We study the composite sequential quantum hypothesis testing (SQHT) problem, where the objective is to distinguish a null quantum state from a 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 strategy simultaneously achieves the Type-I and (worst-case) Type-II error exponents, 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 states.

翻译：研究复合序贯量子假设检验（SQHT）问题，目标是从零假设量子态与一组备选量子态中做出区分。我们提出了一种混合序贯量子概率比检验方法，该方法基于对备选集当前混合估计自适应选择测量，并在混合对数似然比首次跨越阈值时停止。在期望样本量约束下，我们证明所提出的策略能同时实现由零假设与备选集之间最小测量相对熵刻画的I型与（最坏情况）II型误差指数。进一步地，我们建立了匹配的逆命题，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SQHT中实现可忽略的误差概率所需期望样本复杂度至少等同于两个固定态之间序贯检验的样本复杂度。