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Bell formulated local realism as follows: The probability of a coincidence between separated measurements of particles with correlated (e.g. identical or opposite) orientation properties can be written as

$$P(a,b)=\int{d\lambda\cdot \rho(\lambda)\cdot p_A(a,\lambda)\cdot p_B(b,\lambda)}\enspace .$$

To get a better understanding of the terms "local" and "realistic", I'm trying to adapt this formula. So I'd say a theory thatrealistic, butnot necessarily local, would satisfy

$$P(a,b)=\int{d\lambda\cdot \rho(\lambda)\cdot p_{AB}(a,b,\lambda)}\enspace ,$$

i.e. ##p_{AB}(a,b,\lambda)## is not necessarily a product distribution. As far as I can see quantum expectation values satisfy this probability distribution.

How would this formula look like for anonrealistic(or not necessarily realistic), butlocaltheory? Or is local realism not something that can be split up into locality and realism?

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# I Mathematical formulation of local non-realism

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