We consider the classical shadows task for pure states in the setting of both joint and independent measurements. The task is to measure few copies of an unknown pure state $\rho$ in order to learn a classical description which suffices to later estimate expectation values of observables. Specifically, the goal is to approximate $\mathrm{Tr}(O \rho)$ for any Hermitian observable $O$ to within additive error $\epsilon$ provided $\mathrm{Tr}(O^2)\leq B$ and $\lVert O \rVert = 1$. Our main result applies to the joint measurement setting, where we show $\tilde{\Theta}(\sqrt{B}\epsilon^{-1} + \epsilon^{-2})$ samples of $\rho$ are necessary and sufficient to succeed with high probability. The upper bound is a quadratic improvement on the previous best sample complexity known for this problem. For the lower bound, we see that the bottleneck is not how fast we can learn the state but rather how much any classical description of $\rho$ can be compressed for observable estimation. In the independent measurement setting, we show that $\mathcal O(\sqrt{Bd} \epsilon^{-1} + \epsilon^{-2})$ samples suffice. Notably, this implies that the random Clifford measurements algorithm of Huang, Kueng, and Preskill, which is sample-optimal for mixed states, is not optimal for pure states. Interestingly, our result also uses the same random Clifford measurements but employs a different estimator.

翻译：在设定联合和独立的测量时, 我们考虑纯状态的经典阴影任务。 任务是测量一个未知的纯状态 $\ rho$ 的少量副本, 以便学习一个古典描述, 足以以后估计可见值的预期值。 具体地说, 目标是, 任何赫米提人观测到的美元范围内, 任何埃米提亚人观测到的美元美元到添加误差的美元范围内, $\ epsilon$ (O2)\ leble B$ 和 $\ lVert O\rVert = 1$ 。 我们的主要结果应用到一个联合测量设置, 在那里我们显示 $\ tdelde\ thesetailal deal deal deal rq} (sqrtrt{B\\ lipslon_ 1} +\ epsilonlon% 2} 美元样本是必需的, 并且非常有可能成功。 。 上层是先前为这一问题所知道的精度复杂性复杂性复杂性复杂性的改进。 。 。 我们发现, 瓶内我们无法快速的测量 。