I'm trying to answer the following question about the BB84 protocol from Nielsen and Chuang's Introduction to Quantum Information.
As I understand it, the string $b$ is determining whether we are measuring in the computational basis or the Hadamard basis. I decided to test the claim in the question with an example.
Let $\vert\phi\rangle = \alpha \vert 0 \rangle + \beta \vert 1\rangle$ and assume we are measuring in the $\{\vert - \rangle, \vert + \rangle\}$ basis. That means our measurement is
$(\vert - \rangle \langle - \vert + \vert + \rangle \langle + \vert)\vert \phi \rangle = \frac{\alpha}{\sqrt{2}}\vert 0 \rangle + \frac{\beta}{\sqrt{2}}\vert 1 \rangle$
To me this is clearly wrong. Not only does it suggest we can figure something out about the state which $\vert \phi \rangle$ was prepared in, but we don't even have that $\frac{1}{\sqrt{2}}(\alpha + \beta) = 1$. Could I have guidance as to what I am doing incorrectly, or pointers about how to go about this proof?
EDIT: Applying these projections yields $$\frac{\alpha + \beta}{\sqrt{2}}\vert + \rangle + \frac{\alpha - \beta}{\sqrt{2}}\vert - \rangle$$
We can take probabilities like $|\langle + \vert \phi \rangle|^{2}$, and I am still unsure how to prove that they are uncorrelated.
EDIT2: I have a solution and will update it soon.