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 Schrödinger problem. We show the \(\Gamma \)-convergence of the \(\varepsilon \)-Schrödinger problem towards the geodesic problem for the Fisher-Rao space, and, as a byproduct, the strict geodesic convexity of the entropy.

我们对矩阵值非交换概率测度的Fisher-Rao空间以及相关的Hellinger空间进行了全面的研究。我们的非可交换Fisher-Rao空间是概率测度的经典可交换Fisher-Rao空间和Hermitian正定矩阵的Bures-Wasserstein空间的自然概括。我们引入和证明非交换Fisher-Rao空间上的典范熵，它不同于冯·诺依曼熵。因此，我们得出了热流，Fisher信息和动力学Schrödinger问题的类似物。我们显示\（\ varepsilon \）的\（\ Gamma \）-收敛-Schrödinger问题朝向Fisher-Rao空间的测地线问题，以及作为副产品的熵的严格测地线凸性。