Operational interpretation of weight-based resource quantifiers in convex quantum resource theories

We introduce the resource quantifier of weight of resource for convex quantum resource theories of states and measurements with arbitrary resources. We show that it captures the advantage that a resourceful state (measurement) offers over all possible free states (measurements), in the operational task of exclusion of subchannels (states). Furthermore, we introduce information-theoretic quantities related to exclusion for quantum channels, and find a connection between the weight of resource of a measurement, and the exclusion-type information of quantum-to-classical channels. The results found in this article apply to the resource theory of entanglement, in which the weight of resource is known as the best-separable approximation or Lewenstein-Sanpera decomposition, introduced in 1998. Consequently, the results found here provide an operational interpretation to this 21 year-old entanglement quantifier.