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.

状态对，或“盒子”是非对称可区分性资源理论中的基本对象（Wang and Wilde in Phys Rev Res 1(3):033170, 2019. https://doi.org/10.1103/PhysRevResearch.1.033170） ，其中自由操作是应用于两种状态的任意量子通道。从这个角度来看，假设检验被视为提取可区分性标准形式的过程。受更普遍的量子状态鉴别问题的启发，我们考虑了固定有限数量状态的盒子，并研究了对这些物体的相对次大化预序的扩展。在这种关系中，如果存在完全正的迹非增映射，其中第一个元组的图像相对于另一个元组满足某些半定约束，则正运算符的元组大于另一个。该前序表征了在针对简单替代假设测试复合零假设的情况下的错误概率，以及状态辨别中的某些错误概率。我们提出了盒子之间存在催化转化的充分条件，以及相关渐近前序的表征，两者都以夹心 Rényi 散度表示。