The Global Geometry of the Gaussian Bures Manifold: Admissible Domain, Spectral Boundary, and Asymptotic Classicality

Abstract

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.