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Bell's theorem proves that no local hidden variable theory can reproduce quantum mechanical predictions for entangled states. Bell inequalities give bounds on correlations in any local realistic theory; quantum mechanics violates these bounds. Experiments have confirmed quantum predictions, ruling out local hidden variables. This settles the foundational debate about completeness of quantum mechanics.
The central mystery of quantum entanglement is that two particles can remain correlated even when separated by large distances — measuring one instantly determines something about the other. Before Bell's work, the natural skeptical response was: perhaps the particles simply carry predetermined "hidden" answers with them from the moment they were created, like two gloves placed in separate boxes. Before opening either box you don't know which is left and which is right, but nothing spooky is happening; the information was always there. This is the local hidden variable hypothesis: each particle carries complete information about what outcomes it will produce for any measurement, determined locally without any faster-than-light influence.
Bell's genius was to show that this seemingly reasonable hypothesis makes a testable prediction. Consider measuring spin components of two entangled particles along different angles. A local hidden variable theory must assign each particle definite (hidden) values for every possible measurement direction. The correlations that result from combining those answers must satisfy certain algebraic bounds — these are the Bell inequalities. Quantum mechanics, on the other hand, predicts correlations that violate these bounds for certain measurement choices. The violation is not subtle: quantum mechanics predicts correlations roughly 40% stronger than any local hidden variable theory can produce.
The brilliant simplicity of Bell's argument is that it requires only two assumptions: locality (the measurement choice at one detector doesn't affect the outcome at the other) and realism (particles have definite properties before measurement). Both assumptions together imply the Bell inequalities. Quantum mechanics violates those inequalities, so at least one assumption must fail. You cannot have a theory that is simultaneously local and realistic — hence the phrase "no local hidden variable theory."
Experiments beginning with Clauser and Freedman (1972) and culminating in loophole-free tests in 2015 have confirmed quantum predictions to high precision. The violations are real. The philosophical implication is profound: the correlations between entangled particles are not explained by pre-shared information. Either locality fails (the measurement choice at one end somehow influences the other) or realism fails (particles don't have definite properties before measurement). Bell's theorem ensures that no comfortable hidden-variable escape route exists — the strangeness of quantum mechanics is not a failure of our imagination but a feature of the world.