  Is it always possible to explain random stochastic transitions between states
of a finite-dimensional system as arising from the deterministic quantum
evolution of the system? If not, then what is the minimal amount of randomness
required by quantum theory to explain a given stochastic process? Here, we
address this problem by studying possible coherifications of a quantum channel
$\Phi$, i.e., we look for channels $\Phi^{\mathcal{C}}$ that induce the same
classical transitions $T$, but are "more coherent". To quantify the coherence
of a channel $\Phi$ we measure the coherence of the corresponding
Jamio{\l}kowski state $J_{\Phi}$. We show that the classical transition matrix
$T$ can be coherified to reversible unitary dynamics if and only if $T$ is
unistochastic. Otherwise the Jamio{\l}kowski state $J_\Phi^{\mathcal{C}}$ of
the optimally coherified channel is mixed, and the dynamics must necessarily be
irreversible. To assess the extent to which an optimal process
$\Phi^{\mathcal{C}}$ is indeterministic we find explicit bounds on the entropy
and purity of $J_\Phi^{\mathcal{C}}$, and relate the latter to the unitarity of
$\Phi^{\mathcal{C}}$. We also find optimal coherifications for several classes
of channels, including all one-qubit channels. Finally, we provide a
non-optimal coherification procedure that works for an arbitrary channel $\Phi$
and reduces its rank (the minimal number of required Kraus operators) from
$d^2$ to $d$.
