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zonde
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In this paper http://dx.doi.org/10.1103/PhysRevA.47.R747 Eberhard derives Bell type inequality from these assumptions:
A theory is defined as being "local" if it predicts that, among these possible sequences of events [with the same number of events N], one can find four sequences (one for each setup [(α1,β1), (α2,β1), (α1,β2), (α2,β2)]) satisfying the following conditions:
(i) The fate of photon a is independent of the value of β, i.e., is the same in an event of the sequence corresponding to setup (α1,β1) as in the event with the same event number k for (α1,β2); also same fate for a in (α2,β1) and (α2,β2); this is true for all k's for these carefully selected sequences.
(ii) The fate of photon b is independent of the value of α, i.e., is the same in an event k of the sequences (α1,β1) and (α2,β1); also same fate for b in (α1,β2) and (α2,β2).
(iii) Among all sets of four sequences that one has been able to find with conditions (i) and (ii) satisfied, there are some for which all averages and correlations differ from the expectation values predicted by the theory by less than, let us say, ten standard deviations.
Now the question I am trying to answer is how we could violate condition (i) or (ii) without opening possibility of FTL communication.
Say if we can not find two sequences (α1,β1) and (α1,β2) that are identical at Alice's end then we can communicate by making up a catalog of possible sequences that Alice can see when Bob sets his detector at β1 and another catalog of sequences that Alice can see for Bob's setting β2.
Even if Alice sees some sequences more often for Bob's β1 and some other for Bob's β2 we still can communicate FTL.
A theory is defined as being "local" if it predicts that, among these possible sequences of events [with the same number of events N], one can find four sequences (one for each setup [(α1,β1), (α2,β1), (α1,β2), (α2,β2)]) satisfying the following conditions:
(i) The fate of photon a is independent of the value of β, i.e., is the same in an event of the sequence corresponding to setup (α1,β1) as in the event with the same event number k for (α1,β2); also same fate for a in (α2,β1) and (α2,β2); this is true for all k's for these carefully selected sequences.
(ii) The fate of photon b is independent of the value of α, i.e., is the same in an event k of the sequences (α1,β1) and (α2,β1); also same fate for b in (α1,β2) and (α2,β2).
(iii) Among all sets of four sequences that one has been able to find with conditions (i) and (ii) satisfied, there are some for which all averages and correlations differ from the expectation values predicted by the theory by less than, let us say, ten standard deviations.
Now the question I am trying to answer is how we could violate condition (i) or (ii) without opening possibility of FTL communication.
Say if we can not find two sequences (α1,β1) and (α1,β2) that are identical at Alice's end then we can communicate by making up a catalog of possible sequences that Alice can see when Bob sets his detector at β1 and another catalog of sequences that Alice can see for Bob's setting β2.
Even if Alice sees some sequences more often for Bob's β1 and some other for Bob's β2 we still can communicate FTL.