  We present a self-contained and comprehensive study of the Fisher-Rao space
of matrix-valued non-commutative probability measures, and of the related
Hellinger space. Our non-commutative Fisher-Rao space is a natural
generalization of the classical commutative Fisher-Rao space of probability
measures and of the Bures-Wasserstein space of Hermitian positive-definite
matrices. We introduce and justify a canonical entropy on the non-commutative
Fisher-Rao space, which differs from the von Neumann entropy. We consequently
derive the analogues of the heat flow, of the Fisher information, and of the
dynamical Schr\"odinger problem. We show the $\Gamma$-convergence of the
$\epsilon$-Schr\"odinger problem towards the geodesic problem for the
Fisher-Rao space, and, as a byproduct, the strict geodesic convexity of the
entropy.
