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In this paper we consider the description by a Bell non-local hidden variabletheory of bipartite quantum states with two observables per sub-system. Wederive Bell inequalities of the Collins-Gisin.-Liden-Massar-Popescu type whichinvolve combinations of the probabilities of related outcomes for measurementsfor the four pairs of sub-system observables. It is shown that thecorresponding quantum theory expressions violate the Bell inequalities in thecase of the maximally entangled state of the bipartitite system. This showsthat quantum theory can not be underpinned by a non-local hidden variabletheory. So as local hidden variable theory has already been shown to conflictwith quantum theory, it follows that quantum theory can not be understood interms of any hidden variable theory (which Einstein originally had expected).

Can quantum theory be underpinned by a non-local hidden variable theory ?