• Stefan Hollands
  • Ko SandersEmail author
Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 34)


Entanglement measures quantify the amount of entanglement between parts of a system, but a considerable part of the literature in Quantum Information Theory has focussed on quantum systems with finitely many degrees of freedom. In this volume, we will focus on the question whether qualitatively new features can arise due to the presence of infinitely many degrees of freedom.


  1. 1.
    M. Bell, K. Gottfried, M. Veltman, John S. Bell on the Foundations of Quantum Mechanics (World Scientific Publishing, Singapore, 2001)CrossRefGoogle Scholar
  2. 2.
    J.S. Bell, On the Einstein Podolsky Rosen paradox. Physics 1, 195 (1964)CrossRefMathSciNetGoogle Scholar
  3. 3.
    M.B. Plenio, S. Virmani, An introduction to entanglement measures. Quant. Inf. Comput. 7, 1 (2007)zbMATHMathSciNetGoogle Scholar
  4. 4.
    M.J. Donald, M. Horodecki, O. Rudolph, The uniqueness theorem for entanglement measures. J. Math. Phys. 43, 4252 (2002)ADSCrossRefMathSciNetGoogle Scholar
  5. 5.
    V. Vedral, M.B. Plenio, Entanglement measures and purification procedures. Phys. Rev. A. 57, 3 (1998)CrossRefGoogle Scholar
  6. 6.
    H.F. Chau, C.-H. Fred Fung, H.-K. Lo, No Superluminal Signaling Implies Unconditionally Secure Bit Commitment, arXiv:1405.0198
  7. 7.
    J. Kaniewski, M. Tomamichel, E. Hänggi, S. Wehner, Secure bit commitment from relativistic constraints. IEEE Trans. Inf. Theory 59, 4687–4699 (2013)CrossRefMathSciNetGoogle Scholar
  8. 8.
    A. Kent, Quantum bit string commitment. Phys. Rev. Lett. 90, 237901 (2003)ADSCrossRefGoogle Scholar
  9. 9.
    R. Haag, D. Kastler, An algebraic approach to quantum field theory. J. Math. Phys. 5, 848 (1964)ADSCrossRefMathSciNetGoogle Scholar
  10. 10.
    R. Haag, Local Quantum Physics: Fields, Particles, Algebras (Springer, Berlin, 1992)CrossRefGoogle Scholar
  11. 11.
    F.J. Murray, J. von Neumann, On rings of operators. Ann. Math. 37(1), 116–229 (1936)CrossRefMathSciNetGoogle Scholar
  12. 12.
    A. Connes, Classification of injective factors. Ann. Math. Second Ser. 104(1), 73–115 (1976)Google Scholar
  13. 13.
    D. Buchholz, K. Fredenhagen, C. D’Antoni, The universal structure of local algebras. Commun. Math. Phys. 111, 123 (1987)ADSCrossRefMathSciNetGoogle Scholar
  14. 14.
    K. Fredenhagen, On the modular structure of local algebras of observables. Commun. Math. Phys. 97, 79–89 (1985)ADSCrossRefMathSciNetGoogle Scholar
  15. 15.
    R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras (Academic Press, New York, I 1983, II 1986)Google Scholar
  16. 16.
    O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics (Springer, I 1987, II 1997)Google Scholar
  17. 17.
    S. Doplicher, R. Longo, Standard and split inclusions of von Neumann algebras. Invent. Math. 75, 493 (1984)ADSCrossRefMathSciNetGoogle Scholar
  18. 18.
    D. Buchholz, E.H. Wichmann, Causal independence and the energy level density of states in local quantum field theory. Commun. Math. Phys. 106, 321 (1986)ADSCrossRefMathSciNetGoogle Scholar
  19. 19.
    J. Yngvason, Localization and engtanglement in relativistic quantum physics, in The Message of Quantum Science, eds. by Ph. Blanchard, J. Fröhlich. Lecture Notes in Physics, vol. 899 (Springer, Berlin, 2015), pp. 325–348Google Scholar
  20. 20.
