Bell test and CHSH inequality simulator

A Bell test asks whether measurement results on two distant particles can be explained by pre-existing local hidden values. Quantum mechanics says no, and the CHSH inequality is the standard way to check.

The CHSH quantity

Two observers each pick between two measurement angles (a, a′ and b, b′) and record correlations E. The combination S = E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′) is bounded by |S| ≤ 2 for any local hidden-variable model.

The quantum result

For polarization-entangled photons in the state (|HH⟩ + |VV⟩)/√2, the correlation is E(a,b) = cos 2(a − b). With angles 0°, 45° for one side and 22.5°, 67.5° for the other, S reaches 2√2 ≈ 2.83, the Tsirelson bound. That clearly exceeds the classical limit of 2.

Why it matters

Experiments of this type, awarded the 2022 Nobel Prize in Physics, rule out local hidden-variable explanations and underpin quantum key distribution and quantum networks.

Try it

Open the simulator, adjust the four measurement angles, and watch S cross the classical bound of 2.