Minimal assumption derivation of a weak Clauser–Horne inequality

Abstract

According to Bell's theorem a large class of hidden-variable models obeying Bell's notion of local causality conflict with the predictions of quantum mechanics. Recently, a Bell-type theorem has been proven using a weaker notion of local causality, yet assuming the existence of perfectly correlated event types. Here we present a similar Bell-type theorem without this latter assumption. The derived inequality differs from the Clauser-Horne inequality by some small correction terms, which render it less constraining.

Bell-Type Inequalities from Separate Common Causes

“…Together with the assumption of separate common causes (and traditional locality and independence assumptions), an inequality was derived that is of the same form as the Clauser-Horne inequality (Clauser and Horne 1974) but less restrictive by essentially the amount . Quantum mechanical predictions contradict the predictions of a common-cause model obeying the assumptions of Portmann and Wüthrich (2007) for ≤ 2.689 · 10 −5 . It is, however, extremely difficult to obtain such high precision in experiments confirming perfect anticorrelation, but only then is the inequality violated by the quantum mechanical predictions and experimental data.…”

“…The minimality of the set of assumptions of Graßhoff et al (2005) is, therefore, only relative to the group of derivations that do not assume perfect anticorrelations. A derivation of a Bell-type inequality from separate common causes without the assumption of perfect anticorrelations was published in April 2006 as an electronic preprint before being published in print (Portmann and Wüthrich 2007). A similar derivation is forthcoming (in Hofer-Szabó forthcoming).…”

“…In Portmann and Wüthrich (2007) a small deviation from perfect anticorrelation was allowed, as for instance: p(L + |R − M i N i ) = 1 − . Together with the assumption of separate common causes (and traditional locality and independence assumptions), an inequality was derived that is of the same form as the Clauser-Horne inequality (Clauser and Horne 1974) but less restrictive by essentially the amount .…”

“…It is, however, extremely difficult to obtain such high precision in experiments confirming perfect anticorrelation, but only then is the inequality violated by the quantum mechanical predictions and experimental data. Thus, common-cause models obeying the assumptions of Portmann and Wüthrich (2007) can hardly be ruled out by empirical data.…”

“…A principal outcome of the project is that a Bell-type inequality can be derived from the assumption of separate common causes (Graßhoff, Portmann and Wüthrich 2005), even without the assumption of perfectly anticorrelating event types (Portmann and Wüthrich 2007). We also address the critique that in Graßhoff et al (2005) we implicitly assume a common common cause (Hofer-Szabó forthcoming).…”