Local hidden variable theory

A local hidden variable theory in the interpretation of quantum mechanics is a hidden variable theory that has the added requirement of being consistent with local realism. It refers to all types of the theory that attempt to account for the probabilistic features of quantum mechanics by the mechanism of underlying inaccessible variables, with the additional requirement from local realism that distant events be independent, ruling out instantaneous (i.e. 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 S Bell. Bell's 1964 paper (see Bell's theorem) showed that local hidden variables of certain type cannot reproduce the quantum measurement correlations that quantum mechanics predicts.

The theory of quantum entanglement predicts that separated particles can briefly share common properties and respond to certain types of measurement as if they were a single particle. In particular, a measurement on one particle in one place can alter the probability distribution for the outcomes of a measurement on the other particle at a different location. If a measurement setting in one location instantaneously modifies the probability distribution that applies at a distant location, then local hidden variables are ruled out. For an expanded description, see Bell's theorem.

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:

${\displaystyle P(a,b)=\int d\lambda \cdot \rho (\lambda )\cdot p_{A}(a,\lambda )\cdot p_{B}(b,\lambda )}$

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