arXiv:2607.16376v1 Announce Type: new 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 indi