= Solution
Alice conditions on her observed <computational basis>[computational-basis] result. For outcome $0$, occurring with <probability> $|\alpha|^2$, her post-measurement <density matrix> is $|0\rangle\langle0|$; for outcome $1$, occurring with <probability> $|\beta|^2$, it is $|1\rangle\langle1|$. Explicitly,
$$
\boxed{\rho_{A\mid0}=\begin{pmatrix}1&0\\0&0\end{pmatrix},\qquad
\rho_{A\mid1}=\begin{pmatrix}0&0\\0&1\end{pmatrix}.}
$$
These are the conditional <Lüders rule> updates for the two rank-one <projection operators>[projectors]. They assume the corresponding outcome has nonzero <probability> and Alice knows which one occurred.
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