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：我们研究了对一组未知量子态进行同一性检验的问题，其中假设可通过采样方式访问该集合，且每个量子态以已知概率出现。研究表明，对于包含$N$个$d$维量子态的集合，其样本复杂度为$O(\sqrt{N}d/\epsilon^2)$，并在乘法常数范围内达到匹配的下界。该检验方法通过估计量子态间均方希尔伯特-施密特距离实现，这得益于对Bădescu、O'Donnell和Wright（https://dl.acm.org/doi/10.1145/3313276.3316344）所提出的两个未知态间希尔伯特-施密特距离估计器的适当推广。