  We start from Wootter's construction of discrete phase spaces and Wigner
functions for qubits and more generally for finite dimensional Hilbert spaces.
We look at this framework from a non-commutative space perspective and we focus
on the Moyal product and the differential calculus on the discrete phase
spaces. In particular, the qubit phase space provides the simplest example of a
four-point non-commutative phase space. We give an explicit expression of the
Moyal bracket as a differential operator. We then compare the quantum dynamics
encoded by the Moyal bracket to the classical dynamics: we show that the
classical Poisson bracket does not satisfy the Jacobi identity thus leaving the
Moyal bracket as the only consistent symplectic structure. We finally
generalizes our analysis to Hilbert spaces of prime dimensions d and their
associated d*d phase spaces.
