 Solution

Use the fidelity between the original pure state and the measurement output averaged over Calvin's results. Put $p=|\alpha {|}^2$, so $1-p=|\beta {|}^2$. The output is ${\rho }_Z=\mathrm{diag}(p,1-p)$, andHere $r_z=2p-1$ is the input Bloch component. If a particular result is specified instead, the conditional fidelities are $F_0=p$ and $F_1=1-p$; their outcome-weighted mean is precisely the displayed $F$. This distinction fixes what is meant by the fidelity of the measurement procedure.