We study the error exponents in quantum hypothesis testing between two sets of quantum states, extending the analysis beyond the independent and identically distributed case to encompass composite correlated hypotheses. In particular, we introduce and compare two natural extensions of the quantum Hoeffding divergence and anti-divergence to sets of quantum states, establishing their equivalence or quantitative relations. In the error exponent regime, we generalize the quantum Hoeffding bound to stable sequences of convex, compact sets of quantum states, demonstrating that the optimal Type-I error exponent, under an exponential constraint on the Type-II error, is precisely characterized by the regularized quantum Hoeffding divergence between the sets. In the strong converse exponent regime, we provide a general lower bound on the exponent in terms of the regularized quantum Hoeffding anti-divergence and a matching upper bound when the null hypothesis is a singleton. The generality of these results enables applications in various contexts, including (i) refining the generalized quantum Stein's lemma by [Fang, Fawzi & Fawzi, 2024]; (ii) exhibiting counterexamples to the continuity of the regularized Petz Rényi divergence and Hoeffding divergence; (iii) obtaining error exponents for adversarial channel discrimination and resource detection problems.

翻译：我们研究了在两个量子态集合之间进行量子假设检验的误差指数，将分析从独立同分布情形扩展到包含复合相关假设。特别地，我们引入并比较了量子Hoeffding散度和反散度到量子态集合的两种自然推广，确立了它们的等价性或定量关系。在误差指数体系中，我们将量子Hoeffding界推广到稳定序列的凸紧量子态集合，证明在第二类误差的指数约束下，最优的第一类误差指数恰好由集合间的正则化量子Hoeffding散度刻画。在强逆指数体系中，我们基于正则化量子Hoeffding反散度给出了指数的一般下界，并在零假设为单点集时给出了匹配的上界。这些结果的普适性使其可应用于多种场景，包括：（i）改进[Fang, Fawzi & Fawzi, 2024]的广义量子Stein引理；（ii）展示正则化Petz Rényi散度和Hoeffding散度不连续性的反例；（iii）获得对抗性信道辨别和资源检测问题的误差指数。