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，其目标在于判定未知量子态属于两个不确定集合中的哪一个。尽管该问题的渐近误差指数已得到充分研究，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量子态副本数——来弥合这一认知鸿沟。具体而言，我们推导了推广简单量子假设检验样本复杂度的下界，并针对包括有限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互匹配，从而实现了对样本复杂度的紧致刻画。最后，我们将分析扩展至差分隐私场景，建立了隐私保护型复合量子假设检验的样本复杂度理论。