Finite violations of a Bell inequality for high spin: An optical realization

Some years ago Peres Phys. Rev. A 46 4413 (1992) described a gedanken experiment for a pair of spatially spin j particles in a singlet state and showed using with a dichotomic observable (essentially a parity operator) that Bell’s theorem in the form of the Clauser-Home-Shimony-Holt (CHSH) inequality is violated by a constant amount (24%) in the limit j→∞. In this paper we present a scheme for an optical realization of a state that is very close to the spin-j singlet state using two traveling-wave modes of the quantized field using a 50:50 beam splitter with an input number state. A near-singlet states comes about because the binomial output state of the beam splitter can be written as a sum in terms of states in the form ∣j,m⟩₁⊗∣j,−m⟩₂, each state being associated with a Holstein-Primakoff realization of the su(2) spin algebra in terms of the Bose operators of each of the field modes, where j=N∕2, N being the number of photons passing through the beam splitter. The binomial state can violate the CHSH inequality to a greater degree than does the singlet state.