nt - p h / 01 04 13 3 v 5 2 O ct 2 00 1 Bell ’ s theorem without inequalities and only two distant observers

A proof of Bell's theorem without inequalities is given by suitably extending a proof of the Bell-Kochen-Specker theorem due to Mermin. This proof is generalized to obtain an inequality-free proof of Bell's theorem based on an entangled state of 2n qubits (with n odd) shared between two distant observers. A generalized CHSH inequality is formulated for 2n qubits and it is shown that quantum mechanics violates this inequality by an amount that grows exponentially with increasing n. In two recent papers[1,2], Cabello gave a proof of Bell's theorem without inequalities by using a special state of four qubits shared between two distant observers. This improved upon the classic proof of Greenberger, Horne and Zeilinger[3] and Mermin[4] by reducing the number of distant observers from three to two. The purpose of this paper is to describe a variant of Cabello's proof that avoids one of its shortcomings and that can also be generalized to apply to an entangled state of 2n qubits (with n odd) shared between two distant observers. The present proof, like Cabello's[2] and several others before it[5], uses a common framework to prove both the Bell-Kochen-Specker (BKS)[6] and Bell[7] theorems. However, while Cabello proceeds backwards from the stronger (Bell) to the weaker (BKS) theorem, we proceed in the opposite direction. Our approach has the advantage that it makes no use of either entanglement or communication between observers in proving the BKS theorem, and invokes these additional elements only to pass from the BKS to the Bell theorem. The present BKS-Bell proof is similar in overall structure to an earlier proof by Heywood and Redhead[8], although it differs in several specific respects. Both proofs exploit EPR type correlations to derive non-contextuality from locality, and then use a Kochen-Specker argument to establish the inevitability of non-locality. However while Heywood and Redhead use a singlet state of two spin-1 particles and the original Kochen-Specker argument to carry out their proof, we use n Bell states shared between two observers and replace the original Kochen-Specker argument by a more transparent variant due to Mermin[9]. We believe that there is both a gain in simplicity relative to the original Heywood-Redhead 1