Solution
= Solution
In the <computational basis>,
$$
|\psi\rangle\langle\psi|
=\begin{pmatrix}
|\alpha|^2&\alpha\beta^*\\
\alpha^*\beta&|\beta|^2
\end{pmatrix}.
$$
At $p=1/2$, the two opposite off-diagonal contributions cancel, giving
$$
\boxed{\widetilde\rho
=\Lambda_{1/2}(\rho)
=\begin{pmatrix}|\alpha|^2&0\\0&|\beta|^2\end{pmatrix}}.
$$