Metric Analysis of Stochastic Vector Trajectories in Spherical Manifolds via von Neumann Entropy Dynamics

This study explores a geometric framework for mapping high-dimensional vector sequences onto unit spherical manifolds (S^2). We investigate the correlation between the radial modulation of the trajectory vector r(t) and the evolution of von Neumann entropy in the associated reduced density matrix. Leveraging the Balian metric (ds^2 = -d^2S), we propose a formal method for visualizing entropy gradients as curvature in the embedding space. This approach provides a theoretical basis for stability analysis in dynamic systems and dimensionality reduction techniques, linking information loss to geometric contraction without relying on specific application domains.