Qm Bell Models And Measurement Limits

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.

Let $\mu$ be a binary correlation model on the finite setting sets $I,J$ with outcome labels $\mathrm{\Sigma }=\{-1,+1\}$ (Binary correlation models: setting-indexed tables of two-valued outcome laws, marginals and the CHSH functional). No choice principle is used in this definition.

(a) Local-response representation. A local-response representation of $\mu$ is a tuple consisting of a probability space $(\mathrm{\Lambda },{\mathrm{\Sigma }}_{\mathrm{\Lambda }},\mathbb{P})$ (Measure spaces, Probability measures and probability spaces) and two families of response laws $({\alpha }_i{)}_{i\in I},({\beta }_j{)}_{j\in J},$ where for each $i\in I$ the function ${\alpha }_i:\mathrm{\Lambda }\times \mathrm{\Sigma }\to [0,1]$ has the two properties ${\sum }_{\epsilon \in \mathrm{\Sigma }}{\alpha }_i(\lambda ,\epsilon )=1(\lambda \in \mathrm{\Lambda }),\lambda \mapsto {\alpha }_i(\lambda ,+1)\text{ is measurable}$ (A measurable function between measurable spaces), and the family $({\beta }_j{)}_{j\in J}$ has the same two properties; and the representation identity ${\mu }_{ij}(\epsilon ,\delta )={\int }_{\mathrm{\Lambda }}{\alpha }_i(\lambda ,\epsilon ){\beta }_j(\lambda ,\delta )d\mathbb{P}(\lambda )={\mathbb{E}}_{\mathbb{P}}\text{\hspace{0.17em}⁣}[{\alpha }_i(\cdot ,\epsilon ){\beta }_j(\cdot ,\delta )]$ holds for every $(i,j)\in I\times J$ and every $(\epsilon ,\delta )\in \mathrm{\Sigma }\times \mathrm{\Sigma }$. The integrand is a product of two bounded measurable $[0,1]$-valued functions, so the expectation exists as a finite number (Expectation of a nonnegative or integrable random variable, Random elements and real random variables). The functions ${\alpha }_i(\cdot ,-1)=1-{\alpha }_i(\cdot ,+1)$ and likewise for ${\beta }_j$ are measurable as well. The variable $\lambda$ is called the specification of the representation; it is a mathematical index, and no physical interpretation is asserted.

(b) Response functions (deterministic form). Equip the finite outcome set $\mathrm{\Sigma }=\{-1,+1\}$ with its discrete $\sigma$-algebra $\mathcal{P}(\mathrm{\Sigma })$. A local-response representation has response functions when there are measurable maps $a_i:(\mathrm{\Lambda },{\mathrm{\Sigma }}_{\mathrm{\Lambda }})\to (\mathrm{\Sigma },\mathcal{P}(\mathrm{\Sigma }))$ $(i\in I)$ and $b_j:(\mathrm{\Lambda },{\mathrm{\Sigma }}_{\mathrm{\Lambda }})\to (\mathrm{\Sigma },\mathcal{P}(\mathrm{\Sigma }))$ $(j\in J)$ with ${\alpha }_i(\lambda ,\epsilon )={\mathbf{1}}_{\{a_i=\epsilon \}}(\lambda ),{\beta }_j(\lambda ,\delta )={\mathbf{1}}_{\{b_j=\delta \}}(\lambda ),$ in which case the representation identity reads ${\mu }_{ij}(\epsilon ,\delta )=\mathbb{P}(\{a_i=\epsilon \}\cap \{b_j=\delta \})\mathrm{.}$ The deterministic form is a special case of clause (a): it is obtained by setting the response laws to the Dirac laws at $a_i(\lambda )$ and $b_j(\lambda )$. In this form one and the same specification space and the same single-station functions $a_i$ and $b_j$ serve every setting pair, and the response of a station depends on the specification and on that station's own setting only.

(c) Terminology. The two mathematical features named by local response are: (i) a single specification space carrying the probability law used for every setting pair (setting-independence), and (ii) per-station response laws that do not depend on the distant station's setting. A model that admits such a representation is also called a local model of the tables in the literature; the representation is an existence statement about the tables and asserts no determinism, no mechanism, no dynamics and no causal relation. In particular clause (a) permits probabilistic responses and does not require the response functions of clause (b).

(d) Status. Clauses (a)-(c) fix vocabulary. They assert no physical assumption and no property of any laboratory arrangement; whether a declared physical model of a comparison admits a representation of this form is a mathematical question about its tables, and the physical reading of that question is fixed on the page of this pair.

