The Infinitesimal Structure of Quantum Information

Abstract

This paper establishes a rigorous, unified geometric framework for quantum state spaces by constructing smooth, regular embeddings into higher-order dual number algebras ${\mathcal{Q}}_N\equiv \mathbb{R}[\epsilon ]/({\epsilon }^{N^2-1})$, wherein every quantum state is faithfully represented as a non-reduced scheme-theoretic point. We show that under this unified family of truncated rings, the non-linear matrix commutators governing the Liouville-von Neumann dynamics map globally onto flat, linear, and rigid algebraic flows, establishing nilpotent dual algebras as a pristine geometric landscape for higher-dimensional quantum kinematics. As $N\to \infty$, this family converges to a Cauchy-complete power series ring ${\mathcal{Q}}_{\infty }$, where non-Archimedean completion linearizes the phase space, aligning the Fubini-Study geometry with the classical Fisher-Rao manifold.

Placed the persistent scalar trace background 1/N which defines the exact baricentro of the configuration space representing the maximally mixed state across all dimensions N

Topics & keywords