Extension beyond Gaussian distributions

Determine how far the coincidence between the gradient-flow characterization and the Bures–Wasserstein distance extends from Gaussian measures to elliptical distributions and to Bures–Wasserstein distances between density operators.

Background

Theorem 12.5 connects the Bures–Wasserstein distance of Gaussian measures to the minimizer of a log-determinant-type gradient-flow potential. The Gaussian assumption enters through the existence of an explicit affine optimal-transport map.

The underlying Kempf–Ness/Azad–Loeb variational mechanism is not intrinsically Gaussian, suggesting possible extensions. The unresolved issue is whether an analogous distance or flow characterization survives for elliptical laws or quantum density operators.