Suppose we have a pair of spins in the state $\left|\Psi\right>=\frac{1}{\sqrt2}(\left|\uparrow\downarrow\right>-\left|\downarrow\uparrow\right>)$. We know that the measurements of the spins along the $z$-direction must give opposite results as a spin-up result of the first particle "collapses" the state into $\left|\uparrow\downarrow\right>$ and vice versa. But why can't it collapse into, say, $\frac{1}{\sqrt2}(\left|\uparrow\downarrow\right>+\left|\uparrow\uparrow\right>)$ (which is again an eigenstate of the spin of the first particle along the $z$-direction with the same eigenvalue)?
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Your initial state is entangled, which means it exhibits a correlation between the spin of the two particles. Each particle is spin up or spin down with equal probability, but the spins are (anti)correlated so that they cannot have the same spin.
The states $\left|\uparrow \downarrow\right>$ and $\left|\downarrow \uparrow\right>$ are in agreement with this correlation. The state $\left|\uparrow \uparrow\right>$ is not, so it is not a possible result.
If you had begun with the unentangled state $$ \frac{1}{2}\left(\left|\uparrow \uparrow\right> + \left|\uparrow \downarrow\right> + \left|\downarrow \uparrow\right> + \left|\downarrow \downarrow\right>\right) $$ which is factorable into
$$ \frac{1}{2}\left(\left|\uparrow\right>_L + \left|\downarrow\right>_L\right)\left(\left|\uparrow\right>_R + \left|\downarrow\right>_R\right)$$ then the spins would have been independent. Measuring the spin of the left particle would then kill one of the terms for the left particle's spin, but due to the factorable form that does not eliminate any of the right particle's possible states.
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