Generalizations of $2$-dimensional diagonal quantum channels with constant Frobenius norm

We introduce the set of quantum channels with constant Frobenius norm, the set of diagonal channels and the notion of equivalence of one-parameter families of channels. First, we show that all diagonal $2$-dimensional channels with constant Frobenius norm are equivalent. Next, we generalize four one-parameter families of $2$-dimensional diagonal channels with constant Frobenius norm to an arbitrary dimension $n$. Finally, we prove that the generalizations are not equivalent in any dimension $n\ge 3$.