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 $\Psi ={\Phi }^2$ satisfies the linear forced heat equation \beginequation* D_tΨ=Ψ_ss-2, \endequation* 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 \beginequation* g(t)=(1-4t)g_BH, \endequation* with scalar curvature $6/(1-4t)$ and extinction time $T=1/4$. 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 ${\Delta }_{{\mathbb{S}}^3}+3$, with spectrum \beginequation* σ_\ell=-(\ell-1)(\ell+3), \endequation* and spectral gap $5$ after removal of the scaling mode.