Bell Inequality Violation

Physics: CHSH inequality: S = |⟨AB⟩ − ⟨AB'⟩ + ⟨A'B⟩ + ⟨A'B'⟩| ≤ 2 (local realism); quantum maximum S = 2√2 ≈ 2.828 (Tsirelson bound) achieved at optimal angles Δθ = 22.5°. Entangled state: |Φ⁺⟩ = (|00⟩+|11⟩)/√2. Correlation E(a,b) = −cos(2(a−b)). Aspect 1982: first Bell test; Hensen 2015, Giustina 2015, Shalm 2015: loophole-free experiments. Detection loophole requires η > 2/(1+√2) ≈ 82.8%. Locality loophole: spacelike separation. Aspect, Clauser, Zeilinger: Nobel Physics 2022. No local hidden variable theory can mimic entanglement.