  Given two pairs of quantum states, a fundamental question in the resource
theory of asymmetric distinguishability is to determine whether there exists a
quantum channel converting one pair to the other. In this work, we reframe this
question in such a way that a catalyst can be used to help perform the
transformation, with the only constraint on the catalyst being that its reduced
state is returned unchanged, so that it can be used again to assist a future
transformation. What we find here, for the special case in which the states in
a given pair are commuting, and thus quasi-classical, is that this catalytic
transformation can be performed if and only if the relative entropy of one pair
of states is larger than that of the other pair. This result endows the
relative entropy with a fundamental operational meaning that goes beyond its
traditional interpretation in the setting of independent and identical
resources. Our finding thus has an immediate application and interpretation in
the resource theory of asymmetric distinguishability, and we expect it to find
application in other domains.
