The semiring of dichotomies and asymptotic relative submajorization

Abstract: We study quantum dichotomies and the resource theory of asymmetric distinguishability using a generalization of Strassen's theorem on preordered semirings. We find that an asymptotic variant of relative submajorization, defined on unnormalized dichotomies, is characterized by real-valued monotones that are multiplicative under the tensor product and additive under the direct sum. These strong constraints allow us to classify and explicitly describe all such monotones, leading to a rate formula expressed as an optimization involving sandwiched R\'enyi divergences. As an application we give a new derivation of the strong converse error exponent in quantum hypothesis testing.