  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.
