FrediFizzx wrote:Joy Christian wrote:These are all commuting numbers, so mathematically this is quite a different argument.
Is there any way to tell the GAviewer that A's and B's commute for the first F?
Well, the point is that it is possible to show a violation with "dependent" terms. It is a subtle thing on exactly what the dependency is.
I don't think GAViewer will do that but I will investigate it further.
I don't think this is right. Consider the following: Let
Let us for the moment not care about the domain of the function. But it immediately follows that for 4 independent variables in it's domain
It turns out that the calculation often performed by experimentalists using experimental expectations, and also that predicted by using quantum mechanics is similar. It is
Since we already know that those terms relate to statistically independent measurements, we should conclude that the upper bound is 4 as well. However, the variables
which is exactly equivalent to
Therefore we have 3 independent variables only. What then is the upper bound for this expression. With a little algebra, we find that the upper bound is also the maximum of the expression
Which corresponds to 2 sqrt 2 when f(x) = cos(x). (http://www.wolframalpha.com/input/?i=Ma ... os%283x%29)
On the flip side, if we know that the terms are counterfactual as in the CHSH or Bell's inequality, then there is more dependence between the terms than just the settings as explained above, and the upper bound is 2 because
where
therefore the expression reduces to
Which factors as
And obviously the upper bound for this expression is 2. This is the CHSH. However, this analysis does not work for experiments and QM because instead of exactly the same functions
The factorizations do not work, so we are left with only the settings dependency present in the fourth term and must use the previous method to analyze the upper bound, which gives us
In summary:
- Counterfactual dependent terms like in Bell/CHSH gives us
- 4 Independent measurements with 3 independent angle variables gives us a maximum of
- 4 Completely independent measurements at 4 independent settings, gives us a maximum of 4.
Therefore, "violation" is only possible, if the terms are independent but impossible for terms with the same dependence as in the CHSH.
Note that if the 4 variables