Operational characterization of weight-based resource quantifiers via exclusion tasks in general probabilistic theories

The formalism of general probabilistic theories (GPTs), which include classical probability theory and quantum mechanics, has provided us with physical theories beyond quantum mechanics. In this work, we focus on studying the connection between several classes of exclusion tasks and core problems of resource theories-weight-based resource quantifiers and resource manipulations in any GPT. First, we introduce two resourceful quantifiers called the weight of state (WOS) and the weight of measurement (WOM) in any GPT; then, we show that the WOS accurately quantifies the best advantage that a given resource state offers over resourceless states in all channel exclusion tasks. Meanwhile, a similar conclusion can be drawn for the WOM. Second, we introduce the weight-generating power of a channel (WGPC) in any GPT, based on which the resource content of a nonfree channel can be quantified by understanding the number of resources produced by it. It is proven that the WGPC can be considered as the best advantage provided by a given nonfree channel when considering a class of free-state exclusion tasks. In the context of quantum mechanics, we show that the best advantage that a given resource channel provides over resourceless channels in a class of entanglement-assisted state exclusion tasks can be accurately quantified by the weight of channel (WOC). In addition, we introduce the maximum WOC ensemble (MWCE) and find that the MWCE admits an operational interpretation as the best advantage that a given resource channel ensemble provides over free channel ensembles in a class of specific free-channel exclusion tasks. Finally, we show that several classes of channel and state exclusion tasks can constitute complete sets of monotones, completely describing the transformations of states and measurements in any GPT, respectively.

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