Given $n$ copies of an unknown quantum state $\rho\in\mathbb{C}^{d\times d}$, quantum state certification is the task of determining whether $\rho=\rho_0$ or $\|\rho-\rho_0\|_1>\varepsilon$, where $\rho_0$ is a known reference state. We study quantum state certification using unentangled quantum measurements, namely measurements which operate only on one copy of $\rho$ at a time. When there is a common source of shared randomness available and the unentangled measurements are chosen based on this randomness, prior work has shown that $\Theta(d^{3/2}/\varepsilon^2)$ copies are necessary and sufficient. This holds even when the measurements are allowed to be chosen adaptively. We consider deterministic measurement schemes (as opposed to randomized) and demonstrate that ${\Theta}(d^2/\varepsilon^2)$ copies are necessary and sufficient for state certification. This shows a separation between algorithms with and without shared randomness. We develop a unified lower bound framework for both fixed and randomized measurements, under the same theoretical framework that relates the hardness of testing to the well-established L\"uders rule. More precisely, we obtain lower bounds for randomized and fixed schemes as a function of the eigenvalues of the L\"uders channel which characterizes one possible post-measurement state transformation.

翻译：给定未知量子态$\rho\in\mathbb{C}^{d\times d}$的$n$个副本，量子态认证的任务是判定$\rho=\rho_0$或$\|\rho-\rho_0\|_1>\varepsilon$，其中$\rho_0$为已知参考态。我们研究使用无纠缠量子测量（即每次仅对$\rho$的一个副本进行操作的测量）的量子态认证。当存在公共共享随机性源且基于该随机性选择无纠缠测量时，先前工作表明需要$\Theta(d^{3/2}/\varepsilon^2)$个副本（必要且充分），即使允许自适应选择测量方案也是如此。我们考虑确定性测量方案（而非随机方案），并证明对于态认证，${\Theta}(d^2/\varepsilon^2)$个副本是必要且充分的。这表明有无共享随机性的算法之间存在分离。我们在同一理论框架下，开发了适用于固定测量和随机测量的统一下界框架，该框架将测试的难度与已确立的Lüders规则相关联。更精确地说，我们获得了随机与固定方案的下界，这些下界作为Lüders通道特征值的函数，该通道刻画了测量后态的一种可能变换。