达到SIC基准的一致稳定极小外尔-海森伯测量
Uniformly Stable Minimal Weyl--Heisenberg Measurements Approaching the SIC Benchmark
AI总结：
该研究针对极小外尔-海森伯测量，构造不同维度的测量族，证明素幂维下的一致稳定结果，其SIC归一化最小值趋于1，为量子测量的稳定性分析提供关键进展。
AI中文摘要：
信息完备性（IC）保证存在逆映射，但不保证其统计条件良好。对于极小秩-1外尔-海森伯（WH）测量，协方差使非恒等投影格拉姆谱与基准态的模糊强度成正比，特征值为\nd|\u03c7_\u03c6(u)|^2，将稳定性转化为显式的最坏方向设计问题；记\ud706为其最小非恒等特征值。哈尔基准态几乎必然是IC，而\ud835\udc44[\ud835\udefb^(-1)]=\u221e，用于在每维建立平衡信息完备测量的显式几何族，其归一化谱底受指数衰减包络约束。我们构造了极小测量的层级结构：每整数维含恰好\ud835\udc84^2个结果的循环族，奇数\ud835\udc84的谱底为\ud835\udef4(\ud835\udc84^(-3))，偶数\ud835\udc84的谱底为\ud835\udef4(\ud835\udc84^(-5))；\ud835\udef1=2^m的有限域族满足一致约束\ud835\udef6\u22654/9。主要结果针对特征\ud835\udef1\u22655的每个素幂维：平衡单坐标扰动修正三次Alltop态的零模糊轴，给出达到的谱底被正常数一致下界约束，且整个非恒等谱被限制在[\ud835\udc81_\ud835\udc85,\ud835\udc8a_\ud835\udc85]内，其中\ud835\udc8a_\ud835\udc85/\ud835\udc81_\ud835\udc85\u21921；其SIC归一化最小值趋于1，且\ud835\udef6(\ud835\udc66_\ud835\udc85)/\ud835\udc36_\ud835\udc85^u2605\u21921，这里\ud835\udc36_\ud835\udc85^u2605是全局有限域WH极大-极小最优值，且不假设SIC存在。完整谱决定了\ud835\udc44/\ud835\udc84处典型线性逆的精确有限样本希尔伯特-施密特误差，其下界控制局部费舍尔效率和典型影子界。
英文摘要：
Informational completeness (IC) guarantees that an inverse exists, not that it is statistically well conditioned. For minimal rank-one Weyl--Heisenberg (WH) measurements, covariance makes the nonidentity projector-Gram spectrum proportional to the fiducial's ambiguity intensities, with eigenvalues \(d|χ_ϕ(u)|^2\), turning stability into an explicit worst-direction design problem; write \(λ\) for its smallest nonidentity eigenvalue. Haar fiducials are IC almost surely while \(\mathbb E[λ^{-1}]=\infty\), and an explicit geometric family used to establish balanced informationally complete measurements in every dimension has a normalized spectral floor bounded above by an exponentially decaying envelope. We then construct a hierarchy of minimal measurements. A cyclic family with exactly \(d^2\) outcomes in every integer dimension has floors \(Θ(d^{-3})\) for odd \(d\) and \(Θ(d^{-5})\) for even \(d\); a finite-field family for \(q=2^m\) obeys the uniform bound \(λ\ge4/9\). Our main result treats every prime-power dimension of characteristic \(p\ge5\). A balanced one-coordinate perturbation repairs the zero ambiguity axis of a cubic Alltop state, gives an attained floor uniformly bounded below by a positive constant, and confines the entire nonidentity spectrum to \([L_q,U_q]\) with \(U_q/L_q\to1\). Its SIC-normalized minimum tends to one, and \(λ(ϕ_q)/Λ_q^\star\to1\) for the global finite-field WH max--min optimum \(Λ_q^\star\), without assuming SIC existence. The complete spectrum determines the exact finite-sample Hilbert--Schmidt error of canonical linear inversion at \(I/d\), while its lower edge controls local Fisher efficiency and canonical-shadow bounds.
