Homogeneous and genuine Bell inequalities

We discuss homogeneous and inhomogeneous Bell inequalities, following Santos's classification. According to it, homogeneous inequalities entail only coincidence probabilities, whereas inhomogeneous inequalities entail coincidence probabilities together with single probabilities or with numbers. Because of technical limitations, all performed tests of Bell inequalities have been based on homogeneous inequalities whose derivation required additional assumptions besides realism and locality, thereby losing their genuine character. Here we derive, starting from the Clauser-Horne inequality, a homogeneous inequality that was at the basis of an experimental test performed some years ago by Torgerson [Phys. Rev. A 51, 4400 (1995)]. We show that its derivation does not require anything but realism and locality, contrary to what has been previously assumed. It can thus be considered a genuine Bell inequality, appropriate for testing local realism. Similar, homogeneous inequalities can be analogously derived. They constitute a promising family that is likely to serve as a basis for loophole-free tests of local realism. The existence of such a family proves false the assertion that all genuine Bell inequalities must be inhomogeneous.