Bell inequality

A Bell inequality is a precise limit that any theory based on local hidden variables must respect. In 1964 John Bell took the EPR idea seriously and asked: if each particle really did carry predetermined answers, fixed locally before measurement, how strongly could the results on the two sides be correlated? He proved that under those assumptions the correlations cannot exceed a definite mathematical bound, no matter what the hidden details are.

The power of the result is that quantum mechanics predicts correlations that overshoot this bound. For certain choices of measurement angles on an entangled pair, the quantum statistics are stronger than any local, predetermined scheme could ever produce. So the question of whether the world is 'locally real' stopped being philosophy: it became an inequality you could actually test in a laboratory.

Bell's theorem is sometimes summarised as 'no local hidden-variable theory can reproduce all the predictions of quantum mechanics'. It does not by itself say the world is non-local in a spooky, signal-sending sense; rather, it forces a stark choice between giving up locality, giving up the idea of pre-existing definite values, or giving up other quiet assumptions. Whatever you keep, the comfortable classical picture cannot survive intact.