Complementarity and Additivity for Covariant Channels

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.

[1]  Christopher King,et al.  Multiplicativity properties of entrywise positive maps on matrix algebras , 2004 .

[2]  J. Eisert,et al.  Classical information capacity of a class of quantum channels , 2004, quant-ph/0412133.

[3]  Nilanjana Datta,et al.  A quantum channel with additive minimum output entropy , 2004 .

[4]  Christopher King,et al.  Multiplicativity properties of entrywise positive maps , 2004 .

[5]  Alexander Semenovich Holevo,et al.  ADDITIVITY CONJECTURE AND COVARIANT CHANNELS , 2005 .

[6]  Nilanjana Datta.  Multiplicativity of Maximal p-Norms in Werner-Holevo channels for $1 \le p \le 2$ , 2004 .

[7]  R. Werner,et al.  Counterexample to an additivity conjecture for output purity of quantum channels , 2002, quant-ph/0203003.

[8]  M. Fukuda.  Extending additivity from symmetric to asymmetric channels , 2005, quant-ph/0505022.

[9]  P. Shor,et al.  The Capacity of a Quantum Channel for Simultaneous Transmission of Classical and Quantum Information , 2003, quant-ph/0311131.

[10]  Keiji Matsumoto,et al.  Entanglement cost of antisymmetric states and additivity of capacity of some quantum channels , 2003, quant-ph/0306009.

[11]  Christopher King,et al.  Minimal entropy of states emerging from noisy quantum channels , 2001, IEEE Trans. Inf. Theory.

[12]  Christopher King,et al.  Properties of Conjugate Channels with Applications to Additivity and Multiplicativity , 2005 .

[13]  C. King.  The capacity of the quantum depolarizing channel , 2003, IEEE Trans. Inf. Theory.

[14]  O. Hirota,et al.  Quantum Information, Statistics, Probability , 2004 .

[15]  R. Werner,et al.  On Some Additivity Problems in Quantum Information Theory , 2000, math-ph/0003002.

[16]  Nilanjana Datta,et al.  ADDITIVITY FOR TRANSPOSE DEPOLARIZING CHANNELS , 2004 .

[17]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

[18]  Nilanjana Datta,et al.  Complementarity and additivity for depolarizing channels , 2005 .

[19]  A. S. Holevo,et al.  On Weyl-covariant channels , 2008 .

[20]  M. Fannes,et al.  Note on multiple additivity of minimal entropy output of extreme SU(d)-covariant channels , 2004 .

[21]  Seth Lloyd,et al.  Conditions for multiplicativity of maximal ℓp-norms of channels for fixed integer p , 2004 .

[22]  Alfréd Rényi,et al.  Probability Theory , 1970 .

[23]  Christopher King,et al.  Comments on multiplicativity of maximal p -norms when p=2 , 2004 .

[24]  Paolo G. Radaelli,et al.  Orbital ordering in transition-metal spinels , 2005 .

[25]  M. Fannes,et al.  Additivity of minimal entropy output for a class of covariant channels , 2004, quant-ph/0410195.