Skip to main content
Log in

On positive maps in quantum information

  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

To SLAVA

Abstract

Using the Grothendieck approach to the tensor product of locally convex spaces, we review a characterization of positive maps as well as Belavkin-Ohya characterization of PPT states. Moreover, within this scheme, a generalization of the idea of Choi matrices for genuine quantum systems will be presented.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. W. B. Arveson, “Subalgebras of C*-Algebras,” Acta Math 123(141), (1969).

    Google Scholar 

  2. V. P. Belavkin, in Works of the 8th All-Union Conference on Coding Theory and Transmission of Information, Kuibyshev, June 1981 (in Russian).

    Google Scholar 

  3. V. P. Belavkin, and M. Ohya, “Quantum Entropy and Information in Discrete Entangled States,” Infinite analysis, quantum probability and related topics 4(137), (2001).

  4. V. P. Belavkin, and M. Ohya, “Quantum Entanglement and Entangled Mutual Entropy,” Proc. Royal. Soc. London A 458(209), (2002).

    Google Scholar 

  5. V. P. Belavkin, and S. Staszewski, “A Radon-Nikodym Theorem for Completely Positive Maps,” Rep. Math. Phys. 24, 49–55 (1986).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. M.-D. Choi, “Positive Linear Maps,” Proc. Sympos. Pure. Math. 38, 583–590 (1982).

    Article  Google Scholar 

  7. M.-D. Choi, “Completely Positive Maps on Complex Matrices,” Lin. Alg. Appl. 10, 285–290 (1975).

    Article  MATH  Google Scholar 

  8. M.-D. Choi, “Some Assorted Inequalities for Positive Linear Maps on C*-Algebras,” J. Operator Th. 4, 271–285 (1980).

    MATH  Google Scholar 

  9. A. Defant, and K. Floret, Tensor norms and operator ideals (North-Holland, 1993).

    MATH  Google Scholar 

  10. J. Diestel, J. H. Fourie, and J. Swart, The metric theory of tensor product. Grothendieck’s Résumé Revisited (A. M. S., 2008).

    Google Scholar 

  11. V. P. Fonf, J. Lindenstrauss, and R. R. Phelps, Infinite Dimensional Convexity (Chapter 15 in Handbook of Geometry of Banach Spaces 1, Edited by William B. Jonson and Joram Lindenstrauss, Elsevier Science, 2001).

  12. A. Grothendieck, “Products Tensoriels Topologiques et Espaces Nuclearies,” Memoirs of the American Mathematical Society 16, Providence, Rhode Island (1955).

  13. A. Holevo, “Entropy Gain and the Choi-Jamiolkowski Correspondence for Infinite-Dimensional Quantum Evolutions,” Theor. Math. Phys. 166, 123–138 (2011).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. A. Holevo, “The Choi-Jamiolkowski Forms on Quantum Gaussian Channels,” J. Math. Phys. 52, 042202 (2011); doi:10.1063/1.3581879.

    Article  ADS  MathSciNet  Google Scholar 

  15. H. Jarchow, Locally Convex Spaces (B. G. Teubner Stuttgart, 1981).

    Book  MATH  Google Scholar 

  16. V. L. Klee, jr, “Extremal Structure of Convex Sets. II,” Math. Zeitschr. 69, 90–104 (1958).

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Lindenstrauss, “On Operators Which Attains Their Norm,” Israel J. Math. 1, 139–148 (1963).

    Article  MATH  MathSciNet  Google Scholar 

  18. R. V. Kadison, and J. R. Ringrose, “Fundamentals of the Theory of Operator Algebras,” II, Advanced theory; Academic Press, INC (1986).

    Google Scholar 

  19. W. A. Majewski, “Separable and Entangled States of Composite Quantum Systems - Rigorous Description,” Open Sys. & Inf. Dyn. 6, 79–86 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  20. W. A. Majewski, “On the Structure of Positive Maps: Finite-Dimensional Case,” J. Math. Phys. 53, 023515 (2012).

    Article  ADS  MathSciNet  Google Scholar 

  21. W. A. Majewski and T. I. Tylec, “On the Structure of Positive Maps. II. Low Dimensional Matrix Algebras,” J. Math. Phys. 54, 073508 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  22. W. A. Majewski, and M. Marciniak, “On a Characterization of Positive Maps,” J. Phys. A 34, 5863–5874 (2001).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. W. A. Majewski, and M. Marciniak, “On the Structure of Positive Maps Between Matrix Algebras,” Banach Center Publications 78, 249–263 (2007).

    Article  MathSciNet  Google Scholar 

  24. W. A. Majewski, T. Matsuoka, and M. Ohya, “Characterization of Partial Positive Transposition States and Measures of Entanglement,” J. Math. Phys. 50, 113509 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  25. M. Raginsky, “Radon-Nikodym Derivatives of Quantum Operations,” J. Math. Phys. 44, 5003 (2003).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. R. A. Ryan, Introduction to Tensor Products of Banach Spaces (Springer Verlag, 2002).

    Book  MATH  Google Scholar 

  27. R. Schatten, Norm Ideals of Completely Continuous Operators (Springer Verlag, 1970).

    Book  MATH  Google Scholar 

  28. E. Størmer, “Positive Linear Maps on Operator Algebras,” Acta Math. 110, 233–278 (1963).

    Article  MathSciNet  Google Scholar 

  29. E. Størmer, “Extension of Positive Maps,” J. Funct. Anal. 66, 235–254 (1986).

    Article  MathSciNet  Google Scholar 

  30. E. Størmer, “Cones of Positive Maps,” Contemporary Mathematics 62, 345–356 (1987).

    Article  Google Scholar 

  31. E. Størmer, Positive Linear Maps on operator algebras (Springer, 2013).

    Book  Google Scholar 

  32. S. Straszewicz, “Über exponierte Punkte abgeschlossener Punktmengen,” Fund. Math. 24, 139–143 (1935).

    Google Scholar 

  33. M. Takesaki, Theory of Operator Algebras I (Springer Verlag, 1979).

    Book  MATH  Google Scholar 

  34. A. Winter, “The Unboundedness of Quantum Mechanical Matrices,” Phys. Rev. 71, 737–739 (1947).

    Article  ADS  Google Scholar 

  35. H. Wielandt, “Üder der unbeschränktheit der operatoren der Quantum Mechanik,” Math. Ann. 121, pp. 21 (1949).

    Article  MATH  MathSciNet  Google Scholar 

  36. G. Wittstock, “Ordered Normed Tensor Products,” Foundation of Quantum Mechanics and Ordered Linear Spaces, Springer Verlag, 67–84 (1974).

    Chapter  Google Scholar 

  37. A. W. Wickstead, Linear Operators Between Partially Ordered Banach Space and Some Related Topics (PhD Thesis, Chelsea College, University of London, 1973).

    Google Scholar 

  38. S. L. Woronowicz, “Positive Maps of Low Dimensional Algebras,” Rep. Math. Phys. 10, 165–183 (1976).

    Article  ADS  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to W. A. Majewski.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Majewski, W.A. On positive maps in quantum information. Russ. J. Math. Phys. 21, 362–372 (2014). https://doi.org/10.1134/S106192081403008X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S106192081403008X

Keywords

Navigation