= Solution
{c}
Write $c=\langle e_0|e_1\rangle=ve^{i\alpha}$. The <Cauchy-Schwarz inequality> gives $0\leq v\leq1$ because the two environment states are normalized. Immediately after the interaction, the joint <quantum state> is
$$
|\Psi\rangle=\frac{|0\rangle|e_0\rangle+e^{i\varphi}|1\rangle|e_1\rangle}{\sqrt2}.
$$
Applying the last <Hadamard gate> gives
$$
|\Psi_{\rm out}\rangle=\frac12\left[
|0\rangle\bigl(|e_0\rangle+e^{i\varphi}|e_1\rangle\bigr)
+|1\rangle\bigl(|e_0\rangle-e^{i\varphi}|e_1\rangle\bigr)\right].
$$
Since the environment is unobserved, the <Born rule> uses the squared <norm> of the environment-valued coefficient of $|0\rangle$:
$$
P_0=\frac14\left[2+e^{i\varphi}\langle e_0|e_1\rangle
+e^{-i\varphi}\langle e_1|e_0\rangle\right].
$$
Thus the <Hadamard interference with environment overlap> formula is
$$
\boxed{P_0(\varphi,v,\alpha)=\frac{1+v\cos(\varphi+\alpha)}2.}
$$
The plus sign follows from the explicitly chosen <phase gate> and the order $\langle e_0|e_1\rangle$ of the <inner product>. Equivalently, before the last <Hadamard gate> the <partial trace> over the environment gives
$$
\rho=\frac12\begin{pmatrix}
1&v e^{-i(\varphi+\alpha)}\\
v e^{i(\varphi+\alpha)}&1
\end{pmatrix}.
$$
The <complex environment overlap in qubit dephasing> both reduces the <quantum coherence> by $v$ and adds a relative <quantum phase> $\alpha$. Orthogonal environment states have $v=0$ and give $P_0=1/2$ for every $\varphi$. If $v=1$, the <quantum interference> fringe has full contrast and is only shifted; the <qubit> has not become mixed. For $v=0$, $\alpha$ is immaterial.
Moving the interaction to between the first <Hadamard gate> and the <phase gate> gives
$$
\frac{|0\rangle|e_0\rangle+|1\rangle|e_1\rangle}{\sqrt2}
\xrightarrow{P_\varphi\otimes I}
\frac{|0\rangle|e_0\rangle+e^{i\varphi}|1\rangle|e_1\rangle}{\sqrt2},
$$
the identical joint <quantum state>. Therefore \b[the expression is unchanged]. This is the <commutation of computational-basis dephasing and phase gates>; it uses the specified population-preserving interaction with the same environment states, rather than an arbitrary model of <quantum decoherence>.
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