### Re: Joy's derivation of the 2*sqrt(2) bound

Posted:

**Sat Aug 29, 2015 8:30 pm**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 be a function of a single variable, any function, defined such that it has range

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 , any linear combination of the function over those variables is bounded above by 4 and below by -4:

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 in the EPRB scenario are not completely independent. Often the expression is written as:

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 , we have a different context each time for independent measurements and instead we have

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 , which for f(x) = cos(x) is 2 sqrt 2. You can solve this equation to find what settings will give you the maximum values and it turns out to be just the ones used by Bell. But there are many other combinations. See the wolfram alpha link above.

- 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 are indepependent, even if , we will still have an upper bound of 4.