Hoeffding's formulation and solution to the universal hypothesis testing (UHT) problem had a profound impact on many subsequent works dealing with asymmetric hypotheses. In this work, we introduce a quantum universal hypothesis testing framework that serves as a quantum analog to Hoeffding's UHT. Motivated by Hoeffding's approach, which estimates the empirical distribution and uses it to construct the test statistic, we employ quantum state tomography to reconstruct the unknown state prior to forming the test statistic. Leveraging the concentration properties of quantum state tomography, we establish the exponential consistency of the proposed test: the type II error probability decays exponentially quickly, with the exponent determined by the trace distance between the true state and the nominal state.

翻译：Hoeffding对通用假设检验（UHT）问题的提出与解决，对后续许多处理非对称假设的研究产生了深远影响。本文引入了一种量子通用假设检验框架，作为Hoeffding UHT的量子类比。受Hoeffding方法的启发——该方法通过估计经验分布来构建检验统计量——我们采用量子态层析技术，在构建检验统计量之前重构未知量子态。利用量子态层析的集中性性质，我们证明了所提检验的指数一致性：第二类错误概率以指数速率衰减，其指数由真实量子态与名义量子态之间的迹距离决定。