Quantum Entanglement Simulator (Bell / CHSH)

Simulates measurement outcomes of a pair of spin-½ particles in a singlet state. You choose measurement angles for Alice and Bob. The simulator generates outcomes (+1 or −1) with quantum correlations E = -cos(θ_A - θ_B). In CHSH mode, it estimates the Bell S value and shows violation of the local bound.

Single Pair Settings (Alice & Bob)

Angles are in degrees. For the singlet state, quantum mechanics predicts E(θ_A, θ_B) = -cos(θ_A - θ_B). The plot below shows the simulated correlation vs the ideal curve.

Alice angle θ_A (°): Bob angle θ_B (°): Number of pairs N:

Waiting. Choose angles, then run.

Simulated E_est Quantum E_qm

Bars run from -1 to +1. 0 is the center line.

Bell / CHSH Test

Choose two settings for Alice (A, A′) and two for Bob (B, B′). The simulator estimates S = E(A,B) - E(A,B′) + E(A′,B) + E(A′,B′). Local hidden-variable theories obey |S| ≤ 2. Quantum mechanics reaches |S| up to 2√2 ≈ 2.828. The plot shows the four correlations used in S.

θ_A (°): θ_A′ (°): θ_B (°): θ_B′ (°): Pairs per setting:

Waiting. Set angles or use preset, then run.

Simulated E_est Quantum E_qm

Bars show the four correlations E(A,B), E(A,B′), E(A′,B), E(A′,B′). You should see the pattern that gives |S| > 2.