Coherifying quantum channels

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(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 Jamiolkowski 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 Jamiolkowski state J(Phi)(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(c) is indeterministic we find explicit bounds on the entropy and purity of J(Phi)(c), and relate the latter to the unitarity of Phi(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.