Bell Test & CHSH Inequality

Entanglement correlations and nonlocality

Prepare a Bell state and choose measurement angles for Alice and Bob. Run trials to compute correlation functions E(a,b) and the CHSH value S. At the optimal angles, the quantum prediction S = 2√2 violates the classical bound |S| ≤ 2, demonstrating that no local hidden-variable theory can reproduce quantum correlations.

Entangled State

Alice's Settings

a₀ = 0°

a₁ = 90°

Bob's Settings

b₀ = 45°

b₁ = -45°

CHSH Value: S = E(a₀,b₀) + E(a₀,b₁) + E(a₁,b₀) - E(a₁,b₁)

theoretical S = -2.828

Theoretical Correlations

E(a₀,b₀) = -0.7071

E(a₀,b₁) = -0.7071

E(a₁,b₀) = -0.7071

E(a₁,b₁) = 0.7071

Correlation Function E(0°, b) vs Bob's Angle b

S = E(a_0, b_0) + E(a_0, b_1) + E(a_1, b_0) - E(a_1, b_1)

|S_{\text{classical}}| \leq 2 \qquad |S_{\text{quantum}}| \leq 2\sqrt{2}

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