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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?

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Alice doesn't just concatenate her message qubit with her half of the Bell pair, she entangles these qubits in the process of doing the Bell measurement (remember that it involves a CNOT gate before doing computational basis measurement). This means that before the computational basis measurement the information about $\alpha$ and $\beta$ is spread across all three qubits rather than contained only in the message qubit. Now the computational basis measurement "squeezes" that information from the qubits being measured (two in Alice's possession) to the qubit remaining unmeasured (Bob's one). Alice's information about $i$ and $j$ serves only to adjust the mapping of $\alpha$ and $\beta$ to the basis states and their relative sign, not to convey the information about $\alpha$ and $\beta$.

It helps to follow the math of what's happening for a more formal explanation, but I'm not up to spelling it out right now. There are lots of questions about it on this site, and other resources outside of it.

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  • $\begingroup$ I went over the equations a bit, I have no qualms about the linear algebra, but I think I'm just missing the physics intuition here. But your explanation makes sense from a high level... but I think I just need to see what physical action is done to "entangle" a bit! It's easier in classical registers and transistors... thanks! $\endgroup$
    – Y. S.
    Apr 15 at 20:56

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