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Entanglement Measures and Their Properties in Quantum Field Theory

Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 34))

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Abstract

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.

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Notes

  1. 1.

    This stands for “local operations and classical communications”. In this volume, we will actually use an even broader class.

  2. 2.

    In classical physics, if \(\mu \) is a measure on a product phase space \(X=X_A \times X_B\) which is, say, absolutely continuous relative to the Lebesgue measure, then we can approximate it with arbitrary precision by sums of product measures \(\sum _i \mu _{Ai} \times \mu _{Bi}\) (e.g. on the dense subspace of smooth observables).

  3. 3.

    It may or may not be possible/desirable to also have other properties such as convexity under convex linear combinations.

  4. 4.

    In fact, as shown in [26], entanglement measures that are well-behaved in the type I-setting can become ill-defined for type III, as is the case e.g. for the “entanglement of formation”. [26] has also shown that the entanglement entropy \(E_R(\rho _0)\) behaves well under a “nuclearity condition”, a technique to which we will come back in the body of the volume.

  5. 5.

    In the body of this volume we will distinguish, for technical reasons, the expectation value function of a statistical operator \(\omega ( \ . \ ) = \text {Tr}(\ . \ \rho )\) and the statistical operator \(\rho \) itself.

  6. 6.

    Formula (1.13) below suggests that the upper bound can be improved to \(C_\infty (\delta ) e^{-mr(1-\delta )}\) for each \(\delta >0\).

  7. 7.

    We cannot put \(\kappa \) or \(\delta \) to zero, since the asymptotic bound holds, roughly speaking, when \(1/(\delta \kappa ) \lesssim mr\).

  8. 8.

    It is defined as \(H_n(\rho ) = \frac{1}{1-n} \,\text {ln}\, \text {Tr}\rho ^n \ .\)

References

  1. M. Bell, K. Gottfried, M. Veltman, John S. Bell on the Foundations of Quantum Mechanics (World Scientific Publishing, Singapore, 2001)

    Book  Google Scholar 

  2. J.S. Bell, On the Einstein Podolsky Rosen paradox. Physics 1, 195 (1964)

    Article  MathSciNet  Google Scholar 

  3. M.B. Plenio, S. Virmani, An introduction to entanglement measures. Quant. Inf. Comput. 7, 1 (2007)

    MATH  MathSciNet  Google Scholar 

  4. M.J. Donald, M. Horodecki, O. Rudolph, The uniqueness theorem for entanglement measures. J. Math. Phys. 43, 4252 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  5. V. Vedral, M.B. Plenio, Entanglement measures and purification procedures. Phys. Rev. A. 57, 3 (1998)

    Article  Google Scholar 

  6. H.F. Chau, C.-H. Fred Fung, H.-K. Lo, No Superluminal Signaling Implies Unconditionally Secure Bit Commitment, arXiv:1405.0198

  7. J. Kaniewski, M. Tomamichel, E. Hänggi, S. Wehner, Secure bit commitment from relativistic constraints. IEEE Trans. Inf. Theory 59, 4687–4699 (2013)

    Article  MathSciNet  Google Scholar 

  8. A. Kent, Quantum bit string commitment. Phys. Rev. Lett. 90, 237901 (2003)

    Article  ADS  Google Scholar 

  9. R. Haag, D. Kastler, An algebraic approach to quantum field theory. J. Math. Phys. 5, 848 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  10. R. Haag, Local Quantum Physics: Fields, Particles, Algebras (Springer, Berlin, 1992)

    Book  Google Scholar 

  11. F.J. Murray, J. von Neumann, On rings of operators. Ann. Math. 37(1), 116–229 (1936)

    Article  MathSciNet  Google Scholar 

  12. A. Connes, Classification of injective factors. Ann. Math. Second Ser. 104(1), 73–115 (1976)

    Google Scholar 

  13. D. Buchholz, K. Fredenhagen, C. D’Antoni, The universal structure of local algebras. Commun. Math. Phys. 111, 123 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  14. K. Fredenhagen, On the modular structure of local algebras of observables. Commun. Math. Phys. 97, 79–89 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  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. O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics (Springer, I 1987, II 1997)

    Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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. 325–348

    Google Scholar 

  20. E. Witten, notes on some entanglement properties of quantum field theory, arXiv:1803.04993 [hep-th]

  21. M. Florig, S.J. Summers, On the statistical independence of algebras of observables. J. Math. Phys. 38, 1318 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  22. H. Umegaki, Conditional expectations in an operator algebra IV (entropy and information). Kodai Math. Sem. Rep. 14, 59–85 (1962)

