Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 10C / i / Solution

For a unit vector $|\psi\rangle$ and real $\phi$,
$$
R_\psi^\phi=I-(1-e^{i\phi})|\psi\rangle\langle\psi|
$$
multiplies the component parallel to $|\psi\rangle$ by $e^{i\phi}$ and fixes its orthogonal complement. The case $\phi=\pi$ is a <reflection operator>.

A reflection operator is a unitary involution that changes sign on one orthogonal subspace and fixes its orthogonal complement. For a unit vector $|\psi\rangle$, $I-2|\psi\rangle\langle\psi|$ reflects in the hyperplane perpendicular to $|\psi\rangle$.

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 Incoming links: Selective phase rotation