高斯Bures流形的全局几何：允许域、谱边界与渐近经典性
The Global Geometry of the Gaussian Bures Manifold: Admissible Domain, Spectral Boundary, and Asymptotic Classicality
- Trinity College Institute of Neuroscience, Trinity College Dublin（都柏林三一学院神经科学研究所）
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AI总结：
本文从高斯对称对数导数方程构建Bures几何，确定量子允许域，揭示边界各向异性与纯态核，推导势函数并证明大辛本征值下恢复经典Fisher-Rao几何，且发现Bures与Bures-Wasserstein密度的单模连接。
AI中文摘要：
我们从高斯对称对数导数方程出发，构建了居中玻色子高斯态的协方差扇区Bures几何，并确定了其量子允许域。在Williamson不确定性下限处，边界几何是各向异性的：改变谱的径向系数发散，而沿纯高斯轨道的压缩和旋转方向保持有限。对偶地，当m个模变为纯态时，协方差余度量获得一个精确的m²维核，但边界仍处于有限的径向Bures距离处。我们推导出势函数Φ=-½Σₖlog(νₖ²-¼)，该势函数生成固定框架协方差膨胀，并在固定哈密顿量热族上等于βF。在大辛本征值下，相对量子修正为O(ℏ²/ν²)，恢复了协方差Fisher-Rao几何。这一内在经典区域不同于纯态边界，在纯态边界处经典统计需要指定的测量通道。作为次要比较，归一化的Bures、Fisher-Rao和动作匹配的Bures-Wasserstein行列式密度在(x,η)=(√φ,√φ)，x=2ν/ℏ处具有精确的单模连接。该连接是运动学的，其分支解释是有条件的。在给定的有限非零压缩率下，Bures作用量发散而Bures-Wasserstein代价保持有限；有界的Bures预算则强制径向减速。
英文摘要：
We construct the covariance-sector Bures geometry of centered bosonic Gaussian states from the Gaussian symmetric-logarithmic-derivative equation and determine its quantum-admissible domain. At the Williamson uncertainty floor, the boundary geometry is anisotropic: spectrum-changing radial coefficients diverge, whereas squeezing and rotation directions along the pure Gaussian orbit remain finite. Dually, the covariance cometric acquires an exact \(m^2\)-dimensional kernel when \(m\) modes become pure, yet the boundary remains at finite radial Bures distance. We derive the potential \[ Φ=-\tfrac12\sum_k\log(ν_k^2-\tfrac14), \] which generates fixed-frame covariance dilation and equals \(βF\) on fixed-Hamiltonian thermal families. At large symplectic eigenvalue, the relative quantum correction is \(O(\hbar^2/ν^2)\), recovering covariance Fisher--Rao geometry. This intrinsic classical regime differs from the pure-state boundary, where classical statistics require a specified measurement channel. As a secondary comparison, the normalized Bures, Fisher--Rao, and action-matched Bures--Wasserstein determinant densities possess an exact one-mode junction at \[ (x,η)=(\sqrtφ,\sqrtφ),\qquad x=2ν/\hbar. \] This junction is kinematic, and its branch interpretation is conditional. At a prescribed finite nonzero compression rate, the Bures action diverges while the Bures--Wasserstein cost remains finite; a bounded Bures budget instead enforces radial deceleration.
