The CGLMP Bell Inequalities

Quantum non-locality tests have been of interest since the 1960's paper by Bell on the original EPR paradox.The present paper discusses whether the CGLMP (Bell) inequalities are possible tests for showing that quantum theory is not underpinned by local hidden variable theory (LHVT). It is found by applying Fine's theorem that the CGLMP approach involves a LHVT for the joint probabilities associated with the measurement of one observable from each of the two sub-systems, even though the underlying probabilities for joint measurements of all four observables may involve a non-local HVT. The latter HVT probabilities involve simultaneous measurements of observables corresponding to non-commuting quantum operators - allowable in classical theory. Although the CGLMP inequalities involve probabilities for measurements of one observable per sub-system and are compatible with the Heisenberg uncertainty principle, there is no unambiguous quantum measurement process linked to the probabilities in the CGLMP inequalities, and these quantum measurements each have different probabilities. However, violation of a CGLMP inequality based on any one of the possible quantum measurement sequences is sufficient to show that the Collins et al LHVT predicts different results to quantum theory. This occurs for a state considered in their paper - though for observables whose physical interpretation is unclear. In spite of this issue and in spite of the contextuality loophole, it is concluded that the CGLMP inequalities are indeed suitable for ruling out both local (and non-local) hidden variable theories. The state involved could apply to a macroscopic system, so the CGLMP Bell inequalities are important for finding cases of macroscopic violations of Bell locality. Possible experiments in double-well Bose condensates for atoms with two hyperfine components are discussed.