  One of the main goals of any resource theory such as entanglement, quantum
thermodynamics, quantum coherence, and asymmetry, is to find necessary and
sufficient conditions (NSC) that determine whether one resource can be
converted to another by the set of free operations. Here we find such NSC for a
large class of quantum resource theories which we call affine resource theories
(ART). ARTs include the resource theories of athermality, asymmetry, and
coherence, but not entanglement. Remarkably, the NSC can be expressed as a
family of inequalities between resource monotones (quantifiers) that are given
in terms of the conditional min entropy. The set of free operations is taken to
be (1) the maximal set (i.e. consists of all resource non-generating (RNG)
quantum channels) or (2) the self-dual set of free operations (i.e. consists of
all RNG maps for which the dual map is also RNG). As an example, we apply our
results to quantum thermodynamics with Gibbs preserving operations, and several
other ARTs. Finally, we discuss the applications of these results to resource
theories that are not affine, and along the way, provide the NSC that a quantum
resource theory consists of a resource destroying map.
