Bell Inequality and Quantum Entanglement

What you are seeing: a source emits two photons in a polarization singlet. Alice (left) measures at angle $a$ , Bob (right) at $b$ . The singlet correlation is $E(a, b) = -\cos(2(a-b))$ ; the CHSH statistic $S$ is bounded by 2 classically but reaches $2\sqrt{2}$ in quantum mechanics

Figure 1. Bell-CHSH test with polarization-entangled photons. Method: closed-form singlet correlation + Monte Carlo finite-sample E.

a (deg)0

a (deg)45

b (deg)22

b (deg)67

presetopt

WHAT TO TRY

Set the optimal-CHSH preset: the singlet correlation E(a,b) = -cos(a-b) pushes the CHSH combination S up to 2 sqrt 2, the Tsirelson bound. No local hidden-variable theory can exceed S = 2.
Drag the analyzer angles a and b: watch S swing between 2 and 2.83. Whenever S crosses above 2 the verdict flips to a Bell violation, ruling out local realism.
Switch to the aligned preset (a = b): the photons always agree, S drops to 2, and the correlation looks classical. Entanglement only shows its teeth at the carefully chosen optimal angles.