On majorization of quantum channels by probability distributions

Based on the concept of majorization for the probability distribution, a definition of the majorization of a quantum channel by a probability distribution is introduced. It is shown that the proposed approach makes it possible to solve the problem of taking the extremes of convex functions from the output eigenvalues of mixed unitary channels in the case when summation in the definition of the channel is carried out according to the Heisenberg-Weyl group.