Summary
In the interpretation of quantum mechanics, a local hidden-variable theory is a hidden-variable theory that satisfies the condition of being consistent with local realism. This definition restricts all types of those theories that attempt to account for the probabilistic features of quantum mechanics via the mechanism of underlying inaccessible variables with the additional requirement that distant events be independent, ruling out instantaneous (that is, faster-than-light) interactions between separate events. The mathematical implications of a local hidden-variable theory in regard to the phenomenon of quantum entanglement were explored by physicist John Stewart Bell, who in 1964 proved that broad classes of local hidden-variable theories cannot reproduce the correlations between measurement outcomes that quantum mechanics predicts. The most notable exception is superdeterminism. Superdeterministic hidden-variable theories can be local and yet be compatible with observations. Bell's theorem starts with the implication of the principle of local realism, that separated measurement processes are independent. Based on this premise, the probability of a coincidence between separated measurements of particles with correlated (e.g. identical or opposite) orientation properties can be written: where is the probability of detection of particle with hidden variable by detector , set in direction , and similarly is the probability at detector , set in direction , for particle , sharing the same value of . The source is assumed to produce particles in the state with probability . Using (), various Bell inequalities can be derived, which provide limits on the possible behaviour of local hidden-variable models. When John Stewart Bell originally derived his inequality, it was in relation to pairs of entangled spin-1/2 particles, every one of those emitted being detected. Bell showed that when detectors are rotated with respect to each other, local realist models must yield a correlation curve that is bounded by a straight line between maxima (detectors aligned), whereas the quantum correlation curve is a cosine relationship.
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