  CConsider a bipartite quantum system consisting of two subsystems A and B.
The reduced density matrix ofA a is obtained by taking the partial trace with
respect to B. In this work, we will show that the Wigner distribution of this
reduced density matrix is obtained by integrating the total Wigner distribution
with respect to the phase space variables corresponding to subsystem B. The
proof we give is rigorous (as opposed to those found in the literature) and
makes use of the Weyl--Wigner--Moyal phase space formalism. Our main result is
applied to general Gaussian mixed states, of which it gives a particularly
simple and precise description. We also briefly discuss the purification of a
mixed state from the Wigner formalism point of view.
