Clauser-Horne inequality for three-state systems

We show a new Bell-Clauser-Horne inequality for two entangled three-dimensional quantum systems (so-called qutrits). This inequality is not violated by a maximally entangled state of two qutrits observed through a symmetric three-input-and three-output-port beam splitter only if the amount of noise in the system is greater than (11-6root3)/2approximate to0.308. This result is in a perfect agreement with the previous numerical calculations presented in Kaszlikowski et al. [Phys. Rev. Lett. 85, 4418 (2000)]. Moreover, we prove that for noiseless case, the necessary and sufficient condition for the threshold quantum efficiency of detectors below which there is no violation of local realism for the optimal choice of observables is equal to 6(15-4root3)/59approximate to0.821. This efficiency result again agrees with the numerical predictions.