Qm Bell Models And Measurement Limits — Examples

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 over the two declared spin-half factors $V,V$ with basis $(e_{+},e_{-})$. Assume the Axiom of Countable Choice and the Axiom of Choice, as required by the cited singlet, spin-spectrum and arrangement suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Fix the orthonormal vectors $e_1:=(1,0,0)$, $e_2:=(0,1,0)$, $e_3:=(0,0,1)$ and the four unit directions $a_0:=e_3,a_1:=e_1,b_0:=\frac{e_3+e_1}{\sqrt{2}},b_1:=\frac{e_3-e_1}{\sqrt{2}}\mathrm{.}$ Consider two arrangements over these same directions:

• the singlet arrangement, with the singlet preparation ${\rho }_{\mathrm{\Psi }}=T_{\mathrm{\Psi },\mathrm{\Psi }}$, $\mathrm{\Psi }=\frac{1}{\sqrt{2}}(\kappa (e_{+}\otimes e_{-})-\kappa (e_{-}\otimes e_{+}))$;
• the product-mixture arrangement, with the EPR-type mixture $M:=\frac{1}{2}{T}_{e_{+}\otimes e_{-},e_{+}\otimes e_{-}}+\frac{1}{2}{T}_{e_{-}\otimes e_{+},e_{-}\otimes e_{+}},$ a convex combination of the two product preparations of definite opposite spins along $e_3$.

Then:

(a) for the singlet arrangement the four correlations are $E_{00}=E_{01}=E_{10}=-\frac{1}{\sqrt{2}},E_{11}=+\frac{1}{\sqrt{2}},S({\nu }^{{\rho }_{\mathrm{\Psi }}})=-2\sqrt{2};$

(b) for the product-mixture arrangement the four sign tables are ${\nu }_{ij}^M(\epsilon ,\delta )=\frac{1}{4}(1-\epsilon \delta (a_i{)}_3(b_j{)}_3),$ so its correlations are $E_{ij}=-(a_i{)}_3(b_j{)}_3$; at the displayed directions $(a_0{)}_3=1$, $(a_1{)}_3=0$ and $(b_0{)}_3=(b_1{)}_3=1\mathrm{/}\sqrt{2}$, hence $E_{00}=E_{01}=-\frac{1}{\sqrt{2}},E_{10}=E_{11}=0,S({\nu }^M)=-\sqrt{2};$

(c) the mixture's tables admit a local-response model (Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws): on the two-point specification space, the response laws ${\alpha }_i(1,\epsilon )=\frac{1}{2}(1+\epsilon (a_i{)}_3),{\alpha }_i(2,\epsilon )=\frac{1}{2}(1-\epsilon (a_i{)}_3),{\beta }_j(1,\delta )=\frac{1}{2}(1-\delta (b_j{)}_3),{\beta }_j(2,\delta )=\frac{1}{2}(1+\delta (b_j{)}_3)$ with specification weights $\frac{1}{2},\frac{1}{2}$ reproduce ${\nu }^M$; consequently $\mid S({\nu }^M)\mid =\sqrt{2}\le 2$, as the bound for representable models requires (The CHSH bound for every binary correlation model admitting a local-response representation);

(d) both arrangements have the same marginal sign laws ${\nu }^A(\pm )={\nu }^B(\pm )=\frac{1}{2}$ for every direction, and the same reduced preparation ${\mathrm{Tr}}_{B}M=\frac{1}{2}I={\mathrm{Tr}}_B{\rho }_{\mathrm{\Psi }}$ on each factor, although their joint tables and CHSH functionals differ: $\mid S({\nu }^{{\rho }_{\mathrm{\Psi }}})\mid =2\sqrt{2}>2\ge \sqrt{2}=\mid S({\nu }^M)\mid$.

All clauses are conditional computations in the declared framework; no measured correlation is asserted.

