‎The ‎resource ‎theory‎ ‎of coherence for ‎quantum ‎‎‎channels‎

We show the coherence of a quantum channel is equivalent to ‎the ‎quantum ‎coherence ‎of‎ its Choi state. We use the Choi-Jamiolkowski isomorphism, within the framework of ‏‎the‎ resource theory, to define the relative entropy of coherence of quantum channels. Here, the coherence breaking channels are introduced as free operations and their corresponding Choi states as free states. We also show the coherence of quantum channel can be divided into‎ the coherence of open ‎system‎ and‎ the correlated coherence of channel where ‎one can ‎prove the former is zero for both quantum unital channels and quantum incoherent channels. In addition, we demonstrate the coherence of channel is decreasing for any divisible quantum channel provided that the coherence of open system is zero‎. ‎Moreover, a redistribution of the coherence of channel is provided based on the quantum discord of Choi state and we claim the coherence of quantum channels can be considered as the quantumness of channels.