Universal manipulation of a single qubit

We find the optimal universal way of manipulating a single qubit, \psi(theta, phi)], such that (theta, phi)-->(theta-alpha, phi -beta). Such optimal transformations fall into two classes. For 0 less than or equal to alpha less than or equal to pi /2, the optimal map is the identity and the fidelity varies monotonically from 1 (for alpha =0) to 1/2 (for alpha=pi /2). For pi /2s less than or equal to alpha less than or equal to pi, the optimal map is the universal-NOT gate and the fidelity varies monotonically from 1/2 (for alpha=pi /2) to 2/3 (for alpha=pi). The fidelity 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 2/3. For instance, a universal square root of NOT operation is more difficult to approximate than the universal NOT gate itself.