  We argue that a fundamental (conjectured) property of memoryless quantum
channels, namely the strong superadditivity, is intimately related to the
decreasing property of the quantum relative entropy. Using the latter we first
give, for a wide class of input states, an estimation of the output entropy for
phase damping channels and some Weyl quantum channels. Then we prove, without
any input restriction, the strong superadditivity for several quantum channels,
including depolarizing quantum channels, quantum-classical channels and quantum
erasure channels.