• Why the mixture does not distinguish the two routes. The mixture's tables are represented by a probabilistic local-response model with two specifications, not by a deterministic response function of a single specification; this is why clause (a) of Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws is stated for response laws rather than for response functions. The singlet's tables at the displayed directions admit no representation of either form.
• The lesson of the comparison. Identical marginals and identical reduced preparations on both factors do not determine the joint tables: the contrast between $2\sqrt{2}$ and $\sqrt{2}$ is invisible to every single-station statistic and is a property of the correlations alone.

Adopt a declared spin correlation arrangement $(V\widehat{\otimes }V,\rho ,a_0,a_1,b_0,b_1)$ of Declared spin correlation arrangement of the framework: two spin-half factors, four directions, joint sign tables and local-response models with its outcome laws ${\nu }_{ij}^{\rho }(\epsilon ,\delta )$, correlations $E_{ij}=E_{ij}({\nu }^{\rho })$ and CHSH functional $S=S({\nu }^{\rho })$. Let $n\ge 1$ and declare the following sampling model (the statistical model of clause (ii) of the preparation and apparatus postulate P7, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions): on a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ let $({\mathrm{\Sigma }}_{ijk},{\mathrm{T}}_{ijk}{)}_{\begin{array}{c}i,j\in \{0,1\}\\ k\in \{1,\dots ,n\}\end{array}}$ be $4n$ random elements with values in $\mathrm{\Sigma }\times \mathrm{\Sigma }$ that are mutually independent and have the declared laws $\mathbb{P}(({\mathrm{\Sigma }}_{ijk},{\mathrm{T}}_{ijk})=(\epsilon ,\delta ))={\nu }_{ij}^{\rho }(\epsilon ,\delta )\text{for every }i,j,k,\epsilon ,\delta$ (Pairwise and mutual independence of finite-valued random variables, Random elements and real random variables). Define the sample correlations and the sample CHSH functional ${\widehat{E}}_{ij}:=\frac{1}{n}{\sum }_{k=1}^n{\mathrm{\Sigma }}_{ijk}{\mathrm{T}}_{ijk},\widehat{S}:={\widehat{E}}_{00}+{\widehat{E}}_{01}+{\widehat{E}}_{10}-{\widehat{E}}_{11}\mathrm{.}$ Then:

(a) $\mathbb{E}[{\widehat{E}}_{ij}]=E_{ij}$ and $\mathrm{Var}({\widehat{E}}_{ij})\le 1\mathrm{/}n$ for all $(i,j)$;

(b) $\mathbb{E}[\widehat{S}]=S$, $\mathrm{Var}(\widehat{S})\le 4\mathrm{/}n$, and for every $\epsilon >0$ $\mathbb{P}(\mathrm{\mid }\widehat{S}-S\mathrm{\mid }\ge \epsilon )\le \frac{4}{n{\epsilon }^2};$

(c) threshold consequences: if the declared model satisfies $S\le 2-\epsilon$ for some $\epsilon >0$, then $\mathbb{P}(\widehat{S}>2)\le 4\mathrm{/}(n{\epsilon }^2)$; if the declared model is the singlet arrangement with $S=-2\sqrt{2}$ and $\epsilon :=2\sqrt{2}-2$, then $\mathbb{P}(\widehat{S}\ge -2)\le 4\mathrm{/}(n{\epsilon }^2)$. Moreover, for every declared acceptance criterion $C\subseteq (\mathrm{\Sigma }\times \mathrm{\Sigma }{)}^{4n}$ that rejects at least one outcome sequence $(x_{ijk})$ with ${\prod }_{i,j,k}{\nu }_{ij}^{\rho }(x_{ijk})>0$, the failure probability of the criterion is at least ${\prod }_{i,j,k}{\nu }_{ij}^{\rho }(x_{ijk})>0$;

(d) scope: the independence, the identical distribution and the tables ${\nu }^{\rho }$ are declared model assumptions of the sampling model, not consequences of P1-P6; clauses (a)-(c) assert no measured count, no p-value, no error rate, no power, no required sample size and no empirical conclusion.

