  We show that the generalization of the relative entropy of a resource from
states to channels is not unique, and there are at least six such
generalizations. We then show that two of these generalizations are
asymptotically continuous, satisfy a version of the asymptotic equipartition
property, and their regularizations appear in the power exponent of channel
versions of the quantum Stein's Lemma. To obtain our results, we use a new type
of "smoothing" that can be applied to functions of channels (with no state
analog). We call it "liberal smoothing" as it allows for more spread in the
optimization. Along the way, we show that the diamond norm can be expressed as
a D_max distance to the set of quantum channels, and prove a variety of
properties of all six generalizations of the relative entropy of a resource.
