  We study entanglement generated between a charge qubit and a bosonic bath due
to their joint evolution which leads to pure dephasing of the qubit. We tune
the parameters of the interaction, so that the decoherence is quantitatively
independent of the number of bosonic modes taken into account and investigate,
how the entanglement generated depends on the size of the environment. A second
parameter of interest is the mixedness of the initial state of the environment
which is controlled by temperature. We show analytically that for a pure
initial state of the environment, entanglement does not depend on environment
size. For mixed initial states of the environment, the generated entanglement
decreases with the increase of environment size. This effect is stronger for
larger temperatures, when the environment is initially more mixed, but in the
limit of an infinitely large environment, no entanglement is created at any
finite temperature.