• Finite-sample threshold crossings. At the displayed singlet settings, each of the four possible outcome pairs has strictly positive probability. A sequence with product signs $+1$ at the first three setting pairs and $-1$ at the last therefore has positive probability and gives $\widehat{S}=4\ge -2$. For a representable model with $S<2$, a crossing $\widehat{S}>2$ has positive probability only when its declared tables support at least one such sequence, as specified by clause (c); a deterministic table need not support one. The displayed Chebyshev upper bounds remain valid independently of that support condition. More generally, if two specified finite-sample models both assign positive probability to the same record, that record is possible under both and cannot deductively exclude either; a record assigned probability zero by one specified ideal model can exclude that model conditional on its sampling and apparatus assumptions. Thus this item makes no blanket claim that finite samples cannot distinguish all model pairs. No optimality, power or sample-size assertion is made.
• No inference is drawn. The clauses are probabilities inside the declared sampling model. They do not compute a p-value, do not attach a confidence level and do not assert anything about a real experiment; the preparation, detection and selection qualifications remain as stated in Measurement-model boundaries of the singlet-local comparison: what the no-go excludes and what it does not.

• Recorded tables are conditional tables. The arrangement's laws ${\nu }_{ij}^{\rho }$ of Declared spin correlation arrangement of the framework: two spin-half factors, four directions, joint sign tables and local-response models clause (d) describe a declared pair of local sign measurements for every setting pair on the declared preparation. A comparison with recorded data ordinarily uses only the runs in which both stations register an event; the laws of the recorded runs are therefore conditional laws of a larger model, and they are not the arrangement's tables unless a detection and selection model connecting the two is declared. This is part of the preparation and apparatus discipline of P7 clause (i) (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions).

• A sufficient condition for the transfer. If the full tables admit a local-response representation and the recorded runs are exactly those specifications lying in a measurable selection event $D$ of the specification space with $\mathbb{P}(D)>0$ — membership is determined by the specification alone, using the same event for every setting pair — then the conditional tables again admit a local-response representation, with the same response laws over the conditional measure, and hence satisfy the CHSH bound (Conditioning on a setting-independent selection preserves local-response representability and the CHSH bound). This is the mathematical content of a setting-independent detection or fair-sampling hypothesis for the declared model class. It does not require statistical independence of $D$ from the outcomes: reweighting the specifications can change the tables.

• The condition is a hypothesis, not a theorem about detectors. The selection-closure lemma is a conditional statement about the declared model. It does not assert that any laboratory detector, coincidence circuit or post-selection rule realizes a selection event of the required form; identifying a concrete apparatus with such a model is a tested identification with uncertainty under P7 and carries its calibration and control-experiment qualifications. Efficiency, dead time, misalignment, background and drift each require their own declared model.

• No transfer without a selection model. If the recorded sample is selected by a rule that depends on the settings or on the observed outcomes, the selection-closure lemma does not apply, and no claim is made here that the CHSH bound constrains the recorded tables. Correspondingly, a deviation computed from recorded pairs is a statement about those recorded tables and their own declared model; it becomes a statement about the full arrangement only through an explicitly justified detection and selection model; the condition above is sufficient, not necessary. Finite-sample fluctuations are a further, separate matter. The result Finite samples of a declared arrangement: the CHSH estimator, its concentration and the limits of a finite-sample verdict assumes $n$ trials per setting pair, all $4n$ outcome pairs mutually independent, with each trial law equal to the arrangement's table ${\nu }_{ij}^{\rho }$. It applies to recorded pairs only when their declared sampling model satisfies those hypotheses. A postselected law different from ${\nu }_{ij}^{\rho }$, random recorded trial counts or correlations induced by selection require a separate analysis; the cited result does not supply one. The singlet contrast itself is the conditional model computation of The singlet arrangement: directional sign tables, CHSH value 2 sqrt 2 and the failure of local-response models at violating settings.

