Quantum Information Processing

, Volume 5, Issue 3, pp 179–207 | Cite as

Complementarity and Additivity for Covariant Channels

Article

This paper contains several new results concerning covariant quantum channels in d ≥ 2 dimensions. The first part, Sec. 3, based on [4], is devoted to unitarily covariant channels, namely depolarizing and transpose-depolarizing channels. The second part, Sec. 4, based on [10], studies Weyl-covariant channels. These results are preceded by Sec. 2 in which we discuss various representations of general completely positive maps and channels. In the first part of the paper we compute complementary channels for depolarizing and transpose-depolarizing channels. This method easily yields minimal Kraus representations from non-minimal ones. We also study properties of the output purity of the tensor product of a channel and its complementary. In the second part, the formalism of discrete noncommutative Fourier transform is developed and applied to the study of Weyl-covariant maps and channels. We then extend a result in [16] concerning a bound for the maximal output 2-norm of a Weyl-covariant channel. A class of maps which attain the bound is introduced, for which the multiplicativity of the maximal output 2-norm is proven. The complementary channels are described which have the same multiplicativity properties as the Weyl-covariant channels.

Keywords

Quantum channel output purity additivity/multiplicativity conjecture complementary channel covariant channel 

Pacs

03.67.HK 03.67.−a 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Alicki and M. Fannes, “Note on multiple additivity of minimal entropy output of extreme SU(d)-covariant channels”, quant-ph/0407033.Google Scholar
  2. 2.
    G. G. Amosov, A. S. Holevo, and R. F. Werner, “On some additivity problems in quantum information theory”, math-ph/0003002.Google Scholar
  3. 3.
    N. Datta, “Multiplicativity of maximal p-norms in Werner-Holevo channels for 1 ≤ p ≤ 2”, quant-ph/0410063.Google Scholar
  4. 4.
    N. Datta and A. S. Holevo, “Complementarity and additivity for depolarizing channels”, quant-ph/0510145.Google Scholar
  5. 5.
    N. Datta, A. S. Holevo, and Yu. M. Suhov, “A quantum channel with additive minimum entropy”, quant-ph/0403072.Google Scholar
  6. 6.
    N. Datta, A. S. Holevo, and Y. M. Suhov, “Additivity for transpose-depolarizing channels”, quant-ph/0412034.Google Scholar
  7. 7.
    I. Devetak and P. Shor, “The capacity of a quantum channel for simultaneous transition of classical and quantum information”, quant-ph/0311131.Google Scholar
  8. 8.
    M. Fannes, B. Haegeman, M. Mosonyi, and D. Vanpeteghem, “Additivity of minimal entropy output for a class of covariant channels”, quant-ph/0410195.Google Scholar
  9. 9.
    Fukuda M. (2005) J. Phys. A 38: L753–L758 quant-ph/0505022.CrossRefMathSciNetMATHADSGoogle Scholar
  10. 10.
    M. Fukuda and A. S. Holevo, “On Weyl-covariant channels”, quant-ph/0510148.Google Scholar
  11. 11.
    V. Giovannetti, S. Lloyd, and M. B. Ruskai, J. Math. Phys. 46, 042105 (2005); quant-ph/0408103.Google Scholar
  12. 12.
    A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, Chapter V (North Holland, 1982).Google Scholar
  13. 13.
    A. S. Holevo, Int. J. Quant. Inform. 3, N1, 41–48 (2005).Google Scholar
  14. 14.
    A. S. Holevo, quant-ph/0509101.Google Scholar
  15. 15.
    C. King, “The capacity of the quantum depolarizing channel”, quant-ph/0204172.Google Scholar
  16. 16.
    C. King, K. Matsumoto, M. Natanson, and M. B. Ruskai, “Properties of conjugate channels with applications to additivity and multiplicativity”, quant-ph/0509126.Google Scholar
  17. 17.
    C. King, M. Nathanson, and M. B. Ruskai, “Multiplicativity properties of entrywise positive maps”, quant-ph/0409181.Google Scholar
  18. 18.
    King C., Ruskai M.B. (2001) IEEE Trans. Info. Theory 47, 192–209 quant-ph/9911079CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    King C., Ruskai M.B. (2004). “Comments on multiplicativity of maximal p-norms when p = 2”. In: Hirota O. (eds). Quantum Information, Statistics, Probability. Rinton Press, Princeton, NJ quant-ph/0401026Google Scholar
  20. 20.
    K. Matsumoto and F. Yura, “Entanglement cost of antisymmetric states and additivity of capacity of some channels”, quant-ph/0306009.Google Scholar
  21. 21.
    A. Rényi, Probability Theory (North Holland, Amsterdam, 1970).Google Scholar
  22. 22.
    Werner R.F., Holevo A.S. (2002). J. Math. Phys. 43, 4353–4357CrossRefMATHADSMathSciNetGoogle Scholar
  23. 23.
    Wolf M., Eisert J. (2005) New J. Phys. 7, 53 quant-ph/0412133CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Statistical Laboratory, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK
  2. 2.Steklov Mathematical InstituteMascowRussia

Personalised recommendations