  Quantum resource theories (QRTs) provide a unified theoretical framework for
understanding inherent quantum-mechanical properties that serve as resources in
quantum information processing, but resources motivated by physics may possess
intractable mathematical structure to analyze, such as non-uniqueness of
maximally resourceful states, lack of convexity, and infinite dimension. We
investigate state conversion and resource measures in general QRTs under
minimal assumptions to figure out universal properties of physically motivated
quantum resources that may have such intractable mathematical structure. In the
general setting, we prove the existence of maximally resourceful states in
one-shot state conversion. Also analyzing asymptotic state conversion, we
discover catalytic replication of quantum resources, where a resource state is
infinitely replicable by free operations. In QRTs without assuming uniqueness
of maximally resourceful states, we formulate the tasks of distillation and
formation of quantum resources, and introduce distillable resource and resource
cost based on the distillation and the formation, respectively. Furthermore, we
introduce consistent resource measures that quantify the amount of quantum
resources without contradicting the rate of state conversion even in QRTs with
non-unique maximally resourceful states. Progressing beyond the previous work
showing a uniqueness theorem for additive resource measures, we prove the
corresponding uniqueness inequality for the consistent resource measures; that
is, consistent resource measures of a quantum state take values between the
distillable resource and the resource cost of the state. These formulations and
results establish a foundation of QRTs applicable to mathematically intractable
but physically motivated quantum resources in a unified way.
