  An $E_0$-semigroup of $B(H)$ is a one parameter strongly continuous semigroup
of $*$-endomorphisms of $B(H)$ that preserve the identity. Every
$E_0$-semigroup that possesses a strongly continuous intertwining semigroup of
isometries is cocycle conjugate to an $E_0$-semigroup induced by the Bhat
induction of a $CP$-flow over a separable Hilbert space $K$. We say an
$E_0$-semigroup $\alpha$ is $q$-pure if the $CP$-subordinates $\beta$ of norm
one (i.e. $\Vert\beta_t(I)\Vert = 1$ and $\alpha_t-\beta_t$ is completely
positive for all $t \geq 0$) are totally ordered in the sense that if $\beta$
and $\gamma$ are two $CP$-subordinates of $\alpha$ of norm one, then $\beta
\geq \gamma$ or $\gamma \geq \beta$. This paper shows how to construct and
classify all $q$-pure $E_0$-semigroups induced by $CP$-flows over a
finite-dimensional Hilbert space $K$ up to cocycle conjugacy.
