  We define the coherence non-generating channel as the completely positive
trace-preserving map which does not generate quantum coherence from an
incoherent state. The incoherent operations are the strict subset of the
coherence non-generating channels. We prove that the relative entropy of
coherence is monotonically decreasing under the coherence non-generating
channels, while the coherence of formation may increase under such channels.
Interestingly, by combining mathematically the coherence of formation to the
entanglement of formation, we show that the coherence of formation of a
single-qubit is never increased by a coherence non-generating channel. This
leads to the superadditivity property for the coherence increasing power of
quantum channels, namely, while two channels can not increase coherence
individually, they may increase the quantum coherence of a composed system.
Further, we derive the general form of the rank-2 coherence non-generating
qubit channels. Our results contribute to the resource theory of quantum
coherence.
