Channels that do not generate coherence

We define the non-coherence-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 non-coherence-generating channels. Although the relative entropy of coherence is monotonically decreasing under the non-coherence-generating channels, we prove that the coherence of formation may increase under such channels. Interestingly, by building a mathematical relation between the coherence of formation and the entanglement of formation, we show that the coherence of formation of a single-qubit state is never increased by a non-coherence-generating channel. This leads to the superadditivity property for the coherence increasing power of quantum channels, namely, while two channels cannot increase coherence individually, they may increase the quantum coherence of a composed system. Further, we derive the general form of the rank-2 non-coherence-generating qubit channels. Our results contribute to the resource theory of quantum coherence.