Cartan KAK Decomposition of SU(4): Three Magic Angles as Minimal Non-Local Content of Two-Qubit Gates — E8 Intelligence Research

FINDING: Cartan decomposition of su(4) ≅ so(6) yields the KAK form U = k₁·A·k₂, reducing arbitrary two-qubit unitaries to 3 real parameters (the "magic" angles c₁, c₂, c₃) — the minimal non-local content of any two-qubit gate. MATH: - Lie algebra: su(4) ≅ so(6), dim = 15. Cartan involution θ(X) = −X† splits 𝔤 = 𝔨 ⊕ 𝔭, with 𝔨 = su(2)⊕su(2)⊕u(1) (dim 7, compact/local) and 𝔭 (dim 8, non-compact/non-local). - Maximal abelian subspace 𝔞 ⊂ 𝔭: dim 𝔞 = 3 (rank of the symmetric space SU(4)/S(U(2)×U(2))). - KAK: U = k₁ · exp(𝔞) · k₂, k₁,k₂ ∈ K = SU(2)⊗SU(2). - Canonical coordinates: A = exp[ i(c₁ σₓσₓ + c₂ σᵧσᵧ + c₃ σ_zσ_z) ], with 0 ≤ c₃ ≤ c₂ ≤ c₁ ≤ π/4 (Weyl chamber). - Weyl group S₃ permutes {c₁,c₂,c₃}; chamber walls at cᵢ = cⱼ and c₃ = 0. - Local equivalence classes: 3-torus [0,π/4]³ / S₃. - CNOT: (π/4, 0, 0). SWAP: (π/4, π/4, π/4). iSWAP: (π/4, π/4, 0). √SWAP: (π/4, π/4, π/8). - Non-local cost / entangling power: Q = c₁ + c₂ + c₃ (or 2(c₁+c₂+c₃)/π normalized). CONNECTION: - Root system: 𝔞 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com