The CGLMP Bell inequalities

Quantum non-locality tests have been of interest since the 1960s paper by Bell on the original EPR paradox. The present paper discusses whether the CGLMP (Bell) inequalities obtained by Collins et al. 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 involve a hidden variable theory which is not required to be local. The latter HVT probabilities involve outcomes of simultaneous measurements of pairs of observables corresponding to non-commuting quantum operators, which is allowed 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. Quantum measurements corresponding to the different classical measurements that give the same CGLMP probability are found to yield different CGLMP 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 does not predict the same results as quantum theory. This is found to occur for a state considered in their paper—though for observables whose physical interpretation is unclear. In spite of the problems of comparing the HVT inequalities with quantum expressions, it is concluded that in spite of the contextuality loophole, the CGLMP inequalities are indeed suitable for ruling out local hidden variable theories and also non-local ones as well. 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 involving atoms with two hyperfine components are discussed.