Density matrices are fundamental mathematical objects for describing quantum states, and their set can be viewed as a subset of the manifold of symmetric positive definite (SPD) matrices. Based on the Riemannian geometric structure on the SPD manifold, this paper analyzes geometric paths and distance functions on the space of density matrices. We adopt the affine-invariant Riemannian metric and review the associated geodesics and Riemannian distance, comparing them with linear interpolation and the Frobenius distance defined in the ambient Euclidean space. Numerical experiments are conducted to visualize geodesic paths between density matrices and to quantitatively compare several commonly used matrix distances, including the Frobenius, Riemannian, and Bures distances. The results demonstrate that distances induced by different geometric structures exhibit fundamentally different behaviors in characterizing discrepancies between density matrices, reflecting the intrinsic non-Euclidean geometry of the space. These analyses and numerical results provide a coherent geometric perspective for understanding the structure and similarity of density matrices.
We consider the Riemannian geometry of the space of nonsingular density matrices D 1 equipped with the Bures metric g B .This space is of certain physical relevance on the background of a generalization of the Berry phase to mixed states.We determine the covariant derivative and the curvature tensor field related to the Levi-Cevita connection of (D 1 , g B ), which allow us to calculate other curvature quantities .It turns out that D 1 is not a space of constant curvature and not even a locally symmetric space, in contrast to what the case of two-dimensional density matrices might suggest.Moreover, we give a local description of D 1 and explicit formulae for the Bures metric in terms of natural matrix operations containing and d only.