I tried another approach to the problem of covariance like in Bell's theorem :(adsbygoogle = window.adsbygoogle || []).push({});

from the definition ##Cov(A,B)=\langle\Psi|A\otimes B|\Psi\rangle-\langle\Psi|A\otimes 1|\Psi\rangle\langle\Psi|1\otimes B|\Psi\rangle## (##A=diag(1,-1)=B##)

we can see that this 'average' is in fact a quadratic form and a quartic form in the wave-function. To put all on equal footage we can write the first term as ##\langle\Psi|\langle\Psi|A\otimes B|\Psi\rangle|\Psi\rangle##

then it's all a quartic form. Thus we could find a kind of extended measurement operator acting on ##|\Psi\rangle|\Psi\rangle## :

##\hat{Cov}(A,B)=A\otimes B\otimes 1_4-A\otimes 1_4\otimes B##

hence a 16x16 symmetric matrix and thus an observable. We have obviously

##\langle\Psi|\langle\Psi\hat{Cov}(A,B)|\Psi\rangle|\Psi\rangle=Cov(A,B)##

But the eigenvalues of this operator are 2,-2 and 0. and in this configuration we get <Cov>=-2/2+0/2=-1 as in the usual calculation.

1) Since A and B can have values only 1,-1 the 0 eigenvalue is obvious, but how is 2 possible.

2) Is it possible to prove that the average of Cov is in [-1,1] since we work only with a product of psi with itself ?

3) the CHSH operator being Cov(A,B-Cov(A,B')+Cov(A',B)+Cov(A',B'), I could find the eigenvalues with a trial version of mathematica to be 4,-4 and 0.

Thus we can see that contrary to a global hidden variable model the CHSH values 2 or -2 never appear.

We can see here that CHSH can get only integer value as opposed to CHSH without subtracting the product of the averages which is 2Sqrt(2).

4) How to find the maximum value of CHSH when averaged, is it 2Sqrt(2) as when one does not make this construction ?

Does this construction makes any sense, and how is the cloning of the wavefunction to be interpreted, or is this just more similar to a crackpottery ?

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# Variation on Quantum cross covariance and CHSH

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