We study the problem of testing identity of a collection of unknown quantum states given sample access to this collection, each state appearing with some known probability. We show that for a collection of $d$-dimensional quantum states of cardinality $N$, the sample complexity is $O(\sqrt{N}d/\epsilon^2)$, {with a matching lower bound, up to a multiplicative constant}. The test is obtained by estimating the mean squared Hilbert-Schmidt distance between the states, thanks to a suitable generalization of the estimator of the Hilbert-Schmidt distance between two unknown states by B\u{a}descu, O'Donnell, and Wright (https://dl.acm.org/doi/10.1145/3313276.3316344).

翻译：我们研究了在给定样本访问权限的情况下，检验一组未知量子态集合的同一性问题，其中每个状态以已知概率出现。研究表明，对于包含 $N$ 个 $d$ 维量子态的集合，样本复杂度为 $O(\sqrt{N}d/\epsilon^2)$，且存在匹配的下界（相差一个乘法常数）。该检验通过估计状态之间的均方希尔伯特-施密特距离实现，其方法基于 Bădescu、O'Donnell 和 Wright (https://dl.acm.org/doi/10.1145/3313276.3316344) 对两个未知量子态间希尔伯特-施密特距离估计器的适当推广。