Characterization and measurement of qubit-environment-entanglement generation during pure dephasing

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 the 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 backaction of the qubit).