In his classical argument, Rao derives the Riemannian distance corresponding to the Fisher metric using a mapping between the space of positive measures and Euclidean space. He obtains the Hellinger distance on the full space of measures and the Fisher distance on the subset of probability measures. In order to highlight the interplay between Fisher theory and quantum information theory, we extend this construction to the space of positive-definite Hermitian matrices using Riemannian submersions and quotient manifolds. The analog of the Hellinger distance turns out to be the Bures-Wasserstein (BW) distance, a distance measure appearing in optimal transport, quantum information, and optimisation theory. First we present an existing derivation of the Riemannian metric and geodesics associated with this distance. Subsequently, we present a novel derivation of the Riemannian distance and geodesics for this metric on the subset of trace-one matrices, analogous to the Fisher distance for probability measures.

翻译：在他的古典论点中,Rao利用积极措施空间与欧几里德空间之间的绘图,得出了与渔业指标相对应的里曼尼距离。他获得了措施空间的完整空间与概率测量子子的费雪尔距离。为了突出费雪理论与量子信息理论之间的相互作用,我们利用里曼尼亚子仪和商数元元体,将这一构造扩大到了赫米提亚正定矩阵空间。海灵格距离的类似点是布雷斯-沃塞斯坦(BW)距离,这是最佳运输、量子信息和优化理论中的一种距离尺度。首先我们介绍了与这一距离相关的里曼度指标和大地测量学的现有出处。随后,我们在微量一矩阵子上介绍了里曼距离和大地测量学的新衍生结果,类似于渔运距离的概率测量。