  How mixed can one component of a bi-partite system be initially and still
become entangled through interaction with a thermalized partner? We address
this question here. In particular, we consider the question of how mixed a
two-level system and a field mode may be such that free entanglement arises in
the course of the time evolution according to a Jaynes-Cