Bell Inequality Violations Under Reasonable and Under Weak Hypotheses

Given a sequence of pairs ($p_i$, ${\overline{p}}_i$) of spin-$1/2$ particles in the singlet state, assume that Alice measures the normalized projections $a_i$ of the spins of the $p_i$’s along vector $\mathit{a}$ while Bob measures the normalized projections $b_i$ of the spins of the ${\overline{p}}_i$’s along vector $\mathit{b}$. Then quantum mechanics (QM) lets one evaluate the correlation $⟨a,b⟩$ as $-\cos ({\theta }_{\mathit{a}}-{\theta }_{\mathit{b}})$ where ${\theta }_{\mathit{v}}$ is the angle between the vector $\mathit{v}$ and a reference vector chosen once and for all and in a fixed plane. Assuming classical microscopic realism (CMR) there exist also normalized projection pairs ($a_i^{\prime }$, $b_i^{\prime }$) of the spins of the pairs ($p_i$, ${\overline{p}}_i$) along (${\mathit{a}}^{\prime }$, ${\mathit{b}}^{\prime }$) so that $⟨a^{\prime },b^{\prime }⟩=-\cos ({\theta }_{{\mathit{a}}^{\prime }}-{\theta }_{{\mathit{b}}^{\prime }})$. Since all projections are in $\{-1,1\}$, $|⟨c,d⟩+⟨c,e⟩|+|⟨f,d⟩-⟨f,e⟩|\le 2$ for $c$, $d$, $e$, and $f$ in $\{a,b,a^{\prime },b^{\prime }\}$. Assuming locality (the impossibility of any effect of an event on another event when said events are spatially separated) beside QM and CMR, Bell’s theory lets one deduce various violations of this inequality at some choices of quadruplets $Q\equiv (\mathit{a},\mathit{b},{\mathit{a}}^{\prime },{\mathit{b}}^{\prime })$. Our main result is the existence of such $Q$’s where at least one of the above inequalities is violated if one only assumes QM, CMR, and some very mild further hypotheses that only concern the behavior of correlations that appear in these inequalities near special $Q$’s.

Figure 1

The family ${\mathcal{F}}_{a,b}$ for $({\theta }_{\mathbf{a}},-{\theta }_{\mathit{b}})\in [0,\pi {]}^2$. The pairs are the values of $⟨a,a^{\prime }⟩$ and $⟨b,b^{\prime }⟩$ and correspond to exact computations if they are framed, and to reasonable expectation of the written values otherwise. For further symbols see the main text.