Joy Christian wrote:Let me note that for the 4-particle GHZS state the condition E(a, b, c, d) = << ABCD >> = +1 or -1 for some specific settings for all runs and thus even for a single run is similar to the familiar condition E(a, b) = << AB >> = +1 or -1 for the 2-particle EPRB state for some specific settings (i.e., for a = b and a = -b, respectively) for all runs and thus even for a single run. In the latter example, it is the condition of perfect correlation (or perfect anti-correlation), which is predicted by quantum mechanics.

For completeness, let me prove this here for the EPRB case, parallelling the proof below which I provided in response to the counterchallenge to me by Tim Maudlin:

For the EPRB case, let us follow the construction of my 3-sphere model presented in this paper: https://arxiv.org/abs/1405.2355. The proof goes through as follows:

What I want to show is AB = -1 for a = +b and AB = +1 for a = -b even for a single run. It would suffice to prove AB = +1 for a = -b since the case AB = -1 for a = +b follows quite similarly. We start with equations (54) and (55) of the above paper, which define the binary valued functions A = +/-1 and B = +/-1, subject to the conservation of the spin-0 defined in equations (65) and (66). The expectation value E(a, b) = < AB > = - a . b is then derived in equations (67) to (75) of the paper using these functions A and B. In fact, the expectation value (75) or (76) follows from the very construction of the functions A and B in the equations (54) and (55), as a geometrical identity within my 3-sphere model. Therefore we can use this geometrical identity to prove that AB = +1 for a = -b. In fact, for the chosen settings this identity reduces simply to E(a, b) = < AB > = +1. But E(a, b) = < AB > = +1 tells us that the average of the number AB is a constant, and it is equal to +1. This is mathematically possible only if AB = +1 for all runs, for a = -b. But if AB = +1 for all runs, then AB = +1 holds also for any given run. Therefore AB = +1 for any single run, for the chosen settings a = -b. QED.

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