The Schrödinger problem on the non-commutative Fisher-Rao space

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