 Solution

Alice conditions on her observed computational-basis result. For outcome $0$, occurring with probability $|\alpha {|}^2$, her post-measurement density matrix is $|0⟩⟨0|$; for outcome $1$, occurring with probability $|\beta {|}^2$, it is $|1⟩⟨1|$. Explicitly,These are the conditional Lüders rule updates for the two rank-one projectors. They assume the corresponding outcome has nonzero probability and Alice knows which one occurred.