  A canonical form for unital qubit channels under local unitary transforms is
obtained. In particular, it is shown that the eigenvalues of the Choi matrix of
a unital quantum channel form a complete set of invariants of the canonical
form. It follows immediately that every unital qubit channel is the average of
four unitary channels. More generally, a unital qubit channel can be expressed
as the convex combination of unitary channels with convex coefficients $p_1,
\dots, p_m$ as long as $2(p_1, \dots, p_m)$ is majorized by the vector of
eigenvalues of the Choi matrix of the channel. A unital qubit channel in the
canonical form will transform the Bloch sphere onto an ellipsoid. We look into
the detailed structure of the natural linear maps sending the Bloch sphere onto
a corresponding ellipsoid.
