= Hadamard-phase-Hadamard qubit rotation
{c}
{title2=$HP_\alpha H=e^{i\alpha/2}e^{-i\alpha\sigma_x/2}$}
For $P_\alpha=\operatorname{diag}(1,e^{i\alpha})$, conjugation by the <Hadamard gate> turns a relative <quantum phase> into an $x$-axis <Bloch vector> <rotation>. In the convention $\rho=(I+\mathbf P\cdot\boldsymbol\sigma)/2$, the map is $(P_x,P_y,P_z)\mapsto(P_x,P_y\cos\alpha-P_z\sin\alpha,P_y\sin\alpha+P_z\cos\alpha)$. An $x$-polarized input is unchanged, whereas a $z$-polarized input gives a sinusoidal population difference. A maximally mixed input remains maximally mixed under every <unitary time evolution>.
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