  We determine the total state dynamics of a dephasing open quantum system
using the standard environment of harmonic oscillators. Of particular interest
are random unitary approaches to the same reduced dynamics and
system-environment correlations in the full model. Concentrating on a model
with an at times negative dephasing rate, the issue of "non-Markovianity" will
also be addressed. Crucially, given the quantum environment, the appearance of
non-Markovian dynamics turns out to be accompanied by a loss of
system-environment correlations. Depending on the initial purity of the qubit
state, these system-environment correlations may be purely classical over the
whole relevant time scale, or there may be intervals of genuine
system-environment entanglement. In the latter case, we see no obvious relation
between the build-up or decay of these quantum correlations and
"Non-Markovianity".
