Quantum Entanglement: Bell Test & Correlation Simulator

Measurement Angles

Qubit A axis θ_A 0°

Qubit B axis θ_B 45°

Pair Model

Run

Live Statistics

Δθ = θ_A − θ_B = 0° → E_theory = −cos(Δθ) = −1.00

Same 50%Opposite 50%

Trials

Measured E

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How it works

The pair starts in the singlet Bell state |Ψ⁻⟩ = (|01⟩ − |10⟩)/√2 — perfectly entangled, with no individual qubit having a definite value on its own. Each apparatus measures its qubit along an axis tilted θ_A or θ_B degrees from vertical. Quantum mechanics predicts each side alone still reads 50/50 at random, but the two outcomes are correlated through the angle between the axes, Δθ = θ_A − θ_B:

P(same outcome)     = sin²(Δθ/2)
P(opposite outcome) = cos²(Δθ/2)
Correlation E(θA,θB) = ⟨A·B⟩ = −cos(Δθ)

Toggle to Local Hidden Variable mode to compare against the classical alternative Einstein favoured: each pair secretly carries a shared random direction λ, and each side just reports sign(cos(λ − axis)). That model is simulated trial-by-trial too, and it converges to a straight line E(Δθ) = 1 − 2Δθ/π instead of the quantum cosine — a measurably different curve.

CHSH Bell test — run four correlation measurements at the standard angle set (θ_A ∈ {0°, 45°}, θ_B ∈ {22.5°, 67.5°}) and combine them:

S = E(a1,b1) − E(a1,b2) + E(a2,b1) + E(a2,b2)
Local realism requires |S| ≤ 2      (Bell's inequality)
Quantum mechanics reaches |S| = 2√2 ≈ 2.828   (Tsirelson bound)

Any local hidden-variable model obeys |S| ≤ 2 no matter what λ-distribution it assumes; the entangled qubits in this simulator violate that bound, which is the actual experimental signature (Aspect, Zeilinger, and the 2022 Nobel-winning experiments) that nature really is entangled and not merely "secretly pre-agreed".

Qubit A

Qubit B

Outcome +1 Outcome −1 Measurement axis

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