= Solution
Use the time-ordered network $\operatorname{CNOT}_{1\to2}$, $\operatorname{CNOT}_{2\to1}$, $\operatorname{CNOT}_{1\to2}$. On bits $(a,b)$ its successive values are
$$
(a,b)\longmapsto(a,a\oplus b)\longmapsto(b,a\oplus b)\longmapsto(b,a).
$$
Thus the network exchanges the two <Computational basis>[computational-basis] states. Since the gates are linear, this basis proof also establishes <SWAP gate> on arbitrary quantum states.
The <Bell states> $|\Phi^\pm\rangle=(|00\rangle\pm|11\rangle)/\sqrt2$ and $|\Psi^\pm\rangle=(|01\rangle\pm|10\rangle)/\sqrt2$ obey
$$
\boxed{S|\Phi^+\rangle=|\Phi^+\rangle,\quad S|\Phi^-\rangle=|\Phi^-\rangle,\quad
S|\Psi^+\rangle=|\Psi^+\rangle,\quad S|\Psi^-\rangle=-|\Psi^-\rangle.}
$$
The antisymmetric <spin singlet state> alone has <eigenvalue> $-1$. Its standalone sign is a global <phase>; a <Fredkin gate> can make that sign observable relative to the control branch, as in part d.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2017-swap-and-fredkin-decompositions.png]
{title=Three controlled-X gates for SWAP, and two decompositions of the <Fredkin gate>}
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