    E. Witten, notes on some entanglement properties of quantum field theory, arXiv:1803.04993 [hep-th]
  21. 21.
    M. Florig, S.J. Summers, On the statistical independence of algebras of observables. J. Math. Phys. 38, 1318 (1997)ADSCrossRefMathSciNetGoogle Scholar
  22. 22.
    H. Umegaki, Conditional expectations in an operator algebra IV (entropy and information). Kodai Math. Sem. Rep. 14, 59–85 (1962)CrossRefMathSciNetGoogle Scholar
  23. 23.
    H. Araki, Relative entropy for states of von Neumann algebras. Publ. RIMS Kyoto Univ. 11, 809–833 (1976)CrossRefMathSciNetGoogle Scholar
  24. 24.
    H. Araki, Relative entropy for states of von Neumann algebras II. Publ. RIMS Kyoto Univ. 13, 173–192 (1977)CrossRefMathSciNetGoogle Scholar
  25. 25.
    J.C. Baez, T. Fritz, A Bayesian characterization of relative entropy. Theory Appl. Categ. 29, 421–456 (2014)zbMATHMathSciNetGoogle Scholar
  26. 26.
    H. Narnhofer, Entanglement, split, and nuclearity in quantum field theory. Rep. Math. Phys. 50, 111 (2002)ADSCrossRefMathSciNetGoogle Scholar
  27. 27.
    P. Calabrese, J. Cardy, E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II. J. Stat. Mech. 1101, P01021 (2011)MathSciNetGoogle Scholar
  28. 28.
    P. Calabrese, J. Cardy, E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory. J. Stat. Mech. 0911, P11001 (2009)CrossRefMathSciNetGoogle Scholar
  29. 29.
    K. Fredenhagen, K.H. Rehren, B. Schroer, Superselection sectors with braid group statistics and exchange algebras. Commun. Math. Phys. 125, 201–226 (1989)ADSCrossRefMathSciNetGoogle Scholar
  30. 30.
    R. Longo, Index of subfactors and statistics of quantum fields. I. Commun. Math. Phys. 126, 217 (1989)ADSCrossRefMathSciNetGoogle Scholar
  31. 31.
    R. Longo, Index of subfactors and statistics of quantum fields. 2: correspondences, braid group statistics and Jones polynomial. Commun. Math. Phys. 130, 285 (1990)ADSCrossRefMathSciNetGoogle Scholar
  32. 32.
    R.M. Wald, Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics (Chicago University Press, Chicago, 1994)zbMATHGoogle Scholar
  33. 33.
    D. Buchholz, C. D’Antoni, R. Longo, Nuclear maps and modular structures. 1. General properties. J. Funct. Anal. 88, 223 (1990)CrossRefMathSciNetGoogle Scholar
  34. 34.
    C. D’Antoni, S. Hollands, Nuclearity, local quasiequivalence and split property for Dirac quantum fields in curved space-time. Commun. Math. Phys. 261, 133 (2006)ADSCrossRefGoogle Scholar
  35. 35.
    J.J. Bisognano, E.H. Wichmann, On the duality condition for quantum fields. J. Math. Phys. 17, 303 (1976)ADSCrossRefMathSciNetGoogle Scholar
  36. 36.
    P.D. Hislop, R. Longo, Modular structure of the local algebras associated with the free massless scalar field theory. Commun. Math. Phys. 84, 71 (1982)ADSCrossRefMathSciNetGoogle Scholar
  37. 37.
    D. Buchholz, C. D’Antoni, R. Longo, Nuclearity and thermal states in conformal field theory. Commun. Math. Phys. 270, 267–293 (2007)ADSCrossRefMathSciNetGoogle Scholar
  38. 38.
    G. Lechner, K. Sanders, Modular nuclearity: a generally covariant perspective. Axioms 5, 5 (2016)CrossRefGoogle Scholar
  39. 39.
    Y. Otani, Y. Tanimoto, Towards entanglement entropy with UV cutoff in conformal nets. Ann. Henri Poincaré 19(6), 1817–1842 (2018)ADSCrossRefMathSciNetGoogle Scholar
  40. 40.
