  We employ frames consisting of rank-one projectors (i.e. pure quantum states)
and their induced informationally complete quantum measurements (IC-POVMs) to
represent generally mixed quantum states by quasi-probability distributions. In
the case of discrete frames on finite dimensional systems this results in a
vector like representation by quasi-probability vectors, while for the
continuous frame of coherent states in continuous variable (CV) systems the
approach directly leads to the celebrated representation by Glauber-Sudarshan
P-functions and Husimi Q-functions. We explain that the Kolmogorov distances
between these quasi-probability distributions lead to upper and lower bounds of
the trace distance which measures the distinguishability of quantum states. We
apply these results to the dynamics of open quantum systems and construct a
non-Markovianity witness based on the Kolmogorov distance of the P- and
Q-functions. By means of several examples we discuss the performance of this
witness and demonstrate that it is useful in the regime of high entropy states
for which a direct evaluation of the trace distance is typically very
demanding. For Gaussian dynamics in CV systems we even find a suitable
non-Markovianity measure based on the Kolmogorov distance between the
P-functions which can alternatively be used as a witness for non-Gaussianity.
