Multipartite fully-nonlocal quantum states

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A joint probability distribution is local whenever it can be written as P ⁡ ( a ​ b | x ​ y ) = ∫ d ​ λ ​ ω ​ ( λ ) ​ P ​ ( a | x , λ ) ​ P ​ ( b | y , λ ) P(ab|xy)=\int d\lambda\omega(\lambda)P(a|x,\lambda)P(b|y,\lambda) where the classical variable λ \lambda is distributed according to the probability distribution ω ⁡ ( λ ) \omega(\lambda) . This definition easily generalizes to an arbitrary number of parties. Those probability distributions which do not admit this decomposition are said to be nonlocal

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A distribution P ⁡ ( a ​ b | x ​ y ) P(ab|xy) is no-signaling if the local distributions for one party do not depend on the choice of measurements by the other party, for instance P ⁡ ( a | x ​ y ) ≡ ∑ b P ⁡ ( a ​ b | x ​ y ) = P ⁡ ( a | x ) P(a|xy)\equiv\sum_{b}P(ab|xy)=P(a|x) . This definition easily generalizes to an arbitrary number of parties

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Actually, our proof also applies for slightly more general protocols in which the local measurements among parties belonging to the same block in the bipartition can be correlated using classical communication. It also applies for situations where parties A 1 A_{1} and A m A_{m} are projected into any fully nonlocal state ( p L = 0 p_{L}=0 )

Cited in the paper.

If P N ​ S 𝒜 , ℬ ​ ( a ~ ​ b ~ | x ~ , y ~ ) = 0 P_{NS}^{\mathcal{A},\mathcal{B}}(\tilde{a}\tilde{b}|\tilde{x},\tilde{y})=0 , the nonlocal distribution is not well-defined but the induced state ψ 2 a ~ ​ b ~ \psi_{2}^{\tilde{a}\tilde{b}} is local and our proof still holds

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