The Quantum Correction to Gaussian Information Geometry is the Killing Form of the Symplectic Algebra

Abstract

On the admissible cone $\mathcal C_\Omega=\{\Sigma\in\mathrm{Sym}^+(2n):\Sigma+i\Omega\ge0\}$ of Gaussian covariance matrices, the classical Bures--Wasserstein transport metric, the Fisher--Rao information metric, and the quantum Bures metric are individually well understood, but their mutual relationship is obscured by the operator equations defining them. Working with the contravariant (dual) metrics, we show that the exact difference between the dual quantum Bures metric and the dual Fisher--Rao metric is independent of the covariance matrix: it is identically the trace form, proportional to the Killing form, of the symplectic algebra $\mathfrak{sp}(2n,\mathbb R)$, pulled back through the isomorphism $X\mapsto\Omega X$. The signature of this form on the Cartan decomposition reproduces the anisotropic stiffening of the quantum metric at the pure-state boundary: it is negative on the compact subalgebra $\mathfrak u(n)$, which carries the divergence, and positive on the noncompact complement. Because the correction is quadratic in the momenta, a no-go lemma shows it cannot arise from minimal coupling to a principal connection. We realize it instead through a Schur complement of a pseudo-Riemannian metric on a phase bundle, whose base reduction is the quantum Bures metric and whose horizontal metric is the classical-limit Fisher--Rao metric. All identities are verified numerically to working precision.