Ricci flow for the Bures--Helstrom qubit metric

The Bures–Helstrom metric is the minimal monotone Riemannian metric on the state space of a qubit. With the quantum Fisher normalization used here, it identifies the Bloch ball with a geodesic hemisphere of the unit round three–sphere. We describe its Ricci flow explicitly. In a general rotationally symmetric gauge the flow is a coupled system for the radial lapse and warping factor; a single scalar equation appears only after a Hamilton–DeTurck gauge choice. In the corresponding moving DeTurck frame the squared warping function $\mathrm{\Psi }={\mathrm{\Phi }}^2$ satisfies the linear forced heat equation \begin{equation*} D_tΨ=Ψ_{ss}-2, \end{equation*} while the fixed-lapse coordinate form contains the associated transport term. Since the Bures–Helstrom metric is Einstein, the geometric flow itself is the homothetic shrinker \begin{equation*} g(t)=(1-4t)g_{$\mathrm{B}\mathrm{H}$}, \end{equation*} with scalar curvature $6\mathrm{/}(1-4t)$ and extinction time $T=1\mathrm{/}4$. Thus the metric remains inside the monotone cone for all $t<T$ and leaves the cone of nondegenerate Riemannian metrics only through the collapsed limit. We also record the volume–normalized flow, for which the Bures–Helstrom metric is a fixed point. Its linearization is the shifted round–sphere Laplacian ${\mathrm{\Delta }}_{{\mathbb{S}}^3}+3$, with spectrum \begin{equation*} σ_\ell=-(\ell-1)(\ell+3), \end{equation*} and spectral gap $5$ after removal of the scaling mode.