Universal manipulation of a single qubit
Lucién Hardy, David D. Song · Physical Review A · 2001
We find the optimal universal way of manipulating a single qubit, $|\ensuremath{\psi}(\ensuremath{\vartheta},\ensuremath{\varphi})〉,$ such that $(\ensuremath{\vartheta},\ensuremath{\varphi})\ensuremath{\rightarrow}(\ensuremath{\vartheta}\ensuremath{-}\ensuremath{\alpha},\ensuremath{\varphi}\ensuremath{-}\ensuremath{\beta}).$ Such optimal transformations fall into two classes. For $0<~\ensuremath{\alpha}<~\ensuremath{\pi}/2,$ the optimal map is the identity and the fidelity varies monotonically from 1 (for $\ensuremath{\alpha}=0)$ to $\frac{1}{2}$ (for $\ensuremath{\alpha}=\ensuremath{\pi}/2).$ For $\ensuremath{\pi}/2<~\ensuremath{\alpha}<~\ensuremath{\pi},$ the optimal map is the universal-NOT gate and the fidelity varies monotonically from $\frac{1}{2}$ (for $\ensuremath{\alpha}=\ensuremath{\pi}/2)$ to $\frac{2}{3}$ (for $\ensuremath{\alpha}=\ensuremath{\pi}).$ The fidelity $\frac{2}{3}$ is equal to the fidelity of measurement. It is therefore rather surprising that for some values of $\ensuremath{\alpha}$ 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.