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Quantum resource theories (QRTs) provide a unified framework to analyze quantum properties as resources for achieving advantages in quantum information processing. The generalized robustness and the weight of resource have been gaining increasing attention as useful resource quantifiers. However, the existing analyses of these measures were restricted to the cases where convexity of the set of free states is assumed, and physically motivated resources do not necessarily satisfy this restriction. In this paper, we give characterizations of robustness- and weight-based measures in general QRTs without convexity restriction through two different yet related approaches. On the one hand, we characterize the generalized robustness and the weight of resource by introducing a nonlinear witness. We show a general construction of new witness observables that detect the resourcefulness of a given state from multiple copies of the state and, using these witnesses, we provide operational interpretations of the above resource measures even without any convexity assumption. On the other hand, we find that the generalized robustness and the weight of resource can also be interpreted as the worst-case maximum advantage in variants of channel-discrimination and channel-exclusion tasks, respectively, where the set of free states consists of several convex subsets corresponding to multiple restrictions. We further extend these results to QRTs for quantum channels and quantum instruments. These characterizations show that every quantum resource exhibits an advantage for the corresponding tasks, even in general QRTs without convexity assumption. Thus, we establish the usefulness of robustness-based and weight-based techniques beyond the conventional scope of convex QRTs, leading to a better understanding of the general structure of QRTs.
Kuroiwa et al. (Mon,) studied this question.