We study the composite sequential quantum hypothesis testing (SQHT) problem, where the objective is to distinguish a null quantum state from a compact, convex set of alternative quantum states. We propose a mixture-sequential quantum probability ratio test that adaptively selects measurements based on the current mixture estimate of the alternative set, and stops upon the first threshold crossing of the mixture log-likelihood ratio. Under an expected sample size constraint, we show that our proposed adaptive strategy simultaneously achieves the optimal Type-I and (worst-case) Type-II error exponents. These exponents are characterized by the minimal measured relative entropies between the null state and the alternative set. We further establish a matching converse, thereby characterizing the optimal error exponent region. Finally, our results show that achieving vanishing error probabilities in composite SQHT requires an expected sample complexity at least as large as that of sequential testing between two fixed quantum states.

翻译：本文研究复合序贯量子假设检验问题，其目标是将一个零假设量子态与一个紧致凸集的备选量子态区分开来。我们提出一种混合序贯量子概率比检验方法，该方法基于备选集当前的混合估计自适应地选择测量，并在混合对数似然比首次超过阈值时停止。在期望样本量约束下，我们证明所提出的自适应策略同时实现了最优的第一类误差指数和（最坏情况下的）第二类误差指数。这些指数由零假设量子态与备选集之间的最小测量相对熵刻画。我们进一步建立了匹配的逆问题，从而刻画了最优误差指数区域。最后，我们的结果表明，在复合序贯量子假设检验中实现渐近消失的误差概率所需的期望样本复杂度至少与两个固定量子态之间的序贯检验相同。