After Fred's recent post about my derivation of the upper bound on CHSH, we have been trying to better understand why the upper bound on CHSH is 2\/2 and not 4 ?
In fact I have already answered this question much more rigorously in this paper, but perhaps a simplified explanation is called for, especially because the underlying geometrical reason for the bound 2\/2 instead of 4 is quite straightforward. In particular, it has nothing to do with any mystical notion such as "quantum non-locality."
As in Fred's post linked above, we begin with the square of the CHSH-type string of the (geometric) products of the standardized variables (or standard scores):
where
Now in Fred's post as well as in my original derivation, the upper bound of 2\/2 on CHSH is derived by using the assumption
So far so good. But can we do better? Is it possible to increase the upper bound on CHSH to its apparent full potential of 4 ? And if not, then why not? Evidently, this seems to be possible if we can arrange a maximum value of
So there. One cannot have all four directions
The maximum value of the upper bound on CHSH within R^3 is 2\/2. Therefore the observed upper bound on CHSH is 2\/2. It has nothing to do with any mystical idieas like quantum entanglement or non-locality. It is simply a numerical constraint arising form the geometrical and topological properties of the physical space we live in.
In fact, |CHSH|
Joy Christian