Bell Inequalities
Bell inequalities are limits obeyed by any local hidden-variable theory; quantum mechanics violates them, ruling out local realism.
The question they answer
Are entanglement correlations just pre-arranged classical agreements — hidden variables set when the pair was created — or something genuinely non-classical? In 1964 John Bell showed the two possibilities give measurably different statistics. Any local hidden-variable theory must satisfy an inequality that quantum mechanics can violate.
The CHSH form
The most used version is the CHSH inequality. Two parties, Alice and Bob, each choose between two measurement settings and record outcomes of +1 or -1. Define a combination S of the four correlation values. Any local realistic theory obeys |S| <= 2. Quantum mechanics with a shared Bell pair reaches |S| = 2 sqrt(2) approximately 2.83, the Tsirelson bound.
- Local hidden variables: |S| <= 2
- Quantum mechanics: |S| up to 2 sqrt(2) ~ 2.83
- Experiments consistently measure S > 2
What the violation means
Repeated experiments — increasingly loophole-free since the 2010s — measure S above 2. The conclusion is that nature cannot be described by any theory that is simultaneously local (no faster-than-light influence) and realistic (measurement outcomes fixed in advance). At least one of those assumptions is false, and the correlations are truly quantum.
Why it is not signalling
Violating a Bell inequality does not let Alice send Bob a message. Each side sees random +1/-1 outcomes with no dependence on the other's setting; the excess correlation only appears when the two data sets are compared afterward. Locality of signalling survives even as local realism fails.
Relevance to computing
Bell violation certifies that a device is producing genuine entanglement rather than classical correlation, which is why CHSH tests are used to validate quantum hardware and to underwrite device-independent protocols in quantum information.