We define the quantum-incoherent relative entropy of coherence (QI REC) of quantum channels in the framework of the resource theory by using the Choi–Jamiolkowski isomorphism. Coherence-breaking channels are introduced as free operations and their corresponding Choi states as free states. We also show the relationship between the coherence of channel and the quantum discord and find that basis-dependent quantum asymmetric discord can never be more than the QI REC for any quantum channels. Also, we prove the QI REC is decreasing for any divisible quantum-incoherent channel and we also claim it can be considered as the quantumness of quantum channels. Moreover, we demonstrate that for qubit channels, the relative entropy of coherence (REC) can be equivalent to the REC of their corresponding Choi states and the basis-dependent quantum symmetric discord can never exceed the coherence.
