Skip to main content
SpringerLink
Log in

Entanglement Measures in QFT

  • Chapter
  • First Online:
Entanglement Measures and Their Properties in Quantum Field Theory

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

Abstract

In this chapter we discuss entanglement in a general setting and we review some quantitative measures of entanglement and their properties.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The general case is unclear to us, but one could modify the definition to make the set of separable states norm-closed.

  2. 2.

    Here the tensor product \(M_N(\mathbb {C}) \otimes \mathfrak {A}_2\) is algebraic, with no completion required. To obtain the (unique) \(C^*\)-norm, one may use the fact that there exists a universal representation \(\pi _u:\mathfrak {A}_2\rightarrow \mathfrak {B}(\mathcal {H}_u)\), which is faithful and hence isometric. One may then represent \(M_N(\mathbb {C}) \otimes \mathfrak {A}_2\) on \(\mathbb {C}^N\otimes \mathcal {H}_u\) and use the operator norm.

  3. 3.

    We thank Marc M. Wilde for pointing out this reference to us.

  4. 4.

    Actually, this is all that is needed in order to define \(E_M\).

  5. 5.

    After this preprint appeared, it was pointed out to the authors that a similar proof of Lemma 4 also appears in the unpublished manuscript [27].

  6. 6.

    For an alternative argument, see Lemma 2.9 of [30], which uses a generalization of the Heinz-Löwner theorem [31] to unbounded operators.

  7. 7.

    A function \(f: \mathbb {R}\rightarrow \mathbb {R}\) is called operator monotone if \(f(A) \le f(B)\) whenever two self-adjoint operators A, B on a Hilbert space \(\mathcal {H}\) satisfy \(A \le B\) on the form domain of B. If \(A=B\) on the form domain of B we obtain \(f(A)\le f(B)\). Notice especially the asymmetry in the assumption on the form domain.

  8. 8.

    We may always pass to a new protocol whose rate is arbitrarily close, so that the \(\limsup \) is actually a \(\lim \).

  9. 9.

    We even get the statement for the “asymptotic” relative entropy of entanglement defined by \(E_R^\infty (\omega ) = \lim _{n \rightarrow \infty } \frac{1}{n} E_R(\omega ^{\otimes n})\). Note that the limit exists: Use (e5) and Lemma 12 of [25].

  10. 10.

    Property (e4) for \(E_M\) has been proven only in a restricted sense, and instead of (e3) we have the opposite: concavity \(\overline{\mathrm{(e3)}}\).

References

  1. R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement. Rev. Mod. Phys. 81(2), 865–942 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  2. G. Vidal, Entanglement monotones. J. Mod. Opt. 47, 355–376 (2000)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  4. V. Vedral, M.B. Plenio, M.A. Rippin, P.L. Knight, Quantifying entanglement. Phys. Rev. Lett. 78, 2275–2279 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  5. G. Lindblad, Entropy, information, and quantum measurements. Commun. Math. Phys. 33, 305–322 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  6. W.F. Stinespring, Positive functions on \(C^*\)-algebras. Proc. Ame. Math. Soc. 6, 211–216 (1955)

    MathSciNet  MATH  Google Scholar 

  7. M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press: Cambridge, 2010)

    Google Scholar 

  8. G. Giedke, J.I. Cirac, The characterization of Gaussian operations and Distillation of Gaussian States. Phys. Rev. A 66, 032316 (2002)

    Article  ADS  Google Scholar 

  9. R. Verch, R.F. Werner, Distillability and positivity of partial transposes in general quantum field systems. Rev. Math. Phys. 17, 545 (2005)

    Article  MathSciNet  Google Scholar 

  10. M. Bell, K. Gottfried, M. Veltman, John S. Bell on the foundations of quantum mechanics (World Scientific Publishing, Singapore, 2001)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. S.J. Summers, R. Werner, Maximal violation of Bell’s inequalities is generic in quantum field theory. Commun. Math. Phys. 110, 247–259 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  13. S.J. Summers, R. Werner, Maximal violation of Bell’s inequalities for algebras of observables in tangent spacetime regions. Ann. Inst. H. Poincaré 49, 2 (1988)

    MathSciNet  MATH  Google Scholar 

  14. S.J. Summers, R. Werner, Bell’s inequalities and quantum field theory. I. General setting. J. Math. Phys. 28, 2440–2447 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  15. B.S. Tsirelson, Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4, 93 (1980)

    Google Scholar 

  16. J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Phys. Rev. Lett. 49, 1804 (1969)

    Google Scholar 

  17. N. Gisin, Hidden quantum nonlocality revealed by local filters. Phys. Lett. A 210, 151–156 (1996)

    Article  ADS  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. J.C. Baez, T. Fritz, A Bayesian characterization of relative entropy. Theor. Appl. Categories 29, 421–456 (2014)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. H. Araki, Relative Hamiltonian for faithful normal states of a von Neumann algebra. Publ. RIMS Kyoto Univ. 9, 165–209 (1973)

    Article  MathSciNet  Google Scholar 

  23. M. Ohya, D. Petz, Quantum Entropy and its Use, Theoretical and Mathematical Physics (Springer, Berlin, Heidelberg, 1993)

    Book  Google Scholar 

  24. A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in a interpolation theory. Commun. Math. Phys. 54, 21–32 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  25. M.J. Donald, M. Horodecki, O. Rudolph, Continuity of relative entropy of entanglement. Phys. Lett. A 264, 257–260 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  26. N. Datta, Min- and max. relative entropies and a new entanglement monotone. IEEE Trans. Inf. Theor. 55, 2816–2826 (2009)

    Article  MathSciNet  Google Scholar 

  27. K. Fredenhagen, F.-C. Greipel, Nuclearity index as a measure for entanglement Unpublished preprint

    Google Scholar 

  28. R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras. (Academic Press: New York, I (1983), II (1986))

    Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  31. F. Hansen, The fast track to Löwner’s theorem. Lin. Alg. Appl. 438, 4557–4571 (2013)

    Article  Google Scholar 

  32. E.A. Carlen, Trace inequalities and entropy: An introductory course. https://www.ueltschi.org/AZschool/notes/EricCarlen.pdf

  33. E.M. Rains, Bound on distillable entanglement. Phys. Rev. A 63, 019902 (2000). Phys. Rev. A 60, 179 (1999); Erratum

    Article  ADS  Google Scholar 

  34. M.M. Wolf, G. Giedke, J.I. Cirac, Extremality of Gaussian quantum states. Phys. Rev. Lett. 96, 080502 (2006)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

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

Authors
  1. Stefan Hollands
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ko Sanders
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ko Sanders .

Rights and permissions

Reprints and permissions

Copyright information

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

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Hollands, S., Sanders, K. (2018). Entanglement Measures in QFT. 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_3

Download citation

Publish with us

Policies and ethics

Search

Navigation