THE INFINITESIMAL STRUCTURE OF QUANTUM INFORMATION

This paper introduces a geometric framework that maps quantum state spaces, typically defined on complex projective spaces, into higher-order nilpotent dual algebras. By representing quantum states as non-reduced scheme-theoretic points within these algebras, the author constructs a bridge between quantum information theory and algebraic geometry. The approach uses density embedding maps to bypass coordinate singularities and non-linear metric distortions inherent in traditional spherical representations, such as the Bloch sphere.

The author establishes that for any N-level quantum system, the state space can be faithfully embedded into a truncated ring of dual numbers. This mapping provides a bijective homothetic isometry between the Fubini-Study metric of the quantum state space and a flat, Euclidean metric in the dual algebra. A key result is the discovery of a universal scaling law where the product of the metric factor and the dynamic coupling constant remains invariant at 4 across all dimensions. This invariance ensures that as system complexity increases, any reduction in statistical distinguishability is compensated by a proportional acceleration in the physical precessional flow, maintaining the conservation of state purity.

By linearizing the non-linear matrix commutators of the Liouville-von Neumann dynamics, this framework offers a chart-free method for analyzing quantum kinematics. It provides a rigorous way to handle higher-dimensional quantum systems, such as qutrits, within a unified geometric landscape. Furthermore, the asymptotic convergence of these algebras toward the ring of formal power series as N approaches infinity suggests a deep connection between quantum information and non-Archimedean geometry, potentially offering new insights into the transition from quantum to classical statistical manifolds.