Bell inequality

A mathematical bound on correlations among measurement outcomes, derived from local hidden-variable theories under specified assumptions. Quantum predictions and experiments can violate these bounds.

Local hidden-variable theory: A model in which measurement outcomes are determined by shared variables and influences do not propagate faster than light. Bell inequalities bound the correlations permitted by this class of models.
Bell's original inequality: A correlation inequality derived by John Bell for a particular local hidden-variable model. This was the first explicit bound separating local hidden-variable predictions from quantum mechanics.
Quantum entanglement: A quantum state in which the parts cannot be described as independent states, producing correlations that may exceed classical bounds. Entangled states can produce the correlations that violate Bell inequalities.
Bell test loopholes: Experimental gaps that could allow local hidden-variable models to mimic an apparent Bell-inequality violation. The history of closing these loopholes clarifies the assumptions behind experimental conclusions.
Bell theorem: A theorem showing that no theory satisfying specified locality and realism conditions reproduces all quantum predictions. The inequalities turn Bell's incompatibility result into testable correlation bounds.
Clauser–Horne–Shimony–Holt inequality: A Bell inequality derived by John Clauser, Michael Horne, Abner Shimony, and Richard Holt in 1969. Its experimentally convenient two-setting form made Bell tests practical.
Device-independent quantum cryptography: Cryptographic methods that certify security from observed quantum correlations without trusting the devices' internal workings. A Bell violation can certify secrecy under explicit assumptions about the devices and experiment.
Detection loophole: A Bell-test loophole arising when detectors fail to register some measurement outcomes. Selective non-detection can bias observed correlations and conceal a local explanation.
CHSH inequality: A Bell inequality involving two measurement settings for each of two separated observers. Its four correlators give the best-known experimentally tested form of the bound.
Freedman–Clauser experiment: A 1972 experiment by Stuart Freedman and John Clauser that tested a Bell inequality using photon polarization correlations. It provided early experimental evidence against local hidden-variable predictions.

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