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Quantum resource theories (QRTs) provide a unified framework to analyzequantum properties as resources for achieving advantages in quantum informationprocessing. The generalized robustness and the weight of resource have beengaining increasing attention as useful resource quantifiers. However, theexisting analyses of these measures were restricted to the cases whereconvexity of the set of free states is assumed, and physically motivatedresources do not necessarily satisfy this restriction. In this paper, we givecharacterizations of robustness- and weight-based measures in general QRTswithout convexity restriction through two different yet related approaches. Onthe one hand, we characterize the generalized robustness and the weight ofresource by introducing a non-linear witness. We show a general construction ofa new witness observable that detects the resourcefulness of a given state frommultiple copies of the state and, using this witness, we provide operationalinterpretations of the above resource measures even without any convexityassumption. On the other hand, we find that the generalized robustness and theweight of resource can also be interpreted as the worst-case maximum advantagein variants of channel-discrimination and channel-exclusion tasks,respectively, where the set of free states consists of several convex subsetscorresponding to multiple restrictions. We further extend these results to QRTsfor quantum channels and quantum instruments. These characterizations show thatevery quantum resource exhibits an advantage for the corresponding tasks, evenin general QRTs without convexity assumption. Thus, we establish the usefulnessof robustness-based and weight-based techniques beyond the conventional scopeof convex QRTs, leading to a better understanding of the general structure ofQRTs.

Robustness and weight resource measures without convexity restriction:  Multicopy witness and operational advantage in static and dynamical quantum  resource theories