# Does the CHSH inequality fully characterise the local polytope?

Consider the standard two-party CHSH scenario. Each party can perform one of two measurements (denoted with $$x,y\in\{0,1\}$$) and observe one of two outcomes (denoted with $$a,b\in\{0,1\}$$).

Let $$P(ab|xy)$$ be the probability of observing outcomes $$a,b$$ when choosing the measurements settings $$x,y$$. Local realistic theories are those that, for some probability distribution over some hidden variable $$\lambda$$, satisfy $$P(ab|xy)=\sum_\lambda q(\lambda)P_\lambda(a|x)P_\lambda(b|y).\tag1$$ Define the local polytope $$\mathcal L$$ as the set of theories that can be written as in (1). Note that we identify here a theory with its set of conditional probabilities: $$\boldsymbol P\equiv (P(ab|xy))_{ab,xy}$$.

Denote with $$E_{xy}$$ the expectation values given by $$E_{xy}=\sum_{ab}(-1)^{a+b}P(ab|xy)$$. We then know that all local realistic theories (i.e. all theories $$\boldsymbol P\in\mathcal L$$) satisfy the CHSH inequality: $$\Big|\sum_{xy}(-1)^{xy} E_{xy}\Big| = |E_{00}+ E_{01} + E_{10} - E_{11}| \le 2.\tag2$$ Is the opposite true? In other words, do all theories satisfying (2) admit local realistic explanations?

• this means that the local polytope has exactly two nontrivial (nontrivial meaning excluding the faces of the form $P(ab|xy)\ge0$) facets, correct? One for $S=2$ and the other one for $S=-2$. I ask because this paper seems to state that the only facet is $S=2$ (see eq. 2) – glS Mar 23 at 9:50