  In this work, we study two different approaches to defining the entropy of a
quantum channel. One of these is based on the von Neumann entropy of the
corresponding Choi-Jamio{\l}kowski state. The second one is based on the
relative entropy of the output of the extended channel relative to the output
of the extended completely depolarizing channel. This entropy then needs to be
optimized over all possible input states. Our results first show that the
former entropy provides an upper bound on the latter. Next, we show that for
unital qubit channels, this bound is saturated. Finally, we conjecture and
provide numerical intuitions that the bound can also be saturated for random
channels as their dimension tends to infinity.
