AI 中文总结
本研究移除了纯态Uhlmann保真度估计需预知哪个态为纯态的先验要求，通过特化改进型算法Uhlmann变换，在仅知两态有一纯态的条件下实现了最优保真度估计。
AI 中文摘要
Uhlmann保真度${\rm F}(ρ_0,ρ_1) = {\rm tr}|\sqrt{ρ_0}\sqrt{ρ_1}|$是量子信息理论中用于量化两个量子态之间接近程度的最基本量之一。要以加性误差$\varepsilon$估计Uhlmann保真度，需要一定数量的态副本，或对其态制备电路的查询，该数量至少线性依赖于$ρ_0$和$ρ_1$中较小的秩。因此，当其中任意一个态为纯态时，这种秩依赖性就会消失，此时查询复杂度和样本复杂度仅以$1/\varepsilon$的多项式形式增长。然而，由Fang和Wang（ESA 2025）提出的已知${\rm F}(ρ,|ψ\rangle\\!\langleψ|)$最优估计器需要预先知道哪个态是纯态。\n在本研究中，我们移除了这一在数学上不必要的先验知识要求，并在仅保证两个态中有一个是纯态、且无需知道具体是哪一个的前提下，建立了${\rm F}(ρ, |ψ\rangle\\!\langleψ|)$的最优估计器。我们的估计器是通过将Utsumi、Nakata、Wang和Takagi（2025）提出的改进型算法Uhlmann变换（algorithmic Uhlmann transform）特化到单态为纯态的情况得到的。在该设定下，可通过以下方式恢复Uhlmann保真度：将${\rm tr}_{\sf A}(|ψ_0\rangle\\!\langleψ_1|)$（或其逆）的酉扩张应用于寄存器${\sf A}$和${\sf R}$上的纯化态$|ψ_1\rangle$（或$|ψ_0\rangle$）的参考寄存器${\sf R}$，分别估计两种情况下对应的平方根振幅，再取两个估计结果的最大值。
英文摘要
The Uhlmann fidelity ${\rm F}(ρ_0,ρ_1) = {\rm tr}|\sqrt{ρ_0}\sqrt{ρ_1}|$ is one of the most fundamental quantities in quantum information theory for quantifying the closeness between two quantum states. Estimating the Uhlmann fidelity to within additive error $\varepsilon$ requires a number of copies of the states, or queries to their state-preparation circuits, that depends at least linearly on the smaller of the ranks of $ρ_0$ and $ρ_1$. Consequently, this rank dependence disappears when either state is pure, in which case the query and sample complexities depend only polynomially on $1/\varepsilon$. However, the known optimal estimator for ${\rm F}(ρ,|ψ\rangle\!\langleψ|)$ due to Fang and Wang (ESA 2025) requires prior knowledge of which state is pure. In this work, we remove this mathematically unnecessary prior-knowledge requirement and establish an optimal estimator for ${\rm F}(ρ, |ψ\rangle\!\langleψ|)$ under the sole promise that one of the two states is pure, without knowing which one. Our estimator is obtained by specializing the refined algorithmic Uhlmann transform of Utsumi, Nakata, Wang, and Takagi (2025) to the case where one state is pure. In this setting, the Uhlmann fidelity can be recovered as follows: apply a unitary dilation of ${\rm tr}_{\sf A}(|ψ_0\rangle\!\langleψ_1|)$ (or its inverse) to the reference register $\sf R$ of the purification $|ψ_1\rangle$ (or $|ψ_0\rangle$) on the registers $\sf A$ and $\sf R$, estimate the corresponding square-root amplitude in each case, and take the maximum of the resulting two estimates.
Comments11 pages, 1 algorithm, 1 table, 3 figures
