nt - p h / 01 04 13 3 v 3 2 2 Ju n 20 01 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 using an entangled state of 2n qubits (with n odd) shared between two distant observers. 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 in proving the BKS theorem, and invokes these additional elements only to build the bridge proceeding from the BKS to the Bell theorem. Figure 1 shows a 3 x 3 array of observables pertaining to a pair of qubits that was used by Mermin[8] to prove the BKS theorem. Mermin's proof is based on the elementary observations that: (a) each observable has only the eigenvalues ±1, (b) the observables in any row or column of the array form a mutually commuting set, and (c) the product of the observables (and hence their eigenvalues) in any row or column is +1, with the exception of the last column for which the product is –1. Armed with these facts, Mermin's argument proceeds as follows. Suppose an experimenter, Alice, who has two qubits in her possession carries out the measurements corresponding to the commuting observables in one of the 1