 Solution

A quarter-wave plate needs a specified axis angle. If its fast axis makes angle $\eta$ to horizontal, choose its Jones matrix $Q=R(\eta )\mathrm{diag}(1,i)R(-\eta )$. ThenAfter the preceding successful lower-path detection the total state is $(aH+bV)|{\downarrow }_1⟩$. The next crystal sends it toFor 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. If no earlier lower-path measurement was actually performed, retain the untouched upper amplitude: the final state is $(|H,{\uparrow }_1⟩+a|H,{\uparrow }_2⟩+b|V,{\downarrow }_2⟩)/\sqrt{2}$, 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.