Orthomodular Bell-Kochen-Specker Theorem

D. M. Appleby

Jose Luis Cereceda

Adán Cabello, JoséM. Estebaranz, Guillermo García Alcaine

James D. Malley, Arthur I. Fine

D. A. Smith

Partial Boolean algebras, first studied by Kochen and Specker in the 1960s, provide the structure for Bell–Kochen–Specker theorems which deny the existence of non-contextual hidden variable theories. In this paper, we study partial Boolean algebras which are ‘algebraic’ in the sense that their elements have coordinates in an algebraic number field. Several of these algebras have been discussed recently in a debate on the validity of Bell–Kochen–Specker theorems in the context of finite precision measurements. The main result of this paper is that every algebraic finitely-generated partial Boolean algebra B ( T ) is finite when the underlying space ℋ is three-dimensional, answering a question of Kochen and showing that Conway and Kochen's infinite algebraic partial Boolean algebra has minimum dimension. This result contrasts the existence of an infinite (non-algebraic) B ( T ) generated by eight elements in an abstract orthomodular lattice of height 3. We then initiate a study of higher-dimensional algebraic partial Boolean algebras. First, we describe a restriction on the determinants of the elements of B ( T ) that are generated by a given set T . We then show that when the generating set T consists of the rays spanning the minimal vectors in a real irreducible root lattice, B ( T ) is infinite just if that root lattice has an A 5 sublattice. Finally, we characterize the rays of B ( T ) when T consists of the rays spanning the minimal vectors of the root lattice E 8 .

P. K. Aravind

A proof of the generalized Kochen-Specker theorem in two dimensions due to Cabello and Nakamura [A. Cabello, Phys. Rev. Lett. 90, 190401 (2003)] is extended to all higher dimensions. A set of 18 states in four dimensions is used to give closely related proofs of the generalized Kochen-Specker, Kochen-Specker, and Bell theorems that shed some light on the relationship between these three theorems.