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Bell's formulation of local realism is $$P(a,b)=\int\ d\lambda\cdot\rho(\lambda)p_A(a,\lambda)p_B(b,\lambda)\enspace.$$

Let's for simplicity assume there's only a finite number of states, so this becomes $$P(a,b)=\sum_{i} p_i\cdot\ p_A(a,i)p_B(b,i)\enspace.$$

I'm trying to translate this into density operator notation and then show that it implies that the state needs to be separable. So my ansatz is

$$P(a,b)=tr(\hat{\rho}\hat{A}(a)\otimes\hat{B}(b))\enspace,$$

where ##\hat{A}(a)## and ##\hat{B}(b)## are observables with spectrum ##\{1,0\}## (detecting or not detecting a photon). I'm trying to show by comparing the last two equations that ##\hat{\rho}## must have the form

$$\hat{\rho}=p_i\cdot\hat{\rho}_A ^i\otimes\hat{\rho}_B ^i$$

where ##\hat{\rho}_A ^i## and ##\hat{\rho}_B ^i## are density operators on their respective subsystems. However I can't see how to do this.

Showing that separable states satisfiy local realism is trivial, is the converse even true in general? If yes, how do you do this? Or is my ansatz nonsense? I'm unsure because I had to pick observables with eigenvalues and if ##\{1,0\}## was the right choice.

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# Does local realism imply separability?

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