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And operators $$A=B=\textrm{diag}(1,-1)$$

I saw in a lecture that we can consider $$A\otimes B$$, it has multiple eigenvalues 1 and -1.

It was then said : we can choose orthonormal basis vectors in each eigenspace.

Hence in this case $$p(+-)$$ and $$p(-+)$$ could be chosen as different.

But then I thought : Whereas if we diagonalize before the tensor product the eigenvalues were not multiple and we get always equiprobability : $$p(+-)=p(-+)$$ and $$p(++)=p(--)$$, here we cannot choose the vectors.

So which order is the right one ?