Von Neumann Entropy in QFT

Abstract

In the framework of Quantum Field Theory, we provide a rigorous, operator algebraic notion of entanglement entropy associated with a pair of open double cones \(O \subset {\widetilde{O}}\) of the spacetime, where the closure of O is contained in \({\widetilde{O}}\). Given a QFT net \({\mathcal {A}}\) of local von Neumann algebras \({\mathcal {A}}(O)\), we consider the von Neumann entropy \(S_{\mathcal {A}}(O, {\widetilde{O}})\) of the restriction of the vacuum state to the canonical intermediate type I factor for the inclusion of von Neumann algebras \({\mathcal {A}}(O)\subset {\mathcal {A}}({\widetilde{O}})\) (split property). We show that this canonical entanglement entropy \(S_{\mathcal {A}}(O, {\widetilde{O}})\) is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). To this end, we first study the notion of von Neumann entropy of a closed real linear subspace of a complex Hilbert space, that we then estimate for the local free fermion subspaces. We further consider the lower entanglement entropy \({\underline{S}}_{\mathcal {A}}(O, {\widetilde{O}})\), the infimum of the vacuum von Neumann entropy of \({\mathcal {F}}\), where \({\mathcal {F}}\) here runs over all the intermediate, discrete type I von Neumann algebras. We prove that \({\underline{S}}_{\mathcal {A}}(O, {\widetilde{O}})\) is finite for the local chiral conformal net generated by finitely many commuting U(1)-currents.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Araki, H.: A lattice of von Neumann algebras associated with the quantum field theory of a free Bose field. J. Math. Phys. 4, 1343–1362 (1963)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Araki, H.: On quasi-free states of CAR and Bogoliubov automorphisms. Publ. RIMS 6, 385–442 (1970)

    Article  Google Scholar 

  3. 3.

    Buchholz, D.: Product states for local algebras. Commun. Math. Phys. 36, 287–304 (1974)

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    Buchholz, D., Wichmann, E.H.: Causal independence and the energy-level density of states in local quantum field theory. Commun. Math. Phys. 106, 321–344 (1986)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    Calabrese, P., Cardy, J., Doyon, B.: Entanglement entropy in extended quantum systems. J. Phys. A 42, 500301 (2009)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Casini, H., Huerta, M.: A \(c\)-theorem for entanglement entropy. J. Phys. A 40(25), 7031–7036 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  7. 7.

    Casini, H., Huerta, M.: Reduced density matrix and internal dynamics for multicomponent regions. Class. Quantum Grav. 26, 185005 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Chamseddine, A.H., Connes, A., van Suijlekom, W.D.: Entropy and the spectral action. Commun. Math. Phys. (2019). https://doi.org/10.1007/s00220-019-03297-8

    Article  MATH  Google Scholar 

  9. 9.

    Dixmier, J.: Positions relative de deux varietés lineaires fermées dans un espace de Hilbert. Rev. Sci. 86, 387–399 (1948)

    MATH  Google Scholar 

  10. 10.

    Doplicher, S., Longo, R.: Standard and split inclusions of von Neumann algebras. Invent. Math. 75, 493–536 (1984)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

    Dutta, S., Faulkner, T.: A canonical purification for the entanglement wedge cross-section. arXiv:1905.00577

  12. 12.

    Figliolini, F., Guido, D.: On the type of second quantization factors. J. Oper. Theory 31, 229–252 (1994)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Foit, J.J.: Abstract twisted duality for quantum free Fermi fields. Publ. RIMS Kyoto Univ. 19, 729–74 (1983)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Halmos, P.R.: Two subspaces. Trans. Am. Math. Soc 144, 381–389 (1969)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Haag, R.: Local Quantum Physics: Fields, Particles, Algebras, 2nd edn. Springer, New York (1996)

    Google Scholar 

  16. 16.

    Harlow, D., Ooguri, H.: Symmetries in quantum field theory and quantum gravity. arXiv:1810.05338 [hep-th]

  17. 17.

    Hollands, S., Sanders, K.: Entanglement Measures and Their Properties in Quantum Field Theory. SprinerBriefs in Mathematical Physics, vol. 34. Springer, Cham (2019)

    Google Scholar 

  18. 18.

    Howland, J.: Trace class Hankel operators. Q. J. Math. Oxf. (2) 22, 147–50 (1971)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Kawahigashi, Y., Longo, R.: Noncommutative spectral invariants and black hole entropy. Commun. Math. Phys. 75(257), 193–225 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    Leyland, P., Roberts, J.E., Testard, D.: Duality for the free electromagnetic field. Marseille (1976) (unpublished)

  21. 21.

    Longo, R.: Real Hilbert subspaces, modular theory, SL(2, R) and CFT. In: Von Neumann algebras in Sibiu, 33-91, Theta Series in Advanced Mathematics, vol. 10. Theta, Bucharest (2008)

  22. 22.

    Longo, R., Martinetti, P., Rehren, K.H.: Geometric modular action for disjoint intervals and boundary conformal field theory. Rev. Math. Phys. 22, 331–354 (2010)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Longo, R., Xu, F.: Relative entropy in CFT. Adv. Math. 337, 139–170 (2018)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Morinelli, V., Tanimoto, Y., Weiner, M.: Conformal covariance and the split property. Commun. Math. Phys. 357, 379–406 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    Narnhofer, H.: Entanglement, split and nuclearity in quantum field theory. Rep. Math. Phys. 50, 111–123 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  26. 26.

    Ohya, M., Petz, D.: Quantum Entropy and Its Use. Texts and Monographs in Physics. Springer, Berlin (1993)

    Google Scholar 

  27. 27.

    Otani, Y., Tanimoto, Y.: Towards entanglement entropy with UV cutoff in conformal nets. Ann. H. Poincaré 19, 1817–1842 (2018)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Peller, V.: Hankel Operators and Their Applications. Springer Monographs in Mathematics. Springer, New York (2003). ISBN 978-0-387-21681-2

  29. 29.

    Rehren, K.H., Tedesco, G.: Multilocal fermionization. Lett. Math. Phys. 103, 19–36 (2013)

    ADS  MathSciNet  Article  Google Scholar 

  30. 30.

    Rieffel, M., Van Daele, A.: A bounded operator approach to Tomita–Takesaki theory. Pac. J. Math. 69, 187–221 (1977)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Wassermann, A.: Operator algebras and conformal field theory III. Invent. Math. 133, 467–538 (1998)

    ADS  MathSciNet  Article  Google Scholar 

  32. 32.

    Witten, E.: APS medal for exceptional achievement in research: invited article on entanglement properties of quantum field theory. Rev. Mod. Phys. 90, 045003 (2018)

    ADS  Article  Google Scholar 

  33. 33.

    Xu, F.: Strong additivity and conformal nets. Pac. J. Math. 221, 167–199 (2005)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

Our collaboration took place, in particular, during the CERN workshop on “Advances in Quantum Field Theory” in March–April 2019 and the program “Operator Algebras and Quantum Physics” at the Simons Center for Geometry and Physics at Stony Brook in June 2019. We are grateful to both institutions for the invitations. R.L. acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Roberto Longo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

R. Longo: Supported by the ERC Advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models”, MIUR FARE R16X5RB55W QUEST-NET and GNAMPA-INdAM

F. Xu: Supported in part by NSF Grant DMS-1764157.

Communicated by Y. Kawahigashi

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Longo, R., Xu, F. Von Neumann Entropy in QFT. Commun. Math. Phys. 381, 1031–1054 (2021). https://doi.org/10.1007/s00220-020-03702-7

Download citation