Abstract
Pairs of states, or “boxes” are the basic objects in the resource theory of asymmetric distinguishability (Wang and Wilde in Phys Rev Res 1(3):033170, 2019. https://doi.org/10.1103/PhysRevResearch.1.033170), where free operations are arbitrary quantum channels that are applied to both states. From this point of view, hypothesis testing is seen as a process by which a standard form of distinguishability is distilled. Motivated by the more general problem of quantum state discrimination, we consider boxes of a fixed finite number of states and study an extension of the relative submajorization preorder to such objects. In this relation, a tuple of positive operators is greater than another if there is a completely positive trace nonincreasing map under which the image of the first tuple satisfies certain semidefinite constraints relative to the other one. This preorder characterizes error probabilities in the case of testing a composite null hypothesis against a simple alternative hypothesis, as well as certain error probabilities in state discrimination. We present a sufficient condition for the existence of catalytic transformations between boxes, and a characterization of an associated asymptotic preorder, both expressed in terms of sandwiched Rényi divergences. This characterization of the asymptotic preorder directly shows that the strong converse exponent for a composite null hypothesis is equal to the maximum of the corresponding exponents for the pairwise simple hypothesis testing tasks.
Highlights
Resource theories provide a unique viewpoint within numerous areas in quantum information theory and physics, such as entanglement theory and quantum thermodynamics [8]
Building upon the work of Matsumoto [15], Wang and Wilde [25] carried out a systematic development of the resource-theoretic approach to hypothesis testing in the form of a resource theory of asymmetric distinguishability
The task of hypothesis testing is interpreted as distillation of standard pairs, “bits of asymmetric distinguishability,” the quantum min- and max-divergences [10] as well as the quantum relative entropy [14,19] emerge as the distillable distinguishability and distinguishability costs in single-shot and asymptotic settings
Summary
Resource theories provide a unique viewpoint within numerous areas in quantum information theory and physics, such as entanglement theory and quantum thermodynamics [8]. The proof of our result on asymptotic relative submajorization uses some of the ideas from [20], where boxes of pairs are studied, but deviates substantially from it in two key steps (in addition to the more obvious differences in the classification of real-valued monotones and the tropical ones that were not considered there). The first difference is that in the case of pairs of states it was possible to find a set of multipliers (the one-dimensional pairs) with the property that every pair can be multiplied by a suitable element in such a way that the product is bounded from above and from below with respect to the natural numbers This in turn made it possible to use [24, Theorem 2], showing that monotone semiring homomorphisms characterize the asymptotic preorder. With multiple states such a set does not exist, and we must take a different route, effectively applying [24, Corollary 1] to the semifield of fractions
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