  Pairs of states, or "boxes" are the basic objects in the resource theory of
asymmetric distinguishability (Wang and Wilde, 2019), 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 R\'enyi 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.
