Reflection operator

For a unit vector $|\psi ⟩$, the reflection $I-2|\psi ⟩⟨\psi |$ negates the component parallel to $|\psi ⟩$ and fixes its orthogonal complement.

For an orthogonal projection, $P^k=P$ for $k\ge 1$. Summing the exponential series proves the displayed identity. At $\theta =\pi$, the operator is $I-2P$, which changes the sign on the range and fixes its orthogonal complement. In Grover search algorithm, the marked-state reflection and the uniform-state reflection compose to the search iterate up to an irrelevant global phase.

For a unit vector $|\psi ⟩$ and real $\varphi$,multiplies the component parallel to $|\psi ⟩$ by $e^{i\varphi }$ and fixes its orthogonal complement. The case $\varphi =\pi$ 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 $|\psi ⟩$, $I-2|\psi ⟩⟨\psi |$ reflects in the hyperplane perpendicular to $|\psi ⟩$.