This may be a bit of an elementary question, but I think understanding it will help me understand fundamentally what makes quantum algorithms a bit magical.
Alice starts with a message she wants to send Bob, in the form of $|\phi \rangle = \alpha |0\rangle + \beta |1\rangle$. She concatenates it with a bell state (say, $\Psi^{00}$), of which she sees the first bit. Bob has access to the second bit.
Now Alice performs a Bell measurement, using a concatenation of $|\phi \rangle$ and her first $\Psi^{00}$ bit. But, to do the measurement, $\alpha$ and $\beta$ will be lost and we will be left with $i\in \{0,1\}$ and $j\in \{0,1\}$, two classical bits.
Now Alice sends $i$ and $j$ to Bob, and Bob also gets the second half of the Bell state. I can see that he can see the result of measuring $|\phi\rangle$, but how would he have access to $\alpha$ and $\beta$, even in a probabilistic sense?
Again, sorry if it sounds elementary, but my classical brain is not wrapping around this concept. No one told Bob anything about $\alpha$ and $\beta$, how is that information conveyed?
