= Solution
Linearity of the detector's <unitary time evolution> maps the two input branches separately, giving the <quantum premeasurement> state
$$
|\Phi^C\rangle=\frac{|\uparrow d_\uparrow\rangle+|\downarrow d_\downarrow\rangle}{\sqrt2}.
$$
In the ordered <tensor product> basis $(|\uparrow d_\uparrow\rangle,|\uparrow d_\downarrow\rangle,|\downarrow d_\uparrow\rangle,|\downarrow d_\downarrow\rangle)$, its pure <density matrix> is
$$
\rho_C=\frac12\begin{pmatrix}1&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&1\end{pmatrix}.
$$
A nonselective <projective measurement> in the indicated detector basis applies $\rho\mapsto\sum_j(I_S\otimes\Lambda_j)\rho(I_S\otimes\Lambda_j)$, removing the two off-diagonal terms:
$$
\boxed{\rho_{\rm up/down}=\frac12\left(|\uparrow d_\uparrow\rangle\langle\uparrow d_\uparrow|
+|\downarrow d_\downarrow\rangle\langle\downarrow d_\downarrow|\right)
=\operatorname{diag}\left(\frac12,0,0,\frac12\right).}
$$
This is the usual nonselective meaning of <von Neumann process 1>. It is not equal to the pure projector $|\Phi^C\rangle\langle\Phi^C|$. If “the detector has clicked” means conditioning on an actual up outcome, the normalized postselected state is instead $|\uparrow d_\uparrow\rangle$, with probability $1/2$. The correlated superposition describes the coherent premeasurement before that selection.
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