Bell-Kochen-Specker theorem: A proof with 18 vectors

Abstract

We present a ``state-independent'' proof of the Bell-Kochen-Specker theorem using only 18 four-dimensional vectors, which is a record for this kind of proof. This set of vectors contains subsets which allow us to develop a ``state-specific'' proof with 10 vectors (also a record) and a ``probabilistic'' proof with 7 vectors which reflects the algebraic structure of Hardy's nonlocality theorem.