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## Main Question or Discussion Point

Let's say you have a pair of electrons in the singlet spin state. I thought that Alice measuring the spin of one electron (about the "z axis") corresponded to applying the operator [tex]\hat{S}_z\otimes \hat{I}[/tex] (where [tex]\hat{I}[/tex] is the identity operator) to the singlet state [tex]\frac{1}{\sqrt{2}}(\uparrow \otimes \downarrow - \downarrow \otimes \uparrow )[/tex]. However, I don't see (mathematically) why Alice measuring spin up forces the state to be [tex]\uparrow\otimes\downarrow [/tex].

I say this because the operator [tex]\hat{S}_z\otimes \hat{I}[/tex] has eigenvectors (like [tex]\frac{1}{\sqrt{2}}(\uparrow\otimes\uparrow + \uparrow\otimes\downarrow )[/tex]) which (1) would give Alice the spin up result and (2) have a nonzero inner product with the initial (singlet) state. So why can't the state collapse to another eigenvector like the one I just mentioned (and thus Bob not necessarily measure spin down)? I think I'm missing something simple here.

I say this because the operator [tex]\hat{S}_z\otimes \hat{I}[/tex] has eigenvectors (like [tex]\frac{1}{\sqrt{2}}(\uparrow\otimes\uparrow + \uparrow\otimes\downarrow )[/tex]) which (1) would give Alice the spin up result and (2) have a nonzero inner product with the initial (singlet) state. So why can't the state collapse to another eigenvector like the one I just mentioned (and thus Bob not necessarily measure spin down)? I think I'm missing something simple here.