gill1109 wrote:The deterministic upper bound is 4. However when the N setting pairs which are actually chosen to be measured are indendently sampled with equal probability from the four possibilities a b; a b'; a' b; a' b' then on average, CHSH does not exceed 4.
For 4 separate sets of particles, each of the averages <a1b1>, <a2b2'>, <a3'b3>, <a4'b4'> are independent from each other. Each of those terms has an upper bound of +1 and a lower bound of -1. Of course in any specific set, the averages can have any value between those two extremes but we are only interested in the extremes for deriving the inequality. Obviously then <a1b1> + <a2b2'> + <a3'b3> - <a4'b4'> must have an
upper bound not less than 4, IF we have 4 different independent (aka disjoint) sets of particles. It does not matter how randomly or non-randomly you sample the 4 sets, so long as the 4 terms above are independent, the
upper bound is clearly 4 not less, not more.
The only way the
upper bound of expression the <a1b1> + <a2b2'> + <a3'b3> - <a4'b4'> measured from 4 sets of particles can be less than 4 is if we introduce dependencies between the sets of particles. To see this, notice that when the first three terms are each at their maximum of +1, to have an
upper bound less than 4, the last term must not be at it's minimum of -1. If the last term is at -1, then the only way to have the
upper bound less than 4 is by forcing at least one of the other 3 terms to not be at their maximum of +1. This way we have introduced a dependency between the separate sets of particles which should be independent, so that values in one set of particles now impose conditions on a separate set of particles. It is also clear that it is impossible for this expression to exceed 4 no matter how you sample the individual values, that is what is meant by "upper bound". This simple proof demonstrates that there must be a serious error in any proof claiming to demonstrate an
upper bound of less than 4 for
independent sets of particles (like those measured in EPR experiments). The two are mutually contradictory.
I emphasize "upper bound" many times to make it very clear that the averages can be any value between the extremes but can not exceed the extremes in either direction. This is why I have argued that the results from experiments and QM are being compared with the wrong inequality. The correct inequality should be <a1b1> + <a2b2'> + <a3'b3> - <a4'b4'> <= 4 (which is not the CHSH). I believe my proof above combined with the though experiment using my simulation I presented earlier provides more support for the argument that Bell's theorem is wrong.