On the equivalence of the CH and CHSH inequalities for two three-level systems

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. [Phys. Rev. A 65, 032118 (2002)]. In contrast to this latter CH inequality, the new one is shown to be equivalent to the ClauserHorne-Shimony-Holt (CHSH) inequality for two qutrits given by Collins et al. [Phys. Rev. Lett. 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 party, as first shown by Acin et al. [Phys. Rev. A 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 (9− √ 33)/4 ≈ 0.814 for the CH inequality, whereas it is equal to ( √ −3 + √ 33)/2 ≈ 0.828 for the CHSH inequality.