Bell-type inequalities and orthomodular lattices (Q2776090)

Concluding remarks: (1) For every Boolean algebra and for every state on it all Bell-type inequalities are valid. (2) This property does not characterize the class of Boolean algebras. This means that there exist orthomodular lattices \(L\) with a non-empty state space which have the property that all Bell-type inequalities hold for all states on \(L\), but which are not Boolean algebras. All Bell-type inequalities are valid for a state \(p\) on an orthomodular lattice \(L\) iff \(L\) is distributive with respect to \(p\) (i. e. if \(p\) is distributive) (Theorem 4.1). (3) The original Bell inequality implies all possible Bell-type inequalities of order 2 (Theorem 2.1). (4) There exist orthomodular lattices \(L\) and states \(p\) on \(L\) such that all Bell-type inequalities of order 2 are valid, but not all Bell-type inequalities of order 3 hold (Proposition 3.1). (5) There exist single Bell-type inequalities of order 3 (Theorem 4.1) and also of higher order (Theorem 4.2) which imply all Bell-type inequalities.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00011].