The singlet arrangement: directional sign tables, CHSH value 2 sqrt 2 and the failure of local-response models at violating settings

Adopt the declared spin correlation arrangement of Declared spin correlation arrangement of the framework: two spin-half factors, four directions, joint sign tables and local-response models: the composite $V\widehat{\otimes }V$ of two declared spin-half factors with orthonormal basis $(e_{+},e_{-})$, a declared preparation $\rho \in D(V\widehat{\otimes }V)$, the setting directions $a_0,a_1,b_0,b_1\in {\mathbb{R}}^3$ and the four declared joint sign measurements with outcome laws ${\nu }_{ij}^{\rho }(\epsilon ,\delta )=\mathrm{tr}(\rho (P_{a_i,\epsilon }\otimes P_{b_j,\delta })),P_{a,\epsilon }=\frac{1}{2}(I+\epsilon a\cdot \sigma )\mathrm{.}$ Assume the Axiom of Countable Choice and the Axiom of Choice, the hypotheses of the cited suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice), and take the singlet preparation $\mathrm{\Psi }:=\frac{1}{\sqrt{2}}(\kappa (e_{+}\otimes e_{-})-\kappa (e_{-}\otimes e_{+})),{\rho }_{\mathrm{\Psi }}:=T_{\mathrm{\Psi },\mathrm{\Psi }},$ where $T_{\mathrm{\Psi },\mathrm{\Psi }}$ is the rank-one operator of Rank-one operator calculus. Then:

(a) for every $(i,j)\in \{0,1{\}}^2$ and all $\epsilon ,\delta \in \mathrm{\Sigma }$, ${\nu }_{ij}^{{\rho }_{\mathrm{\Psi }}}(\epsilon ,\delta )=⟨(P_{a_i,\epsilon }\otimes P_{b_j,\delta })\mathrm{\Psi },\mathrm{\Psi }⟩=\frac{1}{4}(1-\epsilon \delta a_i\cdot b_j);$

(b) the correlations of the arrangement are $E_{ij}({\nu }^{{\rho }_{\mathrm{\Psi }}})=-a_i\cdot b_j$ and $S({\nu }^{{\rho }_{\mathrm{\Psi }}})=-[a_0\cdot b_0+a_0\cdot b_1+a_1\cdot b_0-a_1\cdot b_1]=-(a_0\cdot (b_0+b_1)+a_1\cdot (b_0-b_1));$ in particular, for the unit directions $a_0=(0,0,1)$, $a_1=(1,0,0)$, $b_0=(1,0,1)\mathrm{/}\sqrt{2}$ and $b_1=(-1,0,1)\mathrm{/}\sqrt{2}$ one has $S({\nu }^{{\rho }_{\mathrm{\Psi }}})=-2\sqrt{2}$, so $\mid S({\nu }^{{\rho }_{\mathrm{\Psi }}})\mid =2\sqrt{2}>2$;

(c) consequently the correlation table ${\nu }^{{\rho }_{\mathrm{\Psi }}}$ of the arrangement at those directions admits no local-response model in the sense of clause (e) of Declared spin correlation arrangement of the framework: two spin-half factors, four directions, joint sign tables and local-response models: there is no local-response representation of the four tables (Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws) with these laws;

(d) the marginal sign laws of the arrangement are ${\nu }_{ij}^{{\rho }_{\mathrm{\Psi }},A}(\epsilon )=\frac{1}{2}={\nu }_{ij}^{{\rho }_{\mathrm{\Psi }},B}(\delta )$ for every setting pair and every direction, so the table is no-signalling in the sense of clause (c) of Binary correlation models: setting-indexed tables of two-valued outcome laws, marginals and the CHSH functional; and a discarded declared local instrument on either factor leaves the reduced preparation of the other factor and every local outcome law of that factor unchanged (Local instruments and no-signalling: operations on one factor do not change the other factor's local statistics clause (b)).

The item is a conditional model computation: it asserts no measured correlation, no apparatus and no empirical fact, and its exclusion is exactly of the declared local-response class of these four tables. Higher singleton claims about hidden variables in general are not asserted.

• The value $2\sqrt{2}$ is the largest possible for this arrangement. The general bound $\mathrm{\mid }S\mathrm{\mid }\le 2\sqrt{2}$ over every preparation and every choice of directions is proved as Tsirelson bound for every declared spin correlation arrangement of the framework; the present computation shows that the singlet at the displayed directions attains it.
• What is excluded. The four tables of the arrangement do not admit a representation by one setting-independent specification space with per-station response laws. The computation does not address models of these tables outside that class, and it makes no claim that any laboratory procedure realizes the arrangement or that any measured correlation agrees with ${\nu }^{{\rho }_{\mathrm{\Psi }}}$.