发表机构
Shanghai University of Finance and Economics（上海财经大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对固定秩一测量下的量子态熵估计，建立了基于框架展宽的极小极大风险下界，证明均匀精度需Ω(d²)次观测，并指出中间机制未解。
AI 中文摘要
我们研究从固定秩一测量的独立结果中估计 $d$ 维量子态的冯·诺依曼熵的问题。设 $\kappa_\nu\in[1,d+1]$ 表示归一化的迹零框架展宽；对于精确的复射影 $2$-设计，$\kappa_\nu=1$。对于任何固定秩一测量且 $n\le c_2d^2/\kappa_\nu$，极小极大均方误差至少为 $c_1\log^2\{d^2/(\kappa_\nu n)\}$。对于有界框架展宽和每个固定的 $0<a<2$，在整个 $n\lesssim d^{2-a}$ 范围内，极小极大风险的数量级为 $\log^2 d$；因此，固定的均匀精度需要 $\Omega(d^2)$ 次观测。现有的层析成像界在有限等权精确设计和 Haar 协变测量的 $O(d^3\log^3 d)$ 次观测下给出有界风险，而中间的极小极大机制仍然开放。这些结果涉及在副本间独立重复的固定测量；自适应和集体协议，以及跨副本重用公共随机性的协议，不在覆盖范围内。
英文摘要
We study estimation of the von Neumann entropy of a $d$-dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let $κ_ν\in[1,d+1]$ denote the normalized trace-zero frame spread; $κ_ν=1$ for exact complex projective $2$-designs. For any fixed rank-one measurement and $n\le c_2d^2/κ_ν$, the minimax mean-squared error is at least $c_1\log^2\{d^2/(κ_νn)\}$. For bounded frame spread and every fixed $0<a<2$, the minimax risk is of order $\log^2 d$ throughout $n\lesssim d^{2-a}$; hence fixed uniform accuracy requires $Ω(d^2)$ observations. Existing tomography bounds give bounded risk from $O(d^3\log^3 d)$ observations for finite equal-weight exact designs and the Haar covariant measurement, while the intermediate minimax regime remains open. The results concern a fixed measurement repeated independently across copies; adaptive and collective protocols, and protocols that reuse common randomness across copies, are not covered.
