Abstract
We develop a unified framework to characterize one-shot transformations of dynamical quantum resources in terms of resource quantifiers, establishing universal conditions for exact and approximate transformations in general resource theories. Our framework encompasses all dynamical resources represented as quantum channels, including those with a specific structure-such as boxes, assemblages, and measurements-thus immediately applying in a vast range of physical settings. For the particularly important manipulation tasks of distillation and dilution, we show that our conditions become necessary and sufficient for broad classes of important theories, enabling an exact characterization of these tasks and establishing a precise connection between operational problems and resource monotones based on entropic divergences. We exemplify our results by considering explicit applications to quantum communication, where we obtain exact expressions for one-shot quantum capacity and simulation cost assisted by no-signaling, separability-preserving, and positive partial transpose-preserving codes; as well as to nonlocality, contextuality, and measurement incompatibility, where we present operational applications of a number of relevant resource measures.
Highlights
Introduction.—Manipulating different resources under physical restrictions underpins many quantum technologies, and the precise understanding of physically realizable transformations is a fundamental question both from theoretical and practical points of view
Bell nonlocality [44,45] as well as quantum contextuality [46,47] have been investigated within formal resource-theoretic settings [48,49,50,51,52], but the specific restrictions on the structure of channels allowed in these settings prevented them from being integrated into most of the previous general quantum resource frameworks
By allowing arbitrary restrictions on the set of channels under consideration, our results apply to settings previously studied in resource theories of channels—e.g., entanglement, coherence, magic, quantum thermodynamics, and quantum communication—and to other dynamical resources such as Bell nonlocality, contextuality, steering [55,56], measurement incompatibility [57], and many others [53,54], offering a very general quantitative description of the fundamental task of resource manipulation
Summary
Transformations with finite error in terms of fundamental resource measures, valid in general resource theories of quantum channels. If the chosen reference set obeys Rmin;affðOÞðT Þ 1⁄4 Rmax;OðT Þ ∀ T ∈ T , it holds that dεOðEÞ 1⁄4 bRεH;affðOÞðEÞcT This establishes a precise characterization of distillable resource in any resource theory for suitable target channels T , and gives an exact operational meaning to the resource measures RεH;O and RεH;affðOÞ in the task of distillation. Quantum capacity [84,85,86] and simulation cost [86,87,88] are important figures of merit to evaluate the operational capability of quantum channels, and their one-shot characterization received considerable attention recently [29,89,90,91,92,93] These tasks are precisely channel distillation and dilution where the reference resource is the identity channel, and the sets of free channels and free superchannels specify the accessible resources for the sender and receiver. Corollaries 2 and 4 provide a complete characterization of one-shot quantum capacity and simulation cost as
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