  In the theory of quantum information, the mixed-unitary quantum channels, for
any positive integer dimension $n$, are those linear maps that can be expressed
as a convex combination of conjugations by $n\times n$ complex unitary
matrices. We consider the mixed-unitary rank of any such channel, which is the
minimum number of distinct unitary conjugations required for an expression of
this form. We identify several new relationships between the mixed-unitary
rank~$N$ and the Choi rank~$r$ of mixed-unitary channels, the Choi rank being
equal to the minimum number of nonzero terms required for a Kraus
representation of that channel. Most notably, we prove that the inequality
$N\leq r^2-r+1$ is satisfied for every mixed-unitary channel (as is the
equality $N=2$ when $r=2$), and we exhibit the first known examples of
mixed-unitary channels for which $N>r$. Specifically, we prove that there exist
mixed-unitary channels having Choi rank $d+1$ and mixed-unitary rank $2d$ for
infinitely many positive integers $d$, including every prime power $d$. We also
examine the mixed-unitary ranks of the mixed-unitary Werner--Holevo channels.
