Abstract
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 Φ, i.e., we look for channels Φ^C that induce the same classical transitions T, but are "more coherent". To quantify the coherence of a channel Φ we measure the coherence of the corresponding Jamio{\l}kowski state J_Φ. 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_Φ^C of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To assess the extent to which an optimal process Φ^C is indeterministic we find explicit bounds on the entropy and purity of J_Φ^C, and relate the latter to the unitarity of Φ^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 Φ and reduces its rank (the minimal number of required Kraus operators) from d² to d.