control over the input distribution $p(x,y)$ it has become customary to work with the conditional probability distribution $p(a,b \vert x,y)$. Within this context, the fundamental question is to understand what are the constraints implied by different kind of theories on the possible correlations that can be observed in a Bell experiment.

Under the assumption known as local realism, the probability distribution should be decomposable as $$\label{pc}
p(a,b \vert x,y) = \sum_{\lambda}p(\lambda) p(a \vert x, \lambda) p(b \vert y, \lambda),$$ defining a set of correlation $\mathcal{L}$ where we have explicitly imposed the following conditions. First, the “realism” condition implying the existence of a hidden variable $\Lambda$ determining the probabilities of measurement outcomes even of measurement choices that have not been performed. Second, the locality condition stating that only variables in the causal past of the measurement outcomes might have a causal influence over their statistics, that is, $p(a\vert x,y,b,\lambda)= p(a\vert x,\lambda)$ (similarly for Bob). Finally, we also have to impose the measurement independence assumption (also known as “free will”) implying that $p(x,y,\lambda)=p(x,y)p(\lambda)$.

In turn, the quantum description for this experiment, implies via the Born rule that the distribution should be written as $$\label{pq}
p(a,b \vert x,y) = \mathrm{Tr}\left[\left(M_a^x \otimes M_b^y \right) 