Barycentric Bures--Hellinger--Kantorovich Geometry

This paper develops a geometric framework for finite positive matrix-valued measures by combining Bures geometry, Hellinger--Kantorovich (HK) transport, and barycentric identification of quantum ensemble representations. The resulting barycentric Bures--Hellinger--Kantorovich geometry provides an unbalanced optimal transport metric for spatially distributed density operators in which different ensemble decompositions representing the same local operator-valued measure are identified. The induced chain metric is shown to be complete on every complete separable spatial base, to metrize narrow convergence, and to admit a compactness characterization through bounded trace mass and uniform tightness. Finite quantum measurements provide operational lower bounds, while mutually commuting or simultaneously diagonalizable data reduce exactly to independent scalar Hellinger--Kantorovich transport problems. On compact convex Euclidean domains, the paper proves a dynamic representation of the metric in terms of a resolved spatial velocity action coupled to a Bures reaction action, including singular measures and rank-changing curves. The local velocity-resolution problem is identified with the reverse-estimation problem in quantum statistics, and its incompatibility penalty is connected quantitatively to the Nagaoka--Hayashi bound for multiparameter quantum estimation. The dynamic formulation also gives a necessary and sufficient criterion for coincidence with a quadratic matrix-flux optimal transport distance: an optimal flow must admit commuting normalized matrix fluxes. Equality holds for all endpoint pairs in one spatial dimension. In two spatial dimensions, an explicit smooth positive family is constructed for which the two distances are strictly different. A tangent-action limit and a coercive commutator estimate show that this separation is genuinely caused by noncommutative flux incompatibility and remains stable under sufficiently small discretization and noise. The results connect matrix-valued optimal transport, unbalanced optimal transport, Bures/Wasserstein geometry, quantum information geometry, operator-valued measures, quantum statistical estimation, and noncommutative transport, while providing both a rigorous metric theory and a dynamic characterization suitable for further analytical and numerical study. Keywords (expanded for search)Hellinger–Kantorovich; Wasserstein–Fisher–Rao; unbalanced optimal transport; entropy transport; ET distance; Benamou–Brenier; reaction–diffusion; KL divergence; relative entropy; EVI; Evolution Variational Inequality; JKO scheme; minimizing movements; Strang splitting; operator splitting; BCH formula; Petz monotone metric; quantum information geometry; Bures distance; BKM metric; CPTP channel; coarse graining; observation monoid; image EVI; multiscale; micro meso macro; graph inverse limit; finitely ramified graphs; fractal; resistance metric; Sierpinski gasket; Mosco convergence; ΓΓΓ-convergence; AF algebra; UHF algebra; C∗C^∗-algebra; inductive limit; order envelope; C1C^1C1 bounds; δ−3δ⁻³δ−3 scaling; stability; convergence; error budget; reproducibility; metric geometry; gradient flows; measure-valued dynamics.