Kochen-Specker Proofs Look Wrong to Me

[nougiecat]

Can someone explain to me what is wrong with the following argument? There are two parts. First of all, K-S, despite passing reference to hidden variables, doesn't really seem to depend on any interesting properties of HV, but instead appears to be an indictment of QM itself by asserting that QM cannot consistently predict the results of measurements. I'm not going to elaborate on this part, because the more important part is this: the proofs of K-S all depend on some form of the following assertion. That the eigenvectors of projection operators must always be associated with the same value (zero or one) independent of the operator (i.e. observable) that they are associated with. This results in the so-called coloring rule that a given eigenvector must always have the same color. But I'm pretty sure this is simply not true. The value of an observable is not arbitrary or connected only with an eigenvector. The value is determined by the eigenvalue associated with the eigenvector for a given operator. Different operators in general may share one or more eigenvectors, but the eigenvalues are unrelated. As an example, consider the operator Q, in 3D. This will have three eigenvectors, q₁, q₂, and q₃, and three corresponding projection operators, P₁, P₂, and P₃. All three of these have (or can have, by construction, since they are all degenerate) the same three eigenvectors, q₁, q₂, and q₃. But the eigenvalues that go with these are not all the same. In particular, the eigenvalues for P₁ are 1, for q₁, 0, for q₂, and 0 for q₃. Similarly for P₂, they are 0, 1, and 0, and for P₃, 0, 0, and 1. So all three of the q_i have both zero and one as value depending on which projection operator you use. This seems to disprove all of the K-S proofs that I have seen.

I assume someone can explain to me what I am missing here. Thanks.

 

[bhobba]

Without trying to disentange your issue please note that Kochen Specker (it's been a while since I was familiar with the detail) is in fact a simple corrolary to a much deeper result - Gleasons Theroem:
http://kof.physto.se/cond_mat_page/theses/helena-master.pdf

I suggest going through the above link and see if your issue is not resolved.

But it's fairly simple really. If you assume non contextuality it's (plus a few other very reasonable things - but non contextuality is the biggie) then the Born rule follows so that hidden variables are basically not possible because you can't assign states with 0 and 1.

Thanks
Bill