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, ...

Csiszár's f-divergence of two probability distributions was extended to the quantum case by the author in 1985. In the quantum setting positive semidefinite matrices are in the place of probability distributions and the quantum generalization is called quasi-entropy which is related to some other important concepts as covariance, quadratic costs, Fisher information, Cramér-Rao inequality and un...