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Decoherence
Visibility V = exp(-gamma tau) multiplies every quantum correlation.
Model checks
The science
Singlet correlations
For two spin-1/2 particles in the singlet state, measuring along directions in one plane at angles a and b gives outcomes ±1 with E(a,b) = −cos(a − b). Each outcome alone is a fair coin; only the pair is correlated. Noise is modelled as a Werner mixture, E = −V cos(a − b), with visibility V = exp(−γτ).
CHSH inequality
S = E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′). Any local hidden-variable theory obeys |S| ≤ 2; quantum mechanics reaches 2√2 ≈ 2.83 (Tsirelson) at a = 0, a′ = 90, b = 45, b′ = 135 degrees. With visibility V the value is 2√2 V, so the violation vanishes when V ≤ 1/√2 ≈ 0.707.
Honest simplifications: ideal detectors (no detection loophole), perfect setting independence, a single-parameter noise model, planar settings only; the stream of pairs is sampled from the quantum probabilities, not from a full state-vector simulation. Nothing here transmits information faster than light.
What to try
- Press Optimal angles and watch the measured S climb to about 2.8.
- Press Threshold: S sits right at 2 and the verdict flips between runs.
- Press Poor angles: even perfect entanglement stays below the classical limit.