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We develop a framework to extend resource measures from one domain to alarger one. We find that all extensions of resource measures are boundedbetween two quantities that we call the minimal and maximal extensions. Wediscuss various applications of our framework. We show that any relativeentropy (i.e. an additive function on pairs of quantum states that satisfiesthe data processing inequality) must be bounded by the min and max relativeentropies. We prove that the generalized trace distance, the generalizedfidelity, and the purified distance are optimal extensions. And in entanglementtheory we introduce a new technique to extend pure state entanglement measuresto mixed bipartite states.

Optimal Extensions of Resource Measures and their Applications