  We consider the coupling of a qubit in a pure state to an environment in an
arbitrary state, and characterize the possibility of qubit-environment
entanglement generation during the evolution of the joint system, that leads to
pure dephasing of the qubit. We give a simple necessary and sufficient
condition on the initial density matrix of the environment together with the
properties of the interaction, for appearance of qubit-environment
entanglement. Any entanglement created turns out to be detectable by the
Peres-Horodecki criterion. Furthermore, we show that for a large family of
initial environmental states, the appearance of nonzero entanglement with the
environment is necessarily accompanied by a change in the state of the
environment (i.e. by the back-action of the qubit).
