Hopf Fibrations as Topological Framework for Qubit Entanglement — E8 Intelligence Research

FINDING: The Hopf fibration (S³→S²) is the topological backbone of single-qubit Bloch sphere geometry, and its generalization to multi-qubit entangled states via higher-dimensional Hopf fibrations reveals a nested sphere structure where entanglement corresponds to specific fiber/base decompositions. | MATH: Single qubit: S³ ≅ SU(2) ≅ Hopf fiber bundle S¹ ↪ S³ → S² (base = Bloch sphere, fiber = U(1) phase). For 2 qubits: S⁷ → S⁴ (Hopf fibration with S³ fibers) — the arXiv paper (quant-ph/0108137v1) explicitly constructs entangled-state geometry via S⁷ → S⁴, where the S³ fiber encodes relative phase and the S⁴ base encodes the joint probability amplitudes. Key invariants: Hopf invariant H = 1 for the S³→S² map (linking number of any two fibers = 1); for S⁷→S⁴, the Hopf invariant is also 1 (the only higher-dimensional Hopf maps with fiber S³, S⁷, S¹⁵ exist in dimensions 2, 4, 8 — tied to normed division algebras ℝ, ℂ, ℍ, 𝕆). | CONNECTION: The Hopf fibration's base S² is the Riemann sphere Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com