An operational framework for nonlocality

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Actually, the last construction suggests another venue to generate non-locality, namely, to make use of Stochastic Wirings & Classical Communication Prior to the Inputs (SWCCPI). However, it is easy to see that, whenever non-locality can be generated probabilistically with SWCCPI protocols, it can be also activated deterministically via WCCPI. Imagine, for instance, that P 1 ( a 1 , a 3 | x 1 , x 3 ) ≡ P ( a 1 , a 3 | x 1 , x 3 , x 2 = 1 , a 2 = + 1 ) P_{1}(a_{1},a_{3}|x_{1},x_{3})\equiv P(a_{1},a_{3}|x_{1},x_{3},x_{2}=1,a_{2}=+1) is non-local, but P 2 ( a 1 , a 3 | x 1 , x 3 ) ≡ P ( a 1 , a 3 | x 1 , x 3 , x 2 = 1 , a 2 = − 1 ) P_{2}(a_{1},a_{3}|x_{1},x_{3})\equiv P(a_{1},a_{3}|x_{1},x_{3},x_{2}=1,a_{2}=-1) is not. Let c → \vec{c} be such that c → ⋅ L → ≥ 0 \vec{c}\cdot\vec{L}\geq 0 for all bipartite local distributions L → \vec{L} and c → ⋅ P → 1 < 0 \vec{c}\cdot\vec{P}_{1}<0 . In the event a 2 = − 1 a_{2}=-1 , A 1 A_{1} and A 3 A_{3} receive the order of simulating a local box P 2 ′ ( a 1 , a 3 | x 1 , x 3 ) P_{2}^{\prime}(a_{1},a_{3}|x_{1},x_{3}) such that c → ⋅ P → 2 = 0 \vec{c}\cdot\vec{P}_{2}=0 . Then, it is clear that the so-constructed box Q ( a 1 , a 3 | x 1 , x 3 ) ≡ p ( a 2 = 1 | x 2 ) P 1 ( a 1 , a 3 | x 1 , x 3 ) Q(a_{1},a_{3}|x_{1},x_{3})\equiv p(a_{2}=1|x_{2})P_{1}(a_{1},a_{3}|x_{1},x_{3}) + + p ( a 2 = − 1 | x 2 ) P 2 ′ ( a 1 , a 3 | x 1 , x 3 ) p(a_{2}=-1|x_{2})P^{\prime}_{2}(a_{1},a_{3}|x_{1},x_{3}) satisfies c → ⋅ Q → < 0 \vec{c}\cdot\vec{Q}<0 , and, consequently, is non-local

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