AI 中文总结
该研究探究读出方向对可测量量子切空间几何的控制作用，通过量化保留的切质量，发现交叉拟合对齐可显著提升测量性能，读出方向是影响切信息可用性的关键自由度。
AI 中文摘要
固定的量子测量能够揭示出比受限可观测量读出所保留的多得多的切空间信息，本文研究这一限制。对于N维中心得分空间中测量诱导切得分的归一化协方差C（满足C≥0，Tr C=1）以及秩为r的读出投影器P，我们通过R=Tr(PC)量化保留的切质量。比值ρ=R/(r/N)将实际保留的质量与仅基于秩的随机取向参考值区分开来。标准格拉斯曼平均给出Eρ=1，且Var(ρ)≤2/(r d_eff)，其中d_eff=1/Tr(C²)。我们将这一恒等式作为零模型，而非新的随机投影定理。数值上，平衡族的一体和二体读出在n=16时仍接近秩参考值，即使切协方差变得强各向异性。决定性的等秩比较固定了电路、测量记录、读出秩和评估采样预算。对于n=12的Haar-U(4)，交叉拟合对齐使平均方向梯度能量代理比物理一体读出增加9.584倍，有限采样信噪比增加3.111倍，而随机秩匹配子空间仍接近秩基线。半满的U(1)守恒族为通用取向提供了结构化反例：在测试的有限尺寸范围内，物理低权重Z读出已与主导切方向强对齐。我们将这一对称性结果视为案例研究，而非声称U(1)对称性通用地防止贫瘠高原或流体动力学是已确立的机制。这些结果将读出方向孤立为仅靠秩无法察觉的自由度，其直接控制读出限制后可使用的测量切信息的量。
英文摘要
A fixed quantum measurement can expose substantially more tangent information than a restricted observable readout retains. We study this second restriction. For a normalized covariance $C \succeq 0$, $\mathrm{Tr} C = 1$, of measurement-induced tangent scores in an $N$-dimensional centered score space, and a rank-$r$ readout projector $P$, we quantify retained tangent mass by $R=\mathrm{Tr}(PC)$. The ratio $ρ=R/(r/N)$ separates the actual retained mass from a rank-only random-orientation reference. Standard Grassmann averaging gives $\mathbb{E}ρ=1$ and $\mathrm{Var}(ρ) \le 2/(r d_{\mathrm{eff}})$, where $d_{\mathrm{eff}}=1/\mathrm{Tr}(C^2)$. We use this identity as a null model rather than as a new random-projection theorem. Numerically, family-balanced one- and two-body readouts remain close to the rank reference through $n=16$ even as the tangent covariance becomes strongly anisotropic. The decisive equal-rank comparison holds the circuit, measurement record, readout rank, and evaluation shot budget fixed. For Haar-$U(4)$ at $n=12$, cross-fitted alignment increases the mean directional gradient-energy proxy by a factor 9.584 and the finite-shot signal-to-noise ratio by a factor 3.111 relative to the physical one-body readout, while a random rank-matched subspace remains near the rank baseline. A half-filled $U(1)$-conserving family provides a structured counterexample to generic orientation: physical low-weight $Z$ readouts are already strongly aligned with leading tangent directions over the tested finite-size range. We treat this symmetry result as a case study, not as a claim that $U(1)$ symmetry generically prevents barren plateaus or that hydrodynamics is the established mechanism. The results isolate readout orientation as a degree of freedom invisible to rank alone that directly controls how much measured tangent information remains usable after readout restriction.
Comments15 pages, 14 figures. Source code and reproducibility materials are available at: https://github.com/AHDMarwan/Spectral-Geometry-of-Accessible-Quantum-Tangents-Beyond-Isotropic-Readout-Rank-Laws
