can be expressed in the form of a probabilistic process, again with possibly negative transition rates [@Klimov2005a; @Ruzzi2000]. Alternatively, one can make the transition rates non-negative by adding more structure to the discrete phase space (either a new binary variable [@Cohendet1988; @Cohendet1990] or a phase [@Hashimoto2007]).

In this paper we explore further the formulation of quantum evolution in discrete phase space, for two distinct ways of characterizing this evolution. (i) For a general normalization-preserving quantum transformation, that is, for a trace-preserving completely positive map, we express the transformation in terms of (possibly negative) transition probabilities in the discrete phase space. (ii) For the special case of Hamiltonian evolution—that is, for a closed system—we also express the evolution in terms of transition probabilities per unit time (closely related work can be found in Refs. [@Cohendet1988; @Cohendet1990; @Klimov2005a; @Ruzzi2000; @Hashimoto2007]). These transition rates come directly from a discrete version of the Moyal bracket [@Groenewold1946; @Moyal1949]. In both cases, our main goal is to formulate simple criteria for deciding whether a given set of transition probabilities or transition rates corresponds to a legitimate quantum process. That is, it turns out that one is not free to choose any properly normalized (but possibly negative) 