mollabashi, N. Shiba and T. Takayanagi, Entanglement between Two Interacting CFTs and Generalized Holographic Entanglement Entropy, JHEP 04 (2014) 185 [arXiv:1403.1393] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    A. Karch and C.F. Uhlemann, Holographic entanglement entropy and the internal space, Phys. Rev. D 91 (2015) 086005 [arXiv:1501.00003] [INSPIRE].ADSMathSciNetGoogle Scholar
  29. [29]
    K. Skenderis and M. Taylor, Kaluza-Klein holography, JHEP 05 (2006) 057 [hep-th/0603016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    K. Skenderis and M. Taylor, Holographic Coulomb branch vevs, JHEP 08 (2006) 001 [hep-th/0604169] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    K. Skenderis and M. Taylor, Anatomy of bubbling solutions, JHEP 09 (2007) 019 [arXiv:0706.0216] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    R. Lundgren, Momentum-Space Entanglement in Heisenberg Spin-Half Ladders, Phys. Rev. B 93 (2016) 125107 [arXiv:1412.8612] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    O. Aharony, A.B. Clark and A. Karch, The CFT/AdS correspondence, massive gravitons and a connectivity index conjecture, Phys. Rev. D 74 (2006) 086006 [hep-th/0608089] [INSPIRE].ADSMathSciNetGoogle Scholar
  36. [36]
    E. Kiritsis, Product CFTs, gravitational cloning, massive gravitons and the space of gravitational duals, JHEP 11 (2006) 049 [hep-th/0608088] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    F. Aprile and V. Niarchos, Large-N transitions of the connectivity index, JHEP 02 (2015) 083 [arXiv:1410.7773] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    K.A. Intriligator, Maximally supersymmetric RG flows and AdS duality, Nucl. Phys. B 580 (2000) 99 [hep-th/9909082] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    I. Kanitscheider, K. Skenderis and M. Taylor, Holographic anatomy of fuzzballs, JHEP 04 (2007) 023 [hep-th/0611171] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    J.R. David, G. Mandal and S.R. Wadia, Microscopic formulation of black holes in string theory, Phys. Rept. 369 (2002) 549 [hep-th/0203048] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    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].ADSCrossRefzbMATHGoogle Scholar
  45. [45]
    C.R. Graham and E. Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nucl. Phys. B 546 (1999) 52 [hep-th/9901021] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    C.R. Graham, Volume and area renormalizations for conformally compact Einstein metrics, math.DG/9909042 [INSPIRE].
  47. [47]
    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].ADSGoogle Scholar
  48. [48]
    A. Lewkowycz and J.M. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    B. Biran, A. Casher, F. Englert, M. Rooman and P. Spindel, The Fluctuating Seven Sphere in Eleven-dimensional Supergravity, Phys. Lett. B 134 (1984) 179 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    A. Casher, F. Englert, H. Nicolai and M. Rooman, The Mass Spectrum of Supergravity on the Round Seven Sphere, Nucl. Phys. B 243 (1984) 173 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    H.J. Kim, L.J. Romans and P. van Nieuwenhuizen, The Mass Spectrum of Chiral N = 2 D = 10 Supergravity on S 5, Phys. Rev. D 32 (1985) 389 [INSPIRE].ADSGoogle Scholar
  52. [52]
    K. Pilch, P. van Nieuwenhuizen and P.K. Townsend, Compactification of d = 11 Supergravity on S 4 (Or 11 = 7 + 4, Too), Nucl. Phys. B 242 (1984) 377 [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    M. Günaydin, P. van Nieuwenhuizen and N.P. Warner, General Construction of the Unitary Representations of Anti-de Sitter Superalgebras and the Spectrum of the S 4 Compactification of Eleven-dimensional Supergravity, Nucl. Phys. B 255 (1985) 63 [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    P. van Nieuwenhuizen, The Complete Mass Spectrum of d = 11 Supergravity Compactified on S 4 and a General Mass Formula for Arbitrary Cosets M 4, Class. Quant. Grav. 2 (1985) 1 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of SouthamptonSouthamptonU.K.

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