2 00 1 Mermin ’ s pentagram and Bell ’ s theorem

It is shown how a proof of the Bell-Kochen-Specker (BKS) theorem given by Kernaghan and Peres can be experimentally realized using a scheme of measurements derived from a related proof of the same theorem by Mermin. It is also pointed out that if this BKS experiment is carried out independently by two distant observers who repeatedly make measurements on a specially correlated state of six qubits, it provides an inequality-free demonstration of Bell's theorem as well. Beginning with the groundbreaking work of Greenberger, Horne and Zeilinger (GHZ)[1], the last decade has seen several new proofs of the Bell[2] and Bell-Kochen-Specker(BKS)[3] theorems as well as a better understanding of the relationship between these two fundamental theorems[4]. One interesting line of work has focused on obtaining joint proofs of the BKS and Bell theorems, the idea being to first prove the BKS theorem and then use a suitable strategem to convert this proof into a proof of Bell's theorem.[5]. The three-particle GHZ state serves as the springboard for at least three such joint proofs of the BKS and Bell theorems, as detailed in the three scenarios below: Scenario 1. Three qubits are given to three widely separated observers, each of whom can measure two observables on his/her qubit. By considering a set of ten observables pertaining to these qubits, Mermin[4] gave a state-independent proof of the BKS theorem and then showed how to convert it into a proof of Bell's theorem by assuming that the three qubits were in a GHZ state. Scenario 2. The starting point for this variation was provided by Ker-naghan and Peres[6], who extracted a set of 40 states from Mermin's ten ob-servables in Scenario 1 and used them to give a " non-coloring " proof of the BKS theorem (so called because the proof works by showing that it is impossible to assign the color red or green to each of the states in accordance with a simple set of rules). We[5] reinterpreted the 40 states of Kernaghan and Peres as those of a spin-7/2 particle (which also has an eight-dimensional state space), and their proof as a proof of the BKS theorem for such a particle; then, by considering a pair of spin-7/2 particles in a singlet state, we showed how this BKS proof could be converted into a proof of Bell's theorem. Scenario 3. Six qubits are shared, three to each, by two distant observers. …