对已知秩-$r$参考态的保真度估计的样本复杂度为$\boldsymbol{\tilde{\theta}}(r^2/\boldsymbol{\theta}^2)$
Fidelity Estimation to a Known Quantum State Is Nearly Quadratic in the Smaller Rank
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中文总结 AI 辅助
该研究解决了未知态与已知秩-$r$参考态的根乌尔曼保真度估计的样本复杂度问题，确定其为$\boldsymbol{\tilde{\theta}}(r^2/\boldsymbol{\theta}^2)$，还得到量子谱估计的近二次下界并明确了该区域样本复杂度的多项式阶。
中文摘要 AI 辅助
我们确定了估计未知态$\rho$与已知秩-$r$参考态$\boldsymbol{\theta}$之间的根乌尔曼保真度$F(\rho,\boldsymbol{\theta})=\boldsymbol{\text{tr}}\boldsymbol{\theta}$的样本复杂度。将$S(r,\boldsymbol{\theta})$记为加性误差$\boldsymbol{\theta}$下的样本复杂度，我们通过在对数因子内闭合先前已知的下界$\boldsymbol{\text{Ω}}(r/\boldsymbol{\theta}^2)$与上界$O(r^2/\boldsymbol{\theta}^2)$之间的差距，解决了Wang提出的开放问题。我们证明，对于所有$0<\boldsymbol{\theta}\boldsymbol{\theta}_0$（其中$\boldsymbol{\theta}_0>0$为通用常数），$S(r,\boldsymbol{\theta})=\boldsymbol{\tilde{\theta}}(r^2/\boldsymbol{\theta}^2)$。当$\boldsymbol{\theta}$在固定$r$维子空间上最大混合且态族与$\boldsymbol{\theta}$不对易时，下界已在$2r$维系统上成立。证明结合了精确谱矩匹配、径向大小偏置双相关Wishart模型和柯西恒等式，将态不可区分性转化为加权随机排列的长周期估计。直和嵌入与二项稀疏化得到最优的$1/\boldsymbol{\theta}^2$依赖关系。我们还证明了常精度下量子谱估计的近二次下界$\boldsymbol{\tilde{\text{Ω}}}(r^2)$，结合近期的上界$O(r^2(\boldsymbol{\text{log}}\boldsymbol{\text{log}}r/\boldsymbol{\text{log}}r)^2)$，确定了该区域样本复杂度的多项式阶并建立了近二次壁垒。
英文摘要
We study the number of copies needed to estimate the root Uhlmann fidelity between an unknown quantum state and a classically known reference, under collective measurements. If the unknown state has rank at most $s$, we give an estimator using $O(s^2/\varepsilon^2)$ copies, uniformly in the ambient dimension and reference rank. The estimator applies a random-purification channel and a covariant pure-state measurement, then rescales the observed amplitude before evaluating a weighted nuclear norm. Combined with the lower bound established independently in our first version and in concurrent work of Wang, this determines the worst-case complexity under rank bounds $r,s$ as $\min\{r,s\}^2/\varepsilon^2$ up to logarithmic factors in the smaller rank. The same estimator gives the upper bound $O(r(\operatorname{tr}\sqrtσ)^2/\varepsilon^2)$ for a rank-$r$ reference $σ$. We combine spectral truncation with lower bounds obtained by embedding hard instances for the maximally mixed reference into spectral subspaces. For spectra $λ_i\propto i^{-α}$ with fixed $1<α\le2$, as $\varepsilon\downarrow0$ with $r\ge C_α\varepsilon^{-2/(α-1)}$, the bounds determine the complexity as $\widetildeΘ_α(\varepsilon^{-4/(α-1)})$. In particular, inverse-square spectra have accuracy exponent four. The lower-bound construction uses exact moment matching and an explicitly computable Schur measure.
发表机构
- Georgia Institute of Technology（佐治亚理工学院）
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