Entanglement, Bell inequalities & CHSH violation — quantum vs classical
CHSH inequality: |E(a,b)−E(a,b')+E(a',b)+E(a',b')| ≤ 2 for any local hidden variable theory.
Quantum maximum (Tsirelson): 2√2 ≈ 2.828 — achieved by optimal |Φ⁻⟩ at angles 0°,45°,22.5°,67.5°.
Correlation: E(θA,θB) = ⟨σA⊗σB⟩ = cos(θA−θB) for |Φ⁺⟩, −cos(θA−θB) for |Ψ⁻⟩
Top: Correlation E(θA,θB) vs angle difference. Bottom: CHSH sweep showing classical bound (red) vs quantum (blue).