Generalized Bell inequalities and frustrated spin systems

Abstract

We find a close correspondence between generalized Bell inequalities (GBI's) of a special kind and certain frustrated spin systems. For example, the Clauser-Horn-Shimony-Holt inequality corresponds to the frustrated square with the signs $+++-$ for the nearest neighbor interaction between the spins. Similarly, the Pearle-Braunstein-Cave inequality corresponds to a frustrated even ring with the corresponding signs $+\dots +-$. Upon this correspondence, the violation of such inequalities by the entangled singlet state in quantum mechanics is equivalent to the spin system possessing a classical coplanar ground state, the energy of which is lower than the Ising ground state's energy. We propose a scheme which generates new inequalities and give further examples, the frustrated hexagon with additional diagonal bonds and the frustrated hypercubes in $n=3,4,5$ dimensions. Surprisingly, the hypercube in $n=4$ dimensions yields an inequality which is \emph{not} violated by the singlet state. We extend the correspondence to other entangled states and XXZ-models of spin systems.

arXiv commentUpdated and extended version of the 2006 preprint