= Solution
A <Waveplate>[quarter-wave plate] needs a specified axis angle. If its fast axis makes angle $\eta$ to horizontal, choose its <Jones matrix> $Q=R(\eta)\operatorname{diag}(1,i)R(-\eta)$. Then
$$
Q|V\rangle=a|H\rangle+b|V\rangle,\quad a=(1-i)\sin\eta\cos\eta,\quad b=\sin^2\eta+i\cos^2\eta.
$$
After the preceding successful lower-path detection the total state is $(aH+bV)|\downarrow_1\rangle$. The next crystal sends it to
$$
\boxed{a|H,\uparrow_2\rangle+b|V,\downarrow_2\rangle,\quad P(\downarrow_2\mid\downarrow_1)=|b|^2.}
$$
For axes at $45^\circ$, the state before splitting is circular up to phase, $a=(1-i)/2$, $b=(1+i)/2$, and the <probability> is $1/2$. Detection in $\downarrow_2$ always leaves vertical <polarization (waves)>[polarization]. If no earlier lower-path measurement was actually performed, retain the untouched upper <amplitude>: the final state is $(|H,\uparrow_1\rangle+a|H,\uparrow_2\rangle+b|V,\downarrow_2\rangle)/\sqrt2$, and the last <probability> is $|b|^2/2$, or $1/4$ at $45^\circ$. At axis angle zero the <conditional probability> is instead one, a concrete demonstration that an unspecified orientation does not determine a unique number.
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