Quantum Information Processing

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

Complementarity and Additivity for Covariant Channels

  • N. Datta
  • M. Fukuda
  • A. S. Holevo

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.


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


03.67.HK 03.67.−a 


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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

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