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 $O(\sqrt{N}d/\epsilon^2)$, which is optimal up to a 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：我们研究的是一组未知量子的测试身份问题,这些量子的样本可以进入这一集,每个州都有一定的概率出现。我们显示,对于一组基度为N$的量子集,样本复杂性为$O(sqrt{N}d/\epsilon ⁇ 2),这是最高至常数的最佳办法。通过对海尔伯特-施密特之间平均正方形距离进行估计,通过B\u{a}descu、O'Donnell和赖特(https://dl.acm.org/doi/ 10.10.1145/3313276.3316344)对希尔伯特-施密特之间两个未知州之间距离的测量员进行适当的概括,通过估算海尔伯特-施密特之间平均距离(Hilbert-Schmidt)的距离(Hilbert-Schmidt)来进行测试(https://dl.acam.org/doi/10.1145/3313276.3316344)。