  In this paper we show a Clauser-Horne (CH) inequality for two three-level
quantum systems or qutrits, alternative to the CH inequality given by
Kaszlikowski et al. [PRA 65, 032118 (2002)]. In contrast to this latter CH
inequality, the new one is shown to be equivalent to the
Clauser-Horne-Shimony-Holt (CHSH) inequality for two qutrits given by Collins
et al. [PRL 88, 040404 (2002)]. Both the CH and CHSH inequalities exhibit the
strongest resistance to noise for a nonmaximally entangled state for the case
of two von Neumann measurements per site, as first shown by Acin et al. [PRA
65, 052325 (2002)]. This equivalence, however, breaks down when one takes into
account the less-than-perfect quantum efficiency of detectors. Indeed, for the
noiseless case, the threshold quantum efficiency above which there is no local
and realistic description of the experiment for the optimal choice of
measurements is found to be 0.814 for the CH inequality, whereas it is equal to
0.828 for the CHSH inequality.
