Advertisement

Journal of High Energy Physics

, 2015:69 | Cite as

On the entanglement entropy for gauge theories

  • Sudip Ghosh
  • Ronak M Soni
  • Sandip P. Trivedi
Open Access
Regular Article - Theoretical Physics

Abstract

We propose a definition for the entanglement entropy of a gauge theory on a spatial lattice. Our definition applies to any subset of links in the lattice, and is valid for both Abelian and Non-Abelian gauge theories. For \( {\mathbb{Z}}_N \) and U(1) theories, without matter, our definition agrees with a particular case of the definition given by Casini, Huerta and Rosabal. We also argue that in general, both for Abelian and Non-Abelian theories, our definition agrees with the entanglement entropy calculated using a definition of the replica trick. Our definition, however, does not agree with some standard ways to measure entanglement, like the number of Bell pairs which can be produced by entanglement distillation.

Keywords

Lattice Gauge Field Theories Gauge Symmetry 

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]
    H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].ADSGoogle Scholar
  2. [2]
    D. Radicević, Notes on entanglement in abelian gauge theories, arXiv:1404.1391 [INSPIRE].
  3. [3]
    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    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
  5. [5]
    L. De Nardo, D.V. Fursaev and G. Miele, Heat kernel coefficients and spectra of the vector Laplacians on spherical domains with conical singularities, Class. Quant. Grav. 14 (1997) 1059 [hep-th/9610011] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002 [hep-th/0405152] [INSPIRE].
  7. [7]
    P. Calabrese and J.L. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  8. [8]
    H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  9. [9]
    S.N. Solodukhin, Entanglement entropy of black holes, Living Rev. Rel. 14 (2011) 8 [arXiv:1104.3712] [INSPIRE].CrossRefzbMATHGoogle Scholar
  10. [10]
    D.N. Kabat, Black hole entropy and entropy of entanglement, Nucl. Phys. B 453 (1995) 281 [hep-th/9503016] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    A. Hamma, R. Ionicioiu and P. Zanardi, Bipartite entanglement and entropic boundary law in lattice spin systems, Phys. Rev. A 71 (2005) 022315 [quant-ph/0409073].
  12. [12]
    W. Donnelly, Entanglement entropy in loop quantum gravity, Phys. Rev. D 77 (2008) 104006 [arXiv:0802.0880] [INSPIRE].ADSMathSciNetGoogle Scholar
  13. [13]
    P.V. Buividovich and M.I. Polikarpov, Entanglement entropy in gauge theories and the holographic principle for electric strings, Phys. Lett. B 670 (2008) 141 [arXiv:0806.3376] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE].ADSGoogle Scholar
  15. [15]
    W. Donnelly and A.C. Wall, Do gauge fields really contribute negatively to black hole entropy?, Phys. Rev. D 86 (2012) 064042 [arXiv:1206.5831] [INSPIRE].ADSGoogle Scholar
  16. [16]
    S.N. Solodukhin, Remarks on effective action and entanglement entropy of Maxwell field in generic gauge, JHEP 12 (2012) 036 [arXiv:1209.2677] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    W. Donnelly, Entanglement entropy and nonabelian gauge symmetry, Class. Quant. Grav. 31 (2014) 214003 [arXiv:1406.7304] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    J.B. Kogut and L. Susskind, Hamiltonian formulation of Wilsons lattice gauge theories, Phys. Rev. D 11 (1975) 395 [INSPIRE].ADSGoogle Scholar
  19. [19]
    J.B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51 (1979) 659 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    K. Gottfried and T-M Yan, Quantum mechanics: fundamentals, 2nd edition, Springer, Germany (2004), see section 7.5(d).Google Scholar
  21. [21]
    M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge U.K. (2000).zbMATHGoogle Scholar
  22. [22]
    J. Preskill, Lecture notes on quantum computation, http://www.theory.caltech.edu/ preskill/ph229.
  23. [23]
    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
  24. [24]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  25. [25]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Sudip Ghosh
    • 1
  • Ronak M Soni
    • 2
  • Sandip P. Trivedi
    • 2
  1. 1.International Centre for Theoretical SciencesTata Institute of Fundamental Research, IISc CampusBangaloreIndia
  2. 2.Department of Theoretical PhysicsTata Institute of Fundamental ResearchMumbaiIndia

Personalised recommendations