transition probabilities or transition rates, and we want to know what the constraints are. With a specification of these constraints, the Wigner-function formulation of the evolution of finite-state systems becomes self-contained, not requiring any reference to Hilbert space or to probability amplitudes. The identification of the constraints is the main new contribution of this paper.

The paper is organized as follows. In Section II we specify the particular form of the discrete Wigner function that we will be using. In Section III we consider the transition probabilities in phase space for a general normalization-preserving quantum transformation and ask what constraints there are on these probabilities. We also show in that section exactly how these constraints are strengthened when the evolution is unitary. We do a similar analysis in Section IV for the transition [*rates*]{} in the case of continuous Hamiltonian evolution. These transition rates can be computed easily if one expands the Hamiltonian as a linear combination of displacement operators in the discrete phase space, as we explain in Section V. Finally we present our conclusions in Section VI. In Appendix A, we work out the explicit form of the four-point structure function, which figures prominently in the definition of 