Let $\mu$ be a binary correlation model on finite setting sets $I,J$ with outcome labels $\mathrm{\Sigma }=\{-1,+1\}$ (Binary correlation models: setting-indexed tables of two-valued outcome laws, marginals and the CHSH functional), and suppose that $\mu$ admits a local-response representation with response laws $({\alpha }_i)$, $({\beta }_j)$ and probability space $(\mathrm{\Lambda },{\mathrm{\Sigma }}_{\mathrm{\Lambda }},\mathbb{P})$ (Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws). Then:

(a) for every $i\in I$ the marginal law of the first station satisfies ${\mu }_{ij}^A={\mu }_{i{j}^{\mathrm{\prime }}}^A\text{for all }j,j^{\mathrm{\prime }}\in J,$ and for every $j\in J$ the marginal law of the second station satisfies ${\mu }_{ij}^B={\mu }_{i^{\mathrm{\prime }}j}^B$ for all $i,i^{\mathrm{\prime }}\in I$; that is, $\mu$ is no-signalling;

(b) consequently, if a binary correlation model is not no-signalling, then it admits no local-response representation.

No physical assumption and no choice principle is used; the proposition is a statement about the tables and their representation.

• Direction of the implication. Representability is a strictly stronger requirement than no-signalling: this proposition gives only the implication from representability to no-signalling. The converse fails, and the framework's singlet arrangement is an explicit failing case once its four tables violate the CHSH bound of the next item.
• Reading. The two identities are the statement that each station's outcome law is unaffected by the distant setting in the model. This is a property of the declared tables; it is not a statement about any laboratory procedure and it assumes no spatial or temporal structure.

Let $\mu$ be a binary correlation model on the setting sets $I=J=\{0,1\}$ with outcome labels $\mathrm{\Sigma }=\{-1,+1\}$, with correlations $E_{ij}(\mu )$ and CHSH functional $S(\mu )$ as in Binary correlation models: setting-indexed tables of two-valued outcome laws, marginals and the CHSH functional clauses (d), and suppose that $\mu$ admits a local-response representation with response laws $({\alpha }_i{)}_{i\in I}$, $({\beta }_j{)}_{j\in J}$ on a probability space $(\mathrm{\Lambda },{\mathrm{\Sigma }}_{\mathrm{\Lambda }},\mathbb{P})$ (Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws). Then $\mid S(\mu )\mid \le 2.$ The statement is a theorem about the tables and their representation; it asserts no measurement, no dynamics, no spatial or causal structure, and it contains no physics. No choice principle is used.

• Deterministic special case. If the representation has response functions $a_i,b_j:\mathrm{\Lambda }\to \mathrm{\Sigma }$ in the sense of clause (b) of Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws, then the four compositions $a_i{b}_j$ are $\pm 1$-valued random variables on one probability space and the present bound reduces to the classical CHSH inequality for such variables, which is also proved directly in The CHSH bound and the finite-sampling law for independent trials (a). The probabilistic form proved here is the general case: it does not assume determinism of the responses.
• Sharpness. The bound $2$ is not improvable for the class: the model $I=J=\{0,1\}$ with ${\mu }_{ij}$ the point mass at $(\epsilon ,\delta )$ with $\epsilon =\delta =+1$ for all $(i,j)$ admits the deterministic representation with $a_i\equiv +1$, $b_j\equiv +1$ and has $S(\mu )=2$; reversing the sign of every $b_j$ gives $S(\mu )=-2$.

Let $\mu$ be a binary correlation model on finite setting sets $I,J$ with outcome labels $\mathrm{\Sigma }=\{-1,+1\}$ (Binary correlation models: setting-indexed tables of two-valued outcome laws, marginals and the CHSH functional), and suppose that $\mu$ admits a local-response representation with response laws $({\alpha }_i{)}_{i\in I}$, $({\beta }_j{)}_{j\in J}$ on a probability space $(\mathrm{\Lambda },{\mathrm{\Sigma }}_{\mathrm{\Lambda }},\mathbb{P})$ (Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws). Let $D\in {\mathrm{\Sigma }}_{\mathrm{\Lambda }}$ be a selection event with $\mathbb{P}(D)>0$, put ${\mathbb{P}}_D(A):=\frac{\mathbb{P}(A\cap D)}{\mathbb{P}(D)}(A\in {\mathrm{\Sigma }}_{\mathrm{\Lambda }})$ for the conditional probability measure on $D$, and define the $D$-conditional tables ${\mu }_{ij}^D(\epsilon ,\delta ):=\frac{1}{\mathbb{P}(D)}{\int }_{\mathrm{\Lambda }}{\mathbf{1}}_D(\lambda ){\alpha }_i(\lambda ,\epsilon ){\beta }_j(\lambda ,\delta )d\mathbb{P}(\lambda )$ for $(i,j)\in I\times J$ and $(\epsilon ,\delta )\in \mathrm{\Sigma }\times \mathrm{\Sigma }$. Then:

(a) ${\mathbb{P}}_D$ is a probability measure on $(\mathrm{\Lambda },{\mathrm{\Sigma }}_{\mathrm{\Lambda }})$ and ${\mu }^D:=({\mu }_{ij}^D)$ is a binary correlation model on $(I,J)$;

(b) the same response laws represent ${\mu }^D$ over ${\mathbb{P}}_D$: ${\mu }_{ij}^D(\epsilon ,\delta )={\int }_{\mathrm{\Lambda }}{\alpha }_i(\lambda ,\epsilon ){\beta }_j(\lambda ,\delta )d{\mathbb{P}}_D(\lambda )\text{for all }i,j,\epsilon ,\delta ;$

(c) if $I=J=\{0,1\}$, then $\mid S({\mu }^D)\mid \le 2$.

