  Quantum addition channels have been recently introduced in the context of
deriving entropic power inequalities for finite dimensional quantum systems. We
prove a reverse entropy power equality which can be used to analytically prove
an inequality conjectured recently for arbitrary dimension and arbitrary
addition weight. We show that the relative entropic difference between the
output of such a quantum additon channel and the corresponding classical
mixture quantitatively captures the amount of coherence present in a quantum
system. This new coherence measure admits an upper bound in terms of the
relative entropy of coherence and is utilized to formulate a state-dependent
uncertainty relation for two observables. Our results may provide deep insights
to the origin of quantum coherence for mixed states that truly come from the
discrepancy between quantum addition and the classical mixture.
