被动高斯计量学中松弛性的根系结构
Root-system structure of sloppiness in passive Gaussian metrology
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中文总结 AI 辅助
本文通过$C_n$根系刻画被动高斯计量中的松弛性，解决E最优探针设计，并区分局部可识别性、灵敏度优化与同时可达性。
中文摘要 AI 辅助
当量子Fisher矩阵失去秩时，$n$模压缩探针的被动变换变得局部不可识别。对于纯零均值高斯探针，我们证明$\n\nmathrm{Sp}(2n,\mathbb{R})/\mathrm{U}(n)$的$C_{n}$限制根系支配了这一精确的松弛性。在正则基中，对于压缩幅度$r_{1},\ldots,r_{n}$，根$2r_{j}$和$r_{j}\pm r_{k}$标记了局部相位旋转和两个分束器正交分量，它们的消失识别了每一个Fisher零方向。相关的Fisher权重量化了接近每个奇异壁的程度。在固定的正平均光子数下，我们在两个显式生成元归一化下解决了最大化最小被动Fisher特征值的E最优探针设计问题。在正则相位和分束器角度约定下，基本Weyl室中的唯一最优解是一个算术级数，在资源大时接近连续奇数对偶Weyl方向。在不变生成元范数下，最小特征值与被动框架无关，最优解在每个资源下恰好位于该射线上。均匀压缩反而最小化了局部相位A最优成本，同时使每个模式对的其中一个分束器正交分量不可识别。最后，每个可识别模式对的两个正交分量饱和了量子几何不相容性界限，最大的相容被动子模型在正则谱下具有$n(n+1)/2$个参数。由此产生的分类区分了局部可识别性、灵敏度优化和同时可达性。
英文摘要
Passive transformations of an $n$-mode squeezed probe become locally unidentifiable when the quantum Fisher matrix loses rank. For pure zero-mean Gaussian probes, we show that the $C_{n}$ restricted-root system of $\mathrm{Sp}(2n,\mathbb{R})/\mathrm{U}(n)$ governs this exact sloppiness. For squeezing magnitudes $r_{1},\ldots,r_{n}$ in the canonical basis, the roots $2r_{j}$ and $r_{j}\pm r_{k}$ label the local phase rotations and two beam-splitter quadratures, and their vanishing identifies every Fisher-null direction. The associated Fisher weights quantify the approach to each singular wall. At fixed positive mean photon number, we solve the E-optimal probe-design problem of maximizing the smallest passive Fisher eigenvalue in two explicit generator normalizations. In the canonical phase and beam-splitter angle convention, the unique optimum in the fundamental Weyl chamber is an arithmetic progression approaching the consecutive-odd-integer dual-Weyl direction at large resource. With an invariant generator norm, the smallest eigenvalue is independent of the passive frame and the optimum lies exactly on that ray at every resource. Uniform squeezing instead minimizes the local-phase A-optimal cost while leaving one beam-splitter quadrature unidentifiable for every mode pair. Finally, the two quadratures of each identifiable mode pair saturate the quantum-geometric incompatibility bound, and the largest compatible passive submodel has $n(n+1)/2$ parameters at a regular spectrum. The resulting classification separates local identifiability, sensitivity optimization, and simultaneous attainability.
发表机构
- Centro de Investigación y de Estudios Avanzados del IPN（墨西哥国立理工学院高级研究中心）
- Universidad Politécnica Metropolitana de Hidalgo（伊达尔戈都会理工大学）
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