This paper investigates symmetric composite binary quantum hypothesis testing (QHT), where the goal is to determine which of two uncertainty sets contains an unknown quantum state. While asymptotic error exponents for this problem are well-studied, the finite-sample regime remains poorly understood. We bridge this gap by characterizing the sample complexity -- the minimum number of state copies required to achieve a target error level. Specifically, we derive lower bounds that generalize the sample complexity of simple QHT and introduce new upper bounds for various uncertainty sets, including of both finite and infinite cardinalities. Notably, our upper and lower bounds match up to universal constants, providing a tight characterization of the sample complexity. Finally, we extend our analysis to the differentially private setting, establishing the sample complexity for privacy-preserving composite QHT.

翻译：本文研究对称复合二元量子假设检验问题，其目标是判断两个不确定集合中哪一个包含未知量子态。尽管该问题的渐近误差指数已得到充分研究，但有限样本机制仍缺乏深入理解。我们通过刻画样本复杂度——即达到目标误差水平所需的最小态副本数量——来弥合这一研究空白。具体而言，我们推导了下界，该下界推广了简单量子假设检验的样本复杂度，并针对包括有限基数与无限基数在内的各类不确定集合提出了新的上界。值得注意的是，我们的上下界在通用常数范围内相匹配，从而为样本复杂度提供了紧致刻画。最后，我们将分析拓展至差分隐私场景，建立了隐私保护复合量子假设检验的样本复杂度。