Quantum resource theories in the single-shot regime

Abstract: 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.