• Settings choice. In a specified experiment model, let $Q_{ij}$ be the actual conditional probability law of the specification given that setting pair $(i,j)$ is chosen whenever that conditional law is defined. The common-law condition required by a local-response representation is that these actual conditional laws are all one common law, $Q_{00}=Q_{01}=Q_{10}=Q_{11}$ (Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws clause (c)); if a setting pair is never chosen, its law must still be specified by the comparison model before the four-table representation is applied. The possibility that a setting procedure could depend on the specification does not establish that the actual conditional laws differ; an analysis using a local-response representation must establish their equality from its model and the relevant evidence. This page neither assumes nor reports such a setting procedure or its conditional laws.

• What the page contains. Three conditional computations in the declared framework: the worked comparison at the specified directions between the singlet arrangement and the EPR-type product mixture, with both CHSH values and an explicit local-response representation of the mixture (Worked Bell angles: the singlet against the EPR product mixture at the same directions); the finite-sample behaviour of the sample CHSH functional under a declared independent and identically distributed sampling model, with its concentration inequality and positive failure probabilities (Finite samples of a declared arrangement: the CHSH estimator, its concentration and the limits of a finite-sample verdict); and the selection and loss qualification, with the precise hypothesis under which the CHSH bound transfers to recorded pairs (Selection and loss: the precise hypothesis under which the CHSH bound transfers to recorded pairs).

• What the page does not contain. No experiment item, no reported count, no calibration, no detector efficiency, no loss fraction, no background model, no trial-level uncertainty analysis, no p-value and no confidence level. The worked tables, the sample functional and the failure probabilities are probabilities of the declared models; no number on the page is an observation. The identification of any apparatus with the declared measurements and of any repetition procedure with the declared sampling model is a tested identification with uncertainty under P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions).

• Deferred Bell-test experiment destination. The design of the pair reserves an experiment item for a Bell test. It is not built here because none of the sources read for the pair supplies the required trial-level material: a primary or authoritative full report read directly, with the source-specific preparation and apparatus, calibration, measured outputs, trial count where reported, detection efficiency, losses and post-selection, background, the reported statistical treatment and systematics, and the source's omissions. The experimental remarks of the secondary sources — Binney's reference to the 1972 Freedman-Clauser test and its citation of a 1998 photonic test, and the MIT chapter's statement that experiments have confirmed violations — motivate the comparison only and are not used as measured records; no numerical value from them is reproduced. The statistics and primary-report gap named in the design is recorded as an open destination.

• Why a null or shifted sample is not a verdict. The mixture's tables have $S=-\sqrt{2}$, so $\mathrm{\mid }S\mathrm{\mid }=\sqrt{2}$, strictly below the local-response CHSH bound $2$ (The CHSH bound for every binary correlation model admitting a local-response representation) and the singlet's $2\sqrt{2}$ (Worked Bell angles: the singlet against the EPR product mixture at the same directions). At the worked angles, every singlet table cell has probability $\frac{1}{4}(1\pm 1\mathrm{/}\sqrt{2})>0$, and every mixture table cell has probability either $\frac{1}{4}$ or $\frac{1}{4}(1\pm 1\mathrm{/}\sqrt{2})>0$; hence each finite sequence has positive probability under both specified independent-sampling models. Such a record cannot deductively exclude either member of this pair, though statistical evidence can favor one under an explicit comparison procedure and its assumptions. For a different specified pair, an observed sequence assigned probability zero by one ideal model can exclude that model conditional on its sampling assumptions. The worked comparison's spin calculations carry the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) and the Axiom of Choice (The Axiom of Choice) stated in their supplier hypotheses. The finite-sample analysis bounds the probability that the sample functional crosses a stated threshold under its declared model (Finite samples of a declared arrangement: the CHSH estimator, its concentration and the limits of a finite-sample verdict); it gives no universal impossibility result for all finite-sample comparisons. Any comparison also requires the preparation, detection, selection and sampling assumptions, and agreement is evidence about the combined assumptions, not a proof of the framework.

• Model contrast is not an interpretation. The page compares the tables of two declared arrangements and the class of local-response models of those tables (The CHSH bound for every binary correlation model admitting a local-response representation); it makes no claim about the completeness of quantum mechanics, about hidden variables in general, about mechanisms or about any interpretation of the violation. The boundaries of the comparison are recorded on the pair's first page.