We study the sample complexity of the classical shadows task: what is the fewest number of copies of an unknown state you need to measure to predict expected values with respect to some class of observables? Large joint measurements are likely required in order to minimize sample complexity, but previous joint measurement protocols only work when the unknown state is pure. We present the first joint measurement protocol for classical shadows whose sample complexity scales with the rank of the unknown state. In particular we prove $\mathcal O(\sqrt{rB}/\epsilon^2)$ samples suffice, where $r$ is the rank of the state, $B$ is a bound on the squared Frobenius norm of the observables, and $\epsilon$ is the target accuracy. In the low-rank regime, this is a nearly quadratic advantage over traditional approaches that use single-copy measurements. We present several intermediate results that may be of independent interest: a solution to a new formulation of classical shadows that captures functions of non-identical input states; a generalization of a ``nice'' Schur basis used for optimal qubit purification and quantum majority vote; and a measurement strategy that allows us to use local symmetries in the Schur basis to avoid intractable Weingarten calculations in the analysis.

证明了一种经典阴影的联合测量协议其样本复杂度与状态秩成正比，并且在低秩情况下具有几乎二次优势。

Schur基下局部对称性改进的经典阴影