Minimal Assumption Derivation of a Bell-type Inequality

Abstract

John Bell showed that a big class of local hidden-variable models stands in conflict with quantum mechanics and experiment. Recently, there were suggestions that empirical adequate hidden-variable models might exist, which presuppose a weaker notion of local causality. We will show that a Belltype inequality can be derived also from these weaker assumptions.

Minimal assumption derivation of a weak Clauser–Horne inequality

“…In contrast to Graßhoff et al (2005), the weakening is reflected in the derived inequality: We get the Clauser-Horne inequality with small correction terms rendering our inequality less constraining. 2 There are many different Bell-type theorems with different aims, the weakening of the assumptions being one among many objectives.…”

“…In this article we continue the work of Graßhoff, Portmann, and Wüthrich (2005) 1 and prove a Bell-type theorem from a still weaker set of assumptions. In contrast to Graßhoff et al (2005), the weakening is reflected in the derived inequality: We get the Clauser-Horne inequality with small correction terms rendering our inequality less constraining.…”

“…Mathematically, (13) is stronger than (12). Indeed, as already pointed out by Butterfield (1989) (p. 123), to get from (12) to (13) one needs at least one further assumption (see also Graßhoff et al (2005), p. 15 et seq., and Henson (2005), p. 532 et seq.). The additional assumption states that the statistical independence in (12) does not get disrupted if one (additionally to C abAB i ) conditionalizes on other events which are not causally relevant for A a and B b .…”

“…is thus not clear whether a Bell-type theorem can still be proven with PCC. We used PCC as our causality condition in Graßhoff et al (2005) (GPW) for a proof of a Bell-type theorem, but the minimality of the logical strength of the assumptions was only relative (see figure 1), because we also assumed PCORR. Given this assumption, our set of assumptions was minimal.…”

“…As in Graßhoff et al (2005) there is a common cause C abAB for each quadruple of measurement directions and outcomes (a, b, A a , B b ). That is different from other derivations, where a single common cause {C i } i∈I is stipulated for all correlated events:…”