The selection event $D$ is a subset of the specification space: membership in $D$ is the same for every setting pair and does not depend on the settings or on the outcomes. No choice principle is used.

• Setting-independent selection. The lemma conditions on an event of the specification space alone. This is the mathematical form of a setting-independent selection: whether a run enters the recorded sample is determined by the specification and not by the settings or the outcomes. Nothing is claimed here for selection rules that depend on the distant setting, on the local setting or on the observed outcomes; those require their own models and are not covered by the lemma.
• Direction of use. The lemma is used in the converse direction of detection: if the full tables of a comparison admit a local-response representation and the recorded trials are the runs that pass a setting-independent selection, then the recorded tables again admit one, so the CHSH bound holds for the recorded tables too. Without such a selection model no transfer of the bound to a sub-ensemble is asserted.

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the hypothesis of the tensor construction cited below. Let $V={\mathbb{C}}^2$ carry the orthonormal basis $(e_{+},e_{-})$ and let ${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ be the Pauli matrices of clause (e) of Finite spin matrices in the ordered basis $(e_{+},e_{-})$; for $a=(a_x,a_y,a_z)\in {\mathbb{R}}^3$ put ${\sigma }_a:=a_x{\sigma }_x+a_y{\sigma }_y+a_z{\sigma }_z$, and for $\epsilon \in \mathrm{\Sigma }=\{-1,+1\}$ put $P_{a,\epsilon }:=\frac{1}{2}(I+\epsilon {\sigma }_a)\mathrm{.}$ Let $V\widehat{\otimes }V$ be the completed tensor product with inclusion $\kappa$ of Hilbert tensor construction: inner product, completion and product bases and let $\mathrm{\Psi }:=\frac{1}{\sqrt{2}}(\kappa (e_{+}\otimes e_{-})-\kappa (e_{-}\otimes e_{+}))\in V\widehat{\otimes }V,q:=\frac{1}{\sqrt{2}}(\kappa (e_{+}\otimes e_{-})+\kappa (e_{-}\otimes e_{+}))\mathrm{.}$ Then:

(a) for every unit $a\in {\mathbb{R}}^3$ and $\epsilon \in \mathrm{\Sigma }$ the operator $P_{a,\epsilon }$ is an orthogonal projection on $V$, $P_{a,+}+P_{a,-}=I$ and $P_{a,\epsilon }{P}_{a,{\epsilon }^{\mathrm{\prime }}}={\delta }_{\epsilon {\epsilon }^{\mathrm{\prime }}}{P}_{a,\epsilon }$;

(b) $\mathrm{\parallel }\mathrm{\Psi }\mathrm{\parallel }=1$ and the four vectors $\mathrm{\Psi },q,\kappa (e_{+}\otimes e_{+}),\kappa (e_{-}\otimes e_{-})$ are mutually orthogonal, so $(\mathrm{\Psi },q,\kappa (e_{+}\otimes e_{+}),\kappa (e_{-}\otimes e_{-}))$ is an orthonormal basis of $V\widehat{\otimes }V$;

(c) for all unit $a,b\in {\mathbb{R}}^3$ and all $\epsilon ,\delta \in \mathrm{\Sigma }$, $⟨(P_{a,\epsilon }\otimes P_{b,\delta })\mathrm{\Psi },\mathrm{\Psi }⟩=\frac{1}{4}(1-\epsilon \delta a\cdot b);$

(d) the marginals of the four numbers in (c) are ${\sum }_{\delta \in \mathrm{\Sigma }}⟨(P_{a,\epsilon }\otimes P_{b,\delta })\mathrm{\Psi },\mathrm{\Psi }⟩=\frac{1}{2}={\sum }_{\epsilon \in \mathrm{\Sigma }}⟨(P_{a,\epsilon }\otimes P_{b,\delta })\mathrm{\Psi },\mathrm{\Psi }⟩,$ and their correlation is ${\sum }_{\epsilon \in \mathrm{\Sigma }}{\sum }_{\delta \in \mathrm{\Sigma }}\epsilon \delta ⟨(P_{a,\epsilon }\otimes P_{b,\delta })\mathrm{\Psi },\mathrm{\Psi }⟩=-a\cdot b\mathrm{.}$

The lemma is a finite-dimensional computation in the tensor product; it asserts no preparation, measurement or physical interpretation. The Axiom of Countable Choice enters only through the construction of the completed tensor product cited from Hilbert tensor construction: inner product, completion and product bases.