Abstract We investigate the majorization procedure for mixed unitary quantum channels. The conditions under which we get the exact estimation are found. The example based upon projective unitary representations of Abelian groups is considered.
We study mixed unitary channels generated by finite subgroups of the group of all unitary operators in a Hilbert space. Based on the majorization theory we introduce techniques allowing to calculate different characteristics of output states of channels. A class of channels has been allocated for which the use of entangled states doesn't give any advantage under taking supremum and infimum for output characteristics of channels. In particular, lp-norms are multiplicative and the minimal entropy is additive with respect to taking tensor products of channels. As an important application of the obtained results the classical capacity of channel is calculated in the evident form. We compare our techniques with the informational characteristics of Boson quantum channels.
A quantum channel is said to be a mixed-unitary channel if it can be expressed as a convex combination of unitary channels. We prove that, given the Choi representation of a quantum channel Φ , it is NP-hard with respect to polynomial-time Turing reductions to determine whether or not Φ is a mixed-unitary channel. This hardness result holds even under the assumption that Φ is not within an inverse-polynomial distance (in the dimension of the space upon which Φ acts) of the boundary of the mixed-unitary channels.