Universal manipulation of a single qubit

We find the optimal universal way of manipulating a single qubit, $|\psi (\vartheta ,\phi )〉,$ such that $(\vartheta ,\phi )\to (\vartheta -\alpha ,\phi -\beta ).\mathrm{}$ Such optimal transformations fall into two classes. For $0<~\alpha <~\pi /2,$ the optimal map is the identity and the fidelity varies monotonically from 1 (for $\alpha =0)$ to $\frac{1}{2}$ (for $\alpha =\pi /2).\mathrm{}$ For $\pi /2\mathrm{}<~\alpha <~\pi ,$ the optimal map is the universal-NOT gate and the fidelity varies monotonically from $\frac{1}{2}$ (for $\alpha =\pi /2)$ to $\frac{2}{3}$ (for $\alpha =\pi ).\mathrm{}$ The fidelity $\frac{2}{3}$ is equal to the fidelity of measurement. It is therefore rather surprising that for some values of $\alpha$ the fidelity is lower than $\frac{2}{3}.$ For instance, a universal square root of NOT operation is more difficult to approximate than the universal NOT gate itself.