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

In this paper we consider the description by a general Bell-type non-local hidden variable theory (NLHVT) of bipartite quantum states with two observables per sub-system. We derive Bell inequalities of the Collins-Gisin.-Liden-Massar-Popescu (CGLMP) type which involve combinations of the probabilities of related outcomes for measurements for the four pairs of sub-system observables. It is shown that the corresponding quantum theory expressions violate the Bell inequalities in the case of the maximally entangled state of the bipartitite system. The CHSH Bell inequality is also derived from this general CGLMP Bell-type non-local hidden variable theory. This shows that quantum theory can not be underpinned by a Bell-type non-local hidden variable theory. So as a general Bell-type local hidden variable theory has already been shown to conflict with quantum theory, it follows that quantum theory can not be understood in terms of any CGLMP Bell-type hidden variable theory - local or non-local.