AI 中文总结
研究已知参考态迹距离球内的量子态层析，给出极小极大期望风险与置信半径的紧界，并确定副本复杂度，方法结合加权迹范数见证者与改进的纯化过程。
AI 中文摘要
我们研究当未知态位于已知参考态ρ的半径为R的迹距离球内时的量子态层析问题。对于任意的集体测量，我们确定了极小极大期望风险与置信半径，其精度达到普适常数，且结果在维度、中心谱、半径和样本量上一致成立。记Γ=d(Tr√ρ)^2+d^2R。对于d≥2、0<R≤1以及n≥1个副本，期望风险为Θ(min{R,√(Γ/n)})。在失败概率0<δ≤1/3下，极小极大置信半径为Θ(min{R,√((Γ+log(1/δ))/n)})。对于ε≤c0R（其中c0>0为普适常数），副本复杂度为Θ((Γ+log(1/δ))/ε^2)。对于下界，我们使用加权迹范数见证者来保留平方根扰动族中的所有谱尺度。对于上界，我们对一种修改后的现有基于纯化的过程进行了直接的谱与集中性分析。
英文摘要
We study quantum state tomography when the unknown state lies in a trace-distance ball of radius $R$ around a known reference state $ρ$. For arbitrary collective measurements, we determine the minimax expected risk and confidence radius up to universal constants, uniformly over the dimension, center spectrum, radius, and sample size. Write $Γ=d(\mathrm{Tr}\sqrtρ)^2+d^2R$. For $d\ge2$, $0<R\le1$, and $n\ge1$ copies, the expected risk is $Θ(\min\{R,\sqrt{Γ/n}\})$. At failure probability $0<δ\le1/3$, the minimax confidence radius is \[ Θ\!\left(\min\left\{R,\sqrt{\frac{Γ+\log(1/δ)}{n}}\right\}\right). \] For $\varepsilon\le c_0R$ with a universal $c_0>0$, the copy complexity is $Θ((Γ+\log(1/δ))/\varepsilon^2)$. For the lower bound we use a weighted trace-norm witness to retain all spectral scales in a square-root perturbation family. For the upper bound, we give a direct spectral and concentration analysis of a modified existing purification-based procedure.
