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A setting-independent hidden variable fixes responses for every measurement choice, and each response depends only on the local setting. Stochastic local hidden-variable theories can be represented as mixtures of these assignments by including local random seeds in the hidden variable. Deterministic counterfactual assignments are the starting point of the chained modular Bell inequality.

Fix a hidden variable in the deterministic local hidden-variable model, so that every $A_j,B_j$ is an integer. Let $[x{]}_d$ be the representative of $x$ modulo $d$ in $\{0,\dots ,d-1\}$. The terms in the chained modular Bell inequality alternate between the two parties. Before reduction their sum telescopes:After reduction, the sum $T$ is a nonnegative integer congruent to $-1$ modulo $d$. For $d\ge 2$, the smallest possible such integer is $d-1$. Thus $T\ge d-1$ for each hidden variable separately. Averaging over the setting-independent distribution givesHere each expectation is the expectation value of the reduced random variable, not the residue of its expectation. Each term can be measured using one setting at each site. The proof uses their common deterministic assignments rather than any joint quantum measurement of incompatible local settings. Stochastic local hidden-variable theories satisfy the same bound: include their local random seeds in $\lambda$ and average the resulting deterministic assignments.

The printed final coefficient in the definition of the average is typographically incomplete. The expectation used here is the usual $\mathbb{E}X={\sum }_{x=0}^{d-1}xP(X=x)$, with final coefficient $d-1$.

Take ${\alpha }^2+{\beta }^2=1$ and label the first vector of each measurement basis by outcome $0$. Assign the Pauli measurement value $+1$ to outcome $0$ and $-1$ to outcome $1$. The four observables areHere $X$ and $Z$ are the Pauli X gate and Pauli Z gate matrices. These follow by subtracting the two rank-one basis projectors; a real basis rotated through $\gamma$ has observable $\sin (2\gamma )X+\cos (2\gamma )Z$.

The Schmidt-basis Pauli correlation tensor of the state givesConsequently $E_{11}=1/2$, $E_{21}=E_{22}=\sqrt{3}\alpha \beta$ and $E_{12}=-1/2$, where $E_{jk}=⟨A_j\otimes B_k⟩$. For binary outcomes, $\mathbb{E}([A-B{]}_2)=P(A\ne B)=(1-E_{AB})/2$, whereas the final offset term has $\mathbb{E}([B-A-1{]}_2)=P(A=B)=(1+E_{AB})/2$. The chained modular Bell inequality left side is thereforeThe local bound is $I\ge 1$, so the exact violation condition isEquality saturates the bound. The maximal violation for these fixed measurements occurs at $\alpha =\beta =1/\sqrt{2}$ or their common negative, giving $I=(3-\sqrt{3})/2$. Opposite signs do not violate this particular inequality with these fixed bases, although other measurement choices can reveal the state's entanglement. If unnormalized real amplitudes are used, replace $\alpha \beta$ throughout by $\alpha \beta /({\alpha }^2+{\beta }^2)$.