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Local-response representability implies setting-independent marginals

Statement

Let $μ$ be a binary correlation model on finite setting sets $I, J$ with outcome labels $Σ = {- 1, + 1}$ (Binary correlation models: setting-indexed tables of two-valued outcome laws, marginals and the CHSH functional), and suppose that $μ$ admits a local-response representation with response laws $(α_{i})$ , $(β_{j})$ and probability space $(Λ, Σ_{Λ}, P)$ (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 $μ_{i j}^{A} = μ_{i j^{'}}^{A} for all j, j^{'} \in J,$ and for every $j \in J$ the marginal law of the second station satisfies $μ_{i j}^{B} = μ_{i^{'} j}^{B}$ for all $i, i^{'} \in I$ ; that is, $μ$ 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.

Facts & Assumptions

Given: A binary correlation model $μ$ ; a local-response representation $(α_{i})_{i \in I}$ , $(β_{j})_{j \in J}$ on $(Λ, Σ_{Λ}, P)$ ; setting labels $i \in I$ , $j, j^{'} \in J$ ; outcome labels $ε, δ \in Σ$ .

[F1]

The joint laws and marginals are the finite sums $μ_{i j}^{A} (ε) = \sum_{δ \in Σ} μ_{i j} (ε, δ)$ and $μ_{i j}^{B} (δ) = \sum_{ε \in Σ} μ_{i j} (ε, δ)$ , and no-signalling means exactly the pair of setting-independence identities of clause (c) of Binary correlation models: setting-indexed tables of two-valued outcome laws, marginals and the CHSH functional; under the representation hypothesis the joint law is also given by the representation identity $μ_{i j} (ε, δ) = E_{P} [α_{i} (\cdot, ε) β_{j} (\cdot, δ)]$ of clause (a) of Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws.

[F2]

Both response laws are normalized on the two-point set $Σ$ : $\sum_{ε \in Σ} α_{i} (λ, ε) = 1$ for every $λ \in Λ$ and $i \in I$ , and $\sum_{δ \in Σ} β_{j} (λ, δ) = 1$ for every $λ \in Λ$ and $j \in J$ (Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws clause (a)).

[F3]

Expectation is linear: $E [a X + b Y] = a E [X] + b E [Y]$ for integrable real random variables $X, Y$ and scalars $a, b$ (Linearity, monotonicity, and the modulus bound for expectation, Expectation of a nonnegative or integrable random variable); by induction the identity extends to finite sums $\sum_{δ \in Σ}$ over the two-element set $Σ$ (Finite sums and finite products, by recursion).

Proof

technique · direct; sum the representation identity over the distant outcome label, use the normalization of the distant response law and the linearity of expectation, and read off the setting-independence

Given: The data of the statement and [F1]-[F3].

1.1givenF1F2F3algebra

[Marginal of the first station] Fix $i \in I$ , $j \in J$ and $ε \in Σ$ . By [F1] and the definition of the marginal, $μ_{i j}^{A} (ε) = \sum_{δ \in Σ} μ_{i j} (ε, δ) = \sum_{δ \in Σ} E_{P} [α_{i} (\cdot, ε) β_{j} (\cdot, δ)] = E_{P} [\sum_{δ \in Σ} α_{i} (\cdot, ε) β_{j} (\cdot, δ)] = E_{P} [α_{i} (\cdot, ε) \sum_{δ \in Σ} β_{j} (\cdot, δ)] = E_{P} [α_{i} (\cdot, ε)]$ , where the second equality is the representation identity, the third is the finite linearity of [F3], and the last is the normalization of [F2]. The resulting expression involves $i$ and $ε$ only, so $μ_{i j}^{A} (ε) = μ_{i j^{'}}^{A} (ε)$ for every $j^{'} \in J$ ; as $(ε, i, j)$ were arbitrary this is the first-station half of (a).

1.2givenF1F2F3algebra

[Marginal of the second station] The same computation with the roles of the two stations exchanged, summing this time over $ε \in Σ$ and using the normalization $\sum_{ε} α_{i} (λ, ε) = 1$ of [F2], gives $μ_{i j}^{B} (δ) = \sum_{ε} μ_{i j} (ε, δ) = E_{P} [β_{j} (\cdot, δ)]$ for every $δ \in Σ$ , an expression free of $i$ ; hence $μ_{i j}^{B} = μ_{i^{'} j}^{B}$ for all $i, i^{'} \in I$ , the second-station half of (a).

2.1step 1.1step 1.2F1∎

[Assembly and contrapositive] Steps 1.1 and 1.2 give the two setting-independence identities, which by [F1] is exactly no-signalling of $μ$ ; this is (a). For (b), if a model admits a local-response representation then (a) makes it no-signalling, so a model that is not no-signalling admits none.

Remarks

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.

Depends on

Dependency tree · 2 levels

Sources

N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani and S. Wehner, 'Bell nonlocality', Reviews of Modern Physics 86 (2014) 419, Section I.A, PDF page 3: a marginal p(a|x) = sum over b of p(ab|xy) is independent of the distant setting y, and this is identified with the no-signalling property of the local decomposition