Perfectly Spherical Bloch Hyper-spheres from Quantum Matrix Geometry

Exploiting analogies between the precessing quantum spin system and the charge-monopole system, we construct Bloch hyper-spheres with $\it{exact}$ spherical symmetries in arbitrary dimensions. Such Bloch hyper-spheres are realized as a collection of the orbits of a precessing quantum spin. The geometry of Bloch hyper-spheres is exactly equal to the quantum Nambu geometry of higher dimensional fuzzy spheres. The stabilizer group symmetry of the Bloch hyper-sphere necessarily introduces degenerate spin-coherent states, giving rise to the Wilczek-Zee geometric phase of non-Abelian monopoles associated with the hyper-sphere holonomy. The degenerate spin-coherent states induce matrix-valued quantum geometric tensors. While the minimal spin Bloch hyper-spheres exhibit similar properties in even and odd dimensions, their large spin counterparts differ qualitatively depending on the parity of the dimensions. Exact correspondences between spin-coherent states and monopole harmonics in higher dimensions are established. We also investigate density matrices described by Bloch hyper-balls and elucidate their corresponding statistical and geometric properties, such as von Neumann entropies and Bures quantum metrics.