Measurement operator for Chsh experiment

[jk22]

The experiment is described at p26 of http://arxiv.org/abs/quant-ph/0402001

In this experiment we see that we sum measurement results and not measure the sum.

Is then the quantum measurement operator not :

$$S=A\otimes B\otimes\mathbb{1_{64}}-\mathbb{1_{4}}\otimes A\otimes B'\otimes\mathbb{1_{16}}+\mathbb{1_{16}}\otimes A'\otimes B\otimes\mathbb{1_{4}}+\mathbb{1_{64}}\otimes A'\otimes B'$$

Since we have the eigenvalues of S $$eig(S)\in\{-4,-2,0,2,4\}$$ ?

 

[jk22]

In other words it seems that in this experiment we do the sum of measurement results $$v(AB)-v(AB')+v(A'B)+v(A'B')$$

We are searching for the operation $$\diamond$$ such that we have $$v(A)+v(B)=v(A\diamond B)$$

This cannot be the usual sum of operators but $$A\diamond B=A\otimes 1+1\otimes B$$ satisfies the condition above.

So that for four terms it is easily generalized and we get that in fact the measurement operator corresponding to a Bell Chsh 2 channel experiment is a huge 256x256 matrix.

Does this makes any sense ?