The Infinitesimal Structure of Quantum Information

• URL: http://arxiv.org/abs/2607.12559v1
• Date: Tue, 14 Jul 2026 09:28:54 GMT
• Title: The Infinitesimal Structure of Quantum Information
• Abstract summary: This paper establishes a rigorous, unified geometric framework for quantum state spaces.<n>Every quantum state is faithfully represented as a non-reduced scheme-theoretic point.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper establishes a rigorous, unified geometric framework for quantum state spaces by constructing smooth, regular embeddings into higher-order dual number algebras $\mathcal{Q}_N \equiv \mathbb{R}[\varepsilon]/(\varepsilon^{N^2-1})$, 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.

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