Binary correlation models: setting-indexed tables of two-valued outcome laws, marginals and the CHSH functional

This item fixes finite combinatorial vocabulary for outcome laws of a two-station comparison. No physical interpretation, measurement model, dynamics or locality assertion is made here; the physical readings are fixed on the page of this pair. No choice principle is used.

(a) Binary correlation model. Let $I$ and $J$ be finite nonempty sets, the setting labels of the two stations, and put $\mathrm{\Sigma }:=\{-1,+1\}$ for the common outcome labels. A binary correlation model on $(I,J)$ is a family $\mu =({\mu }_{ij}{)}_{(i,j)\in I\times J},{\mu }_{ij}:\mathrm{\Sigma }\times \mathrm{\Sigma }\to [0,1],$ of functions with the normalization ${\sum }_{\epsilon \in \mathrm{\Sigma }}{\sum }_{\delta \in \mathrm{\Sigma }}{\mu }_{ij}(\epsilon ,\delta )=1\text{for every }(i,j)\in I\times J$ (Finite sums and finite products, by recursion). Each ${\mu }_{ij}$ is the joint law of the setting pair $(i,j)$: it is the outcome-weight function of a finite probability space on the four outcomes $\mathrm{\Sigma }\times \mathrm{\Sigma }$ in the sense of Finite probability spaces, outcome weights, events, and event probabilities, and sums over $\mathrm{\Sigma }\times \mathrm{\Sigma }$ are finite sums.

(b) Marginals. For $(i,j)\in I\times J$ the marginal law of the first station and the marginal law of the second station are ${\mu }_{ij}^A(\epsilon ):={\sum }_{\delta \in \mathrm{\Sigma }}{\mu }_{ij}(\epsilon ,\delta ),{\mu }_{ij}^B(\delta ):={\sum }_{\epsilon \in \mathrm{\Sigma }}{\mu }_{ij}(\epsilon ,\delta ),\epsilon ,\delta \in \mathrm{\Sigma }\mathrm{.}$

(c) No-signalling. The model $\mu$ is no-signalling when the marginal law of the first station does not depend on the distant setting, ${\mu }_{ij}^A={\mu }_{i{j}^{\mathrm{\prime }}}^A$ for all $i\in I$ and all $j,j^{\mathrm{\prime }}\in J$, and the marginal law of the second station does not depend on the distant setting, ${\mu }_{ij}^B={\mu }_{i^{\mathrm{\prime }}j}^B$ for all $j\in J$ and all $i,i^{\mathrm{\prime }}\in I$. The word is used as a name for this pair of identities about the tables; no spatial or causal claim is attached to it.

(d) Correlations and the CHSH functional. Assume $I=J=\{0,1\}$. The correlation of the setting pair $(i,j)$ and the CHSH functional of the model are the real numbers $E_{ij}(\mu ):={\sum }_{\epsilon \in \mathrm{\Sigma }}{\sum }_{\delta \in \mathrm{\Sigma }}\epsilon \delta {\mu }_{ij}(\epsilon ,\delta ),S(\mu ):=E_{00}(\mu )+E_{01}(\mu )+E_{10}(\mu )-E_{11}(\mu )\mathrm{.}$

(e) Status. Clauses (a)-(d) introduce finite tables, their marginals, the no-signalling identities and one linear functional of four correlations. They assert no physical assumption, no dynamics, no measurement, no probability law for a real apparatus and no statement about nature; the identification of a laboratory arrangement with a model $\mu$ is a separate physical matter.