We establish an operational characterization of general convex resource theories -describing the resource content of not only states, but also measurements and channels, both within quantum mechanics and in general probabilistic theories (GPTs) -in the context of state and channel discrimination. We find that discrimination tasks provide a unified operational description for quantification and manipulation of resources by showing that the family of robustness measures can be understood as the maximum advantage provided by any physical resource in several different discrimination tasks, as well as establishing that such discrimination problems can fully characterize the allowed transformations within the given resource theory.Specifically, we introduce a quantifier of resourcefulness of a measurement in any GPT, the generalized robustness of measurement, and show that it admits an operational interpretation as the maximum advantage that a given measurement provides over resourceless measurements in all state discrimination tasks. In the special case of quantum mechanics, we connect discrimination problems with single-shot information theory by showing that the generalized robustness of any measurement can be alternatively understood as the maximal increase in one-shot accessible information when compared to free measurements. We introduce two different approaches to quantifying the resource content of a physical channel based on the generalized robustness measures, and show that they quantify the maximum advantage that a resourceful channel can provide in several classes of state and channel discrimination tasks. Furthermore, we endow another measure of resourcefulness of states, the standard robustness, with an operational meaning in general GPTs as the exact quantifier of the maximum advantage that a state can provide in binary channel discrimination tasks. Finally, we establish that several classes of channel and state discrimination tasks form complete families of monotones fully characterizing the transformations of states and measurements, respectively, under general classes of free operations. Our results establish a fundamental connection between the operational tasks of discrimination and core concepts of resource theories -the geometric quantification of resources and resource manipulation -valid for all physical theories beyond quantum mechanics with no additional assumptions about the structure of the GPT

We characterize the distillation of quantum coherence in the one-shot setting, that is, the conversion of general quantum states into maximally coherent states under different classes of quantum operations. We show that the maximally incoherent operations (MIO) and the dephasing-covariant incoherent operations (DIO) have the same power in the task of one-shot coherence distillation. We establish that the one-shot distillable coherence under MIO and DIO is efficiently computable with a semidefinite program, which we show to correspond to a quantum hypothesis testing problem. Further, we introduce a family of coherence monotones generalizing the robustness of coherence as well as the modified trace distance of coherence, and show that they admit an operational interpretation in characterizing the fidelity of distillation under different classes of operations. By providing an explicit formula for these quantities for pure states, we show that the one-shot distillable coherence under MIO, DIO, strictly incoherent operations, and incoherent operations is equal for all pure states.

We introduce a framework unifying the mathematical characterisation of different measures of general quantum resources and allowing for a systematic way to define a variety of faithful quantifiers for any given convex quantum resource theory. The approach allows us to describe many commonly used measures such as matrix norm-based quantifiers, robustness measures, convex roof-based measures, and witness-based quantifiers together in a common formalism based on the convex geometry of the underlying sets of resourcefree states. We establish easily verifiable criteria for a measure to possess desirable properties such as faithfulness and strong monotonicity under relevant free operations, and show that many quantifiers obtained in this framework indeed satisfy them for any considered quantum resource. We derive various bounds and relations between the measures, generalising and providing significantly simplified proofs of results found in the resource theories of quantum entanglement and coherence. We also prove that the quantification of resources in this framework simplifies for pure states, allowing us to obtain more easily computable forms of the considered measures, and show that several of them are in fact equal on pure states. Further, we investigate the dual formulation of resource quantifiers, characterising the dual sets of resource witnesses.We present an explicit application of the results to the resource theories of multi-level coherence, entanglement of Schmidt number k, multipartite entanglement, as well as magic states, providing insight into the quantification of the four resources by establishing novel quantitative relations and introducing new quantifiers, such as a measure of entanglement of Schmidt number k which generalises the convex roof-extended negativity, a measure of k-coherence which generalises the 1 norm of coherence, and a hierarchy of norm-based quantifiers of k-partite entanglement generalising the greatest cross norm.

One of the central problems in the study of quantum resource theories is to provide a given resource with an operational meaning, characterizing physical tasks in which the resource can give an explicit advantage over all resourceless states. We show that this can always be accomplished for all convex resource theories. We establish in particular that any resource state enables an advantage in a channel discrimination task, allowing for a strictly greater success probability than any state without the given resource. Furthermore, we find that the generalized robustness measure serves as an exact quantifier for the maximal advantage enabled by the given resource state in a class of subchannel discrimination problems, providing a universal operational interpretation to this fundamental resource quantifier. We also consider a wider range of subchannel discrimination tasks and show that the generalized robustness still serves as the operational advantage quantifier for several well-known theories such as entanglement, coherence, and magic.Introduction. -A rigorous understanding of quantum resources has been one of the ultimate goals in quantum information science. In addition to the apparent theoretical interest, it also has high relevance to burgeoning quantum information technologies such as quantum communication [1,2], quantum cryptography [3,4], and quantum computation [5,6].Quantum resource theories [7] have recently attracted much attention as powerful tools which offer formal frameworks dealing with quantification and manipulation of intrinsic resources associated with quantum systems. One could consider different theories depending on the relevant physical constraints, and indeed various resource theories have been proposed and analyzed, such as entanglement [8,9], coherence [10][11][12], asymmetry [13,14], quantum thermodynamics [15,16], non-Markovianity [17], magic [18,19], and non-Gaussianity [20][21][22]. Although these resource theories provide deeper insights into their specific physical settings,they do not tell us much about how to understand the individual properties and results in a unified fashion. In particular, despite the generality of the resource theoretical framework, only a small number of results reported in the literature are applicable to wide classes of general quantum resource theories [23][24][25][26][27][28][29]. In this work, we add a fundamental item to this list with regard to one of the central questions asked in the study of resource theories: the operational characterization of quantum states and the resources they possess.An essential building block of a resource theory is the set of free states. It is the set of states that are considered "easy to prepare" in that theory, and any state outside of this set is called a resource state. A common and intuitive assumption is that the set of free states should be convex and closed. Convexity reflects a natural attribute in many physical settings, i.e. the fact that losing information about which free state was prepared, hence resulti...