    S. Doplicher, R. Haag, J.E. Roberts, Local observables and particle statistics. 1. Commun. Math. Phys. 23, 199 (1971)ADSCrossRefMathSciNetGoogle Scholar
  41. 41.
    S. Doplicher, R. Haag, J.E. Roberts, Local observables and particle statistics. 2. Commun. Math. Phys. 35, 49 (1974)ADSCrossRefMathSciNetGoogle Scholar
  42. 42.
    S. Doplicher, J.E. Roberts, Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys. 131, 51 (1990)ADSCrossRefMathSciNetGoogle Scholar
  43. 43.
    S.N. Solodukhin, Entanglement entropy of black holes. Living Rev. Rel. 14, 8 (2011)CrossRefGoogle Scholar
  44. 44.
    T. Nishioka, S. Ryu, T. Takayanagi, Holographic entanglement entropy: an overview. J. Phys. A 42, 504008 (2009)CrossRefMathSciNetGoogle Scholar
  45. 45.
    P. Calabrese, J.L. Cardy, Entanglement entropy and quantum field theory. J. Stat. Mech. 0406, P06002 (2004)zbMATHGoogle Scholar
  46. 46.
    P. Calabrese, J. Cardy, Entanglement entropy and conformal field theory. J. Phys. A 42, 504005 (2009)CrossRefMathSciNetGoogle Scholar
  47. 47.
    L. Bombelli, R.K. Koul, J. Lee, R.D. Sorkin, A quantum source of entropy for black holes. Phys. Rev. D 34, 373–383 (1986)ADSCrossRefMathSciNetGoogle Scholar
  48. 48.
    M. Srednicki, Entropy and area. Phys. Rev. Lett. 71, 666–669 (1993)ADSCrossRefMathSciNetGoogle Scholar
  49. 49.
    L. Susskind, Some speculations about black hole entropy in string theory (1993), arXiv:hep-th/9309145 [hep-th]
  50. 50.
    D. Marolf, A.C. Wall, State-dependent divergences in the entanglement entropy. J. High Energy Phys. 1610, 109 (2016)ADSCrossRefMathSciNetGoogle Scholar
  51. 51.
    H. Casini, M. Huerta, A finite entanglement entropy and the c-theorem. Phys. Lett. B 600, 142–150 (2004)ADSCrossRefMathSciNetGoogle Scholar
  52. 52.
    H. Casini, M. Huerta, A c-theorem for the entanglement entropy. J. Phys. A 40, 7031–7036 (2007)ADSCrossRefMathSciNetGoogle Scholar
  53. 53.
    J. Cardy, E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory. J. Stat. Mech. 1612, 123103 (2016)CrossRefMathSciNetGoogle Scholar
  54. 54.
    M. Headrick, V.E. Hubeny, A. Lawrence, M. Rangamani, Causality and holographic entanglement entropy. J. High Energy Phys. 1412, 162 (2014)ADSCrossRefGoogle Scholar
  55. 55.
    S. Ryu, T. Takayanagi, Aspects of holographic entanglement entropy. J. High Energy Phys. 0608, 045 (2006)ADSCrossRefMathSciNetGoogle Scholar
  56. 56.
    M. Rangamani, T. Takayanagi, Holographic Entanglement Entropy. Springer Lecture Notes in Physics (2017)Google Scholar
  57. 57.
    J. de Boer, M.P. Heller, R.C. Myers, Y. Neiman, Holographic de sitter geometry from entanglement in conformal field theory. Phys. Rev. Lett. 116, 061602 (2016)ADSCrossRefGoogle Scholar
  58. 58.
    J. de Boer, F.M. Haehl, M.P. Heller, R.C. Myers, Entanglement, holography and causal diamonds. J. High Energy Phys. 1608, 162 (2016)ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsUniversity of LeipzigLeipzigGermany
  2. 2.School of Mathematical SciencesDublin City UniversityDublinIreland

Personalised recommendations