We investigate system-environment correlations based on the exact dynamics of
a qubit and its environment in the framework of pure decoherence (phase
damping). We focus on the relation of decoherence and the build-up of
system-reservoir entanglement for an arbitrary (possibly mixed) initial qubit
state. In the commonly employed regime where the qubit dynamics can be
described by a Markov master equation of Lindblad type, we find that for almost
all qubit initial states inside the Bloch sphere, decoherence is complete while
the total state is still separable - no entanglement is involved. In general,
both "separable" and "entangling" decoherence occurs, depending on temperature
and initial qubit state. Moreover, we find situations where classical and
quantum correlations periodically alternate as a function of time in the regime
of low temperatures
