Advertisement

Annales Henri Poincaré

, Volume 20, Issue 7, pp 2353–2375 | Cite as

Relative Entropy Close to the Edge

  • Stefan HollandsEmail author
Article
  • 9 Downloads

Abstract

We show that the relative entropy between the reduced density matrix of the vacuum state in some region A and that of an excited state created by a unitary operator localized at a small distance \(\ell \) of a boundary point p is insensitive to the global shape of A, up to a small correction. This correction tends to zero as \(\ell /R\) tends to zero, where R is a measure of the curvature of \(\partial A\) at p, but at a rate necessarily slower than \(\sim \sqrt{\ell /R}\) (in any dimension). Our arguments are mathematically rigorous and only use model independent, basic assumptions about quantum field theory such as locality and Poincare invariance.

Notes

Acknowledgements

It is a pleasure to thank Centro Atomico Balseiro, Bariloche, Argentina, for hospitality during my visit in March 2018, as well as the Simons Foundation for financially supporting that visit. I have greatly benefited from discussions with H. Casini, M. Huerta, F. Otto, and D. Pontello. I am grateful to the Max-Planck Society for supporting the collaboration between MPI-MiS and Leipzig U., grant Proj. Bez. M.FE.A.MATN0003.

References

  1. 1.
    Araki, H.: Relative Hamiltonian for faithful normal states of a von Neumann algebra. Publ. RIMS Kyoto Univ. 9, 165–209 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Araki, H.: Relative entropy of states of von Neumann algebras. I. Publ. RIMS Kyoto Univ. 11, 809–833 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Araki, H.: Relative entropy of states of von Neumann algebras. II. Publ. RIMS Kyoto Univ. 13, 173–192 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Araki, H.: On quasifree states of the CAR and Bogoliubov automorphisms. Publ. RIMS Kyoto Univ. 6, 385–442 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arias, R., Blanco, D., Casini, H., Huerta, M.: Local temperatures and local terms in modular Hamiltonians. Phys. Rev. D 95(6), 065005 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bisognano, J.J., Wichmann, E.H.: On the duality condition for quantum fields. J. Math. Phys. 17, 303 (1976)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bochner, S.: Vorlesungen über Fouriersche Integrale. Leipzig Akad. Verlag 50, 22 (1932)zbMATHGoogle Scholar
  8. 8.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  10. 10.
    Carpi, S., Kawahigashi, Y., Longo, R., Weiner, M.: From vertex operator algebras to conformal nets and back. arXiv:1503.01260 [math.OA]
  11. 11.
    Casini, H.: Relative entropy and the Bekenstein bound. Class. Quantum Grav. 25, 205021 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    D’Antoni, C., Hollands, S.: Nuclearity, local quasiequivalence and split property for Dirac quantum fields in curved space–time. Commun. Math. Phys. 261, 133 (2006)CrossRefzbMATHGoogle Scholar
  13. 13.
    Fredenhagen, K.: On the modular structure of local algebras of observables. Commun. Math. Phys. 97, 79–89 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gabbiani, F., Fröhlich, J.: Operator algebras and conformal field theories. Commun. Math. Phys. 155, 569–640 (1993)CrossRefzbMATHGoogle Scholar
  15. 15.
    Gearhart, W.B., Schulz, H.S.: The Function \(\sin x/x\). College Math. J. 21, 90–99 (1990)Google Scholar
  16. 16.
    Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Haag, R.: Local Quantum Physics: Fields, Particles, Algebras. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  18. 18.
    Hislop, P.D., Longo, R.: Modular structure of the local algebras associated with the free massless scalar field theory. Commun. Math. Phys. 84, 71 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Longo, R., Xu, F.: Comment on the Bekenstein bound. J. Geom. Phys. 130, 113 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hollands, S., Sanders, K.: Entanglement measures and their properties in quantum field theory. arXiv:1702.04924 [quant-ph]
  21. 21.
    Ingham, A.E.: A note on Fourier transforms. J. Lond. Math. Soc. S1–9, 29–32 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rangamani, M., Takayanagi, T.: Holographic Entanglement Entropy, Springer Lecture Notes in Physics. Springer, Berlin (2017)CrossRefzbMATHGoogle Scholar
  23. 23.
    Reeh, H., Schlieder, S.: Bemerkungen zur Unitäräquivalenz von lorentzinvarianten Feldern. Nuovo Cimento 22, 1051–1068 (1961)CrossRefzbMATHGoogle Scholar
  24. 24.
    Solodukhin, S.N.: Entanglement entropy of black holes. Liv. Rev. Rel. 14, 8 (2011)CrossRefzbMATHGoogle Scholar
  25. 25.
    Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)CrossRefzbMATHGoogle Scholar
  26. 26.
    Witten, E.: Notes on some entanglement properties of quantum field theory. arXiv:1803.04993 [hep-th]

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität LeipzigLeipzigGermany

Personalised recommendations