= Solution
The normalized input <density operator> is
$$
\rho=\begin{pmatrix}|\alpha|^2&\alpha\beta^*\\\alpha^*\beta&|\beta|^2\end{pmatrix}.
$$
A <quantum measurement in the computational basis> has projectors $P_j=|j\rangle\langle j|$. Discarding its recorded outcome gives the <dephasing channel>
$$
\boxed{\rho'=P_0\rho P_0+P_1\rho P_1
=|\alpha|^2|0\rangle\langle0|+|\beta|^2|1\rangle\langle1|.}
$$
The original off-diagonal coherence disappears. If the outcome is retained instead, the conditional state is $|0\rangle$ or $|1\rangle$ and the record carries information about which occurred. Thus discarding the record is an essential part of this channel specification.
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