  We find that when a qubit initialized in a pure state experiences pure
dephasing due to interaction with an environment, separable qubit-environment
states generated during the evolution also have zero quantum discord with
respect to the environment. What follows is that the set of separable states
which can be reached during the evolution has zero volume and hence, such
effects as sudden death of qubit-environment entanglement are very unlikely. In
case of the discord with respect to the qubit, a vast majority of separable
states qubit-environment is discordant, but in specific situations zero-discord
states are possible. This is conceptually important since there is a connection
between the discordance with respect to a given subsystem and the possibility
of describing the evolution of this subsystem using completely positive maps.
Finally, we use the formalism to find an exemplary evolution of an entangled
state of two qubits that is completely positive, occurs solely due to
interaction of only one of the qubits with its environment (so one could guess
that it corresponds to a local operation, since it is local in a physical
sense), but which nevertheless causes the enhancement of entanglement between
the qubits. While this simply means that the considered evolution is completely
positive, but does not belong to LOCC, it shows how much caution has to be
exercised when identifying evolution channels that belong to that class.
