How to quantify a dynamical quantum resource

Understand

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.