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  F 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 Jamiołkowski 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 Jamiołkowski state J  F of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To assess the extent to which an optimal process  F is indeterministic we find explicit bounds on the entropy and purity of J  F , and relate the latter to the unitarity of  F . We also find optimal coherifications for several classes of channels, including all onequbit 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 d 2 to d.

We investigate an original family of quantum distinguishability problems, where the goal is to perfectly distinguish between M quantum states that become identical under a completely decohering map. Similarly, we study distinguishability of M quantum channels that cannot be distinguished when one is restricted to decohered input and output states. The studied problems arise naturally in the presence of a superselection rule, allow one to quantify the amount of information that can be encoded in phase degrees of freedom (coherences), and are related to time-energy uncertainty relation. We present a collection of results on both necessary and sufficient conditions for the existence of M perfectly distinguishable states (channels) that are classically indistinguishable.