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 +… + −. 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 not violated by the singlet state. We extend the correspondence to other entangled states and XXZ-models of spin systems.