Complete Positivity of Two Class of Maps Involving Depolarizing and Transpose-Depolarizing Channels

In this note, we mainly study the necessary and sufficient conditions for the complete positivity of generalizations of depolarizing and transpose-depolarizing channels. Specifically, we define [Formula: see text] and [Formula: see text], where [Formula: see text] (the set of all bounded linear operators on the finite-dimensional Hilbert space [Formula: see text] is given and [Formula: see text] is the transpose of [Formula: see text] in a fixed orthonormal basis of [Formula: see text] First, we show that [Formula: see text] is completely positive if and only if [Formula: see text] is a positive map, which is equivalent to [Formula: see text] Moreover, [Formula: see text] is a completely positive map if and only if [Formula: see text] and [Formula: see text] At last, we also get that [Formula: see text] is a completely positive map if and only if [Formula: see text] with [Formula: see text] for all [Formula: see text] where [Formula: see text] are eigenvalues of [Formula: see text].