Perfectly Spherical Bloch Hyper-spheres from Quantum Matrix Geometry

Leveraging analogies between precessing quantum spin systems and charge-monopole systems, we construct Bloch hyper-spheres with exact spherical symmetries in arbitrary dimensions. Such a Bloch hyper-sphere is realized as a collection of the orbits of precessing quantum spins, and its geometry mathematically aligns with the quantum Nambu geometry of a higher dimensional fuzzy sphere. Stabilizer group symmetry of the Bloch hyper-sphere necessarily introduces degenerate spin-coherent states and gives rise to Wilczek-Zee geometric phases of non-Abelian monopoles associated with the hyper-sphere holonomies. The degenerate spin-coherent states naturally induce matrix-valued quantum geometric tensors also. While the physical properties of Bloch hyper-spheres with minimal spin in even and odd dimensions are quite similar, their large spin counterparts differ qualitatively depending on the parity of 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.