  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.
