AI 中文总结
研究高斯协方差矩阵容许锥上量子Bures度量与Fisher - Rao度量的差异，通过研究逆变度量发现其差异与协方差矩阵无关，是辛代数基灵型，通过相丛上伪黎曼度量舒尔补实现，数值验证恒等式。
AI 中文摘要
在高斯协方差矩阵的容许锥\(\mathcal C_\Omega=\{\Sigma\in\mathrm{Sym}^+(2n):\Sigma+i\Omega\ge0\}\)上，经典的Bures - Wasserstein传输度量、Fisher - Rao信息度量和量子Bures度量各自都被很好地理解，但它们之间的相互关系因定义它们的算子方程而模糊。通过研究逆变（对偶）度量，我们表明对偶量子Bures度量与对偶Fisher - Rao度量之间的精确差异与协方差矩阵无关：它恒为迹形式，与辛代数\(\mathfrak{sp}(2n,\mathbb R)\)的基灵型成比例，通过同构\(X\mapsto\Omega X\)拉回。这种形式在嘉当分解上的符号再现了量子度量在纯态边界处的各向异性硬化：它在携带散度的紧致子代数\(\mathfrak u(n)\)上为负，在非紧致补空间上为正。由于修正在动量上是二次的，一个不可行引理表明它不能由与主联络的最小耦合产生。我们通过相丛上伪黎曼度量的舒尔补来实现它，其基约化是量子Bures度量，其水平度量是经典极限Fisher - Rao度量。所有恒等式都通过数值验证到工作精度。
英文摘要
For centered bosonic Gaussian states, the covariance pullback of the Bures cometric differs from the lift-normalized classical covariance Fisher--Rao cometric by a state-independent bilinear form. Under the canonical identification of symmetric covariance covectors with $\mathfrak{sp}(2N,\mathbb{R})$, this form is the trace form and hence a fixed multiple of the Killing form. We prove that, within the normalized symmetric Petz family, Bures is uniquely selected by requiring such an additive state-independent symplectic correction. We determine the full Williamson-frame spectrum and show that a boundary stratum with $m$ pure modes has an $m^2$-dimensional cometric kernel isomorphic to $\mathfrak u(m)$, while the pure Gaussian orbit remains nondegenerate. For radial one-mode estimation, ideal heterodyne detection accesses the exact fraction $(ν-\hbar/2)/(ν+\hbar/2)$ of the SLD quantum Fisher information. Its vanishing boundary limit reflects finite heterodyne information relative to a divergent radial quantum Fisher information, not zero measurement information. Finally, a minimal Schur realization fixes the auxiliary inertia and identifies the second Schur complement as the true covariant vertical block.
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