发表机构
School of Statistics and Data Science, Shanghai University of Finance and Economics（上海财经大学统计与数据科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究为冯·诺依曼熵估计建立了全局与局部极小极大风险界，提出基于对偶阴影和U统计量的截断估计器，并证明在临界半径下局部风险可忽略，适用于投影2设计及近似框架。
AI 中文摘要
我们针对任意固定秩一POVM的独立结果，建立了冯·诺依曼熵估计的下界。通过旋转平均的van Trees论证，当$d\ge C$且$n\ge Cd$时，全局极小极大风险至少为$(d/n)\log^2\{n/(4d)\}$，且无需投影设计假设。我们还刻画了围绕最大混合态、半径为$r$的算子范数球上的风险。我们允许近似二阶矩：在迹零厄米子空间上，测量框架与紧致投影框架的差异不超过$\varepsilon<1$。基于典型对偶阴影和纯度完整U统计量的截断估计器，其风险至多为$d^3r^2/n+d^4/n^2+d^6r^6$。在相同框架控制下的下界给出局部极小极大速率$d^3r^2/n+d^4/n^2$，当$n\ge Cd^2$且$r$位于显式匹配范围内时成立。对于每个固定的$\varepsilon$上界，近似仅改变常数，不改变$d,n,r$的幂次。在临界半径$r=n^{-1/2}$处，当$n\log^2(n/d)\gg d^3$时，局部风险相对于全局风险渐近可忽略。这一分离适用于投影2设计，包括量子比特维度的全局Clifford测量，以及其均匀良态框架近似。在维度四的有限样本实验展示了临界半径基准和非精确框架的影响。
英文摘要
We establish a lower bound for estimating the von Neumann entropy from independent outcomes of any fixed rank-one POVM. A rotation-averaged van Trees argument gives a global minimax risk of at least $(d/n)\log^2\{n/(4d)\}$ when $d\ge C$ and $n\ge Cd$, without a projective-design assumption. We also characterize risk on an operator-norm ball of radius $r$ around the maximally mixed state. We allow an approximate second moment: on the trace-zero Hermitian subspace, the measurement frame may differ by $\varepsilon<1$ from the tight projective frame. A clipped estimator based on canonical dual shadows and the complete U-statistic for purity has risk at most $d^3r^2/n+d^4/n^2+d^6r^6$. Lower bounds under the same frame control yield the local minimax rate $d^3r^2/n+d^4/n^2$ when $n\ge Cd^2$ and $r$ lies in an explicit matching range. For every fixed upper bound on $\varepsilon$, approximation changes only the constants, not the powers of $d,n,r$. At the critical radius $r=n^{-1/2}$, the local risk is asymptotically negligible relative to the global risk when $n\log^2(n/d)\gg d^3$. This separation holds for projective 2-designs, including global Clifford measurements in qubit dimensions, and for their uniformly well-conditioned frame approximations. Finite-sample experiments in dimension four illustrate the critical-radius benchmark and the effect of a nonexact frame.
Comments21 pages, 1 figure, no tables
