= Solution
Since $(Z_D\otimes X_E)^2=I$, the solution of the <Time-dependent Schrödinger equation> is
$$
U(t)=I_S\otimes\left[\cos A\,I_{DE}-i\sin A\,Z_D\otimes X_E\right],
\qquad \boxed{A(t)=gt/\hbar.}
$$
The two detector eigenvalues generate opposite rotations of the environment. Acting on the initial state gives
$$
\boxed{|\Psi(t)\rangle=\frac1{\sqrt2}\left[
|\uparrow d_\uparrow\rangle(\cos A|E_\uparrow\rangle-i\sin A|E_\downarrow\rangle)
+|\downarrow d_\downarrow\rangle(\cos A|E_\uparrow\rangle+i\sin A|E_\downarrow\rangle)\right].}
$$
These are the indicated $|E_-\rangle$ and $|E_+\rangle$. Their <conditional environment overlap> is $\langle E_+|E_-\rangle=\cos2A$. The <partial trace> therefore multiplies the coherence between $|\uparrow d_\uparrow\rangle$ and $|\downarrow d_\downarrow\rangle$ by $\cos2A$. This is an exactly solvable <single-qubit environment dephasing model>, with fully reversible global evolution.
\Image[/past-exam-of-the-quantum-information-course-of-the-university-of-cambridge/2015/paper-1-finite-environment-revival.png]
{title=Single-qubit environmental records suppress and later restore reduced coherence}
{description=Original ideal unitary model. Zero off-diagonal coherence at one time does not imply irreversible collapse of the full state.}
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