    Article  MathSciNet  Google Scholar 

  23. H. Araki, Relative entropy for states of von Neumann algebras. Publ. RIMS Kyoto Univ. 11, 809–833 (1976)

    Article  MathSciNet  Google Scholar 

  24. H. Araki, Relative entropy for states of von Neumann algebras II. Publ. RIMS Kyoto Univ. 13, 173–192 (1977)

    Article  MathSciNet  Google Scholar 

  25. J.C. Baez, T. Fritz, A Bayesian characterization of relative entropy. Theory Appl. Categ. 29, 421–456 (2014)

    MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  27. P. Calabrese, J. Cardy, E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II. J. Stat. Mech. 1101, P01021 (2011)

    MathSciNet  Google Scholar 

  28. P. Calabrese, J. Cardy, E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory. J. Stat. Mech. 0911, P11001 (2009)

    Article  MathSciNet  Google Scholar 

  29. K. Fredenhagen, K.H. Rehren, B. Schroer, Superselection sectors with braid group statistics and exchange algebras. Commun. Math. Phys. 125, 201–226 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  30. R. Longo, Index of subfactors and statistics of quantum fields. I. Commun. Math. Phys. 126, 217 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  32. R.M. Wald, Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics (Chicago University Press, Chicago, 1994)

    MATH  Google Scholar 

  33. D. Buchholz, C. D’Antoni, R. Longo, Nuclear maps and modular structures. 1. General properties. J. Funct. Anal. 88, 223 (1990)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  35. J.J. Bisognano, E.H. Wichmann, On the duality condition for quantum fields. J. Math. Phys. 17, 303 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  37. D. Buchholz, C. D’Antoni, R. Longo, Nuclearity and thermal states in conformal field theory. Commun. Math. Phys. 270, 267–293 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  38. G. Lechner, K. Sanders, Modular nuclearity: a generally covariant perspective. Axioms 5, 5 (2016)

    Article  Google Scholar 

  39. Y. Otani, Y. Tanimoto, Towards entanglement entropy with UV cutoff in conformal nets. Ann. Henri Poincaré 19(6), 1817–1842 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  40. S. Doplicher, R. Haag, J.E. Roberts, Local observables and particle statistics. 1. Commun. Math. Phys. 23, 199 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  41. S. Doplicher, R. Haag, J.E. Roberts, Local observables and particle statistics. 2. Commun. Math. Phys. 35, 49 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  43. S.N. Solodukhin, Entanglement entropy of black holes. Living Rev. Rel. 14, 8 (2011)

    Article  Google Scholar 

  44. T. Nishioka, S. Ryu, T. Takayanagi, Holographic entanglement entropy: an overview. J. Phys. A 42, 504008 (2009)

    Article  MathSciNet  Google Scholar 

  45. P. Calabrese, J.L. Cardy, Entanglement entropy and quantum field theory. J. Stat. Mech. 0406, P06002 (2004)

    MATH  Google Scholar 

  46. P. Calabrese, J. Cardy, Entanglement entropy and conformal field theory. J. Phys. A 42, 504005 (2009)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  48. M. Srednicki, Entropy and area. Phys. Rev. Lett. 71, 666–669 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  49. L. Susskind, Some speculations about black hole entropy in string theory (1993), arXiv:hep-th/9309145 [hep-th]

  50. D. Marolf, A.C. Wall, State-dependent divergences in the entanglement entropy. J. High Energy Phys. 1610, 109 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  51. H. Casini, M. Huerta, A finite entanglement entropy and the c-theorem. Phys. Lett. B 600, 142–150 (2004)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  53. J. Cardy, E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory. J. Stat. Mech. 1612, 123103 (2016)

    Article  MathSciNet  Google Scholar 

  54. M. Headrick, V.E. Hubeny, A. Lawrence, M. Rangamani, Causality and holographic entanglement entropy. J. High Energy Phys. 1412, 162 (2014)

    Article  ADS  Google Scholar 

  55. S. Ryu, T. Takayanagi, Aspects of holographic entanglement entropy. J. High Energy Phys. 0608, 045 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  56. M. Rangamani, T. Takayanagi, Holographic Entanglement Entropy. Springer Lecture Notes in Physics (2017)

    Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute for Theoretical Physics, University of Leipzig, Leipzig, Sachsen, Germany

    Stefan Hollands

  2. School of Mathematical Sciences, Dublin City University, Dublin, Ireland

    Ko Sanders

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  1. Stefan Hollands
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Hollands, S., Sanders, K. (2018). Introduction. In: Entanglement Measures and Their Properties in Quantum Field Theory. SpringerBriefs in Mathematical Physics, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-319-94902-4_1

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