针对总影响的鲁棒量子态认证与不确定性原理
Robust quantum state certification and uncertainty principles for total influence
查看机构详情
- Paul G. Allen School of Computer Science and Engineering, University of Washington（华盛顿大学保罗·G·艾伦计算机科学与工程学）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
该研究提出非自适应单量子比特泡利测量的量子态检验方法，基于布尔函数总影响加权推广的不确定性原理，实现信息论最优的鲁棒量子态认证。
中文摘要 AI 辅助
我们证明，对于除$2^{-\u03a9(n)}$比例的目标态之外的所有情况，非自适应单量子比特泡利测量足以检验未知的$n$量子比特态$\rho$是否与理想目标态$|\u03c8\rangle$的距离为$\u03b5$-近或$O(\u03b5)$-远。该检验使用$O(\u03b5^{-2}\u00b7\log(1/\delta))$份$\rho$副本以达到$1-\u03b5$的置信度，即使在采用任意联合测量的协议中，这也是信息论最优的。主要技术创新是布尔函数总影响的加权推广形式的不确定性原理。作为简单示例，未加权形式表明$\u0046\u006e\u0066[\u0066]+\u0046\u006e\u0066[\widehat{f}] = \u03a9(n)$，这是海森堡不确定性原理的自然超立方体类比（此处$\widehat{\\,\cdot\\,}$表示以$2^{-n/2}$归一化的傅里叶变换）。加权形式将$\u0046\u006e\u0066[\\,\cdot\\,]$和$\u0046\u006e\u0066[\widehat{\\,\cdot\\,}\\,]$推广为与超立方体上某些对偶测度的格劳伯动力学相关的狄利克雷能量。
英文摘要
We show that nonadaptive single-qubit Pauli measurements suffice to test whether an unknown $n$-qubit state $ρ$ is $\varepsilon$-close to or $O(\varepsilon)$-far from an ideal target state $|ψ\rangle$, for all but a $2^{-Ω(n)}$ fraction of target states. The test uses $O(\varepsilon^{-2}\log(1/δ))$ copies of $ρ$ to achieve confidence $1-δ$, which is information-theoretically optimal even among protocols with arbitrary joint measurements. The main technical innovation is an uncertainty principle for weighted generalizations of the total influence of Boolean functions. As a simple example, the unweighted variant states that $\mathbf{Inf}[f]+\mathbf{Inf}[\widehat{f}] = Ω(n)$, which is a natural hypercube analogue of the Heisenberg uncertainty principle (here $\widehat{\,\cdot\,}$ denotes the $2^{-n/2}$-normalized Fourier transform). The weighted case generalizes $\mathbf{Inf}[\,\cdot\,]$ and $\mathbf{Inf}[\,\widehat{\,\cdot\,}\,]$ to Dirichlet energies associated with Glauber dynamics for certain dual measures on the cube.
