English

The Infinitesimal Structure of Quantum Information

Quantum Physics 2026-07-14 v1 Mathematical Physics Algebraic Geometry Differential Geometry Quantum Algebra

Abstract

This paper establishes a rigorous, unified geometric framework for quantum state spaces by constructing smooth, regular embeddings into higher-order dual number algebras QNR[ε]/(εN21)\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.

Keywords

Cite

@article{arxiv.2607.12559,
  title  = {The Infinitesimal Structure of Quantum Information},
  author = {Luca Barbieri-Viale},
  journal= {arXiv preprint arXiv:2607.12559},
  year   = {2026}
}