Quantum Information Processing, Vol. 5, No. 3, June 2006 (© 2006)
Complementarity and Additivity for Covariant
and A. S. Holevo
Received January 9, 2006; accepted March 29, 2006; Published online May 27, 2006
This paper contains several new results concerning covariant quantum channels
in d ≥ 2 dimensions. The ﬁrst part, Sec. 3, based on , is devoted to unitarily
covariant channels, namely depolarizing and transpose-depolarizing channels. The
second part, Sec. 4, based on , studies Weyl-covariant channels. These results
are preceded by Sec. 2 in which we discuss various representations of general com-
pletely positive maps and channels. In the ﬁrst part of the paper we compute com-
plementary 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  concerning a bound for the maxi-
mal 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 multi-
plicativity properties as the Weyl-covariant channels.
KEY WORDS: Quantum channel; output purity; additivity/multiplicativity
conjecture; complementary channel; covariant channel.
PACS: 03.67.HK; 03.67.−a
This paper contains several new results concerning covariant quantum
channels in d ≥ 2 dimensions. The ﬁrst part, Sec. 3, based on Ref. 4, is
devoted to unitarily covariant channels, namely depolarizing and trans-
pose-depolarizing channels. The second part, Sec. 4, based on Ref. 10,
Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge,
Steklov Mathematical Institute, Mascow, Russia.
To whom correspondence should be addressed. E-mail: firstname.lastname@example.org
1570-0755/06/0600-0179/0 © 2006 Springer Science+Business Media, Inc.