  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 $n \ge 3$.
