  Based on the resource theory for quantifying the coherence of quantum
channels, we introduce a new coherence quantifier for quantum channels via
maximum relative entropy. We prove that the maximum relative entropy for
coherence of quantum channels is directly related to the maximally coherent
channels under a particular class of superoperations, which results in an
operational interpretation of the maximum relative entropy for coherence of
quantum channels. We also introduce the conception of sub-superchannels and
sub-superchannel discrimination. For any quantum channels, we show that the
advantage of quantum channels in sub-superchannel discrimination can be exactly
characterized by the maximum relative entropy of coherence for quantum
channels. Similar to the maximum relative entropy of coherence for channels,
the robustness of coherence for quantum channels has also been investigated. We
show that the maximum relative entropy of coherence for channels provides new
operational interpretations of robustness of coherence for quantum channels and
illustrates the equivalence of the dephasing-covariant superchannels,
incoherent superchannels, and strictly incoherent superchannels in these two
operational tasks.
