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

Abstract

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 $j\ensuremath{\rightarrow}\ensuremath{\infty}$. 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 ${\ensuremath{\mid}j,m⟩}_{1}\ensuremath{\bigotimes}{\ensuremath{\mid}j,\ensuremath{-}m⟩}_{2}$, 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 $j=N∕2,\phantom{\rule{0.2em}{0ex}}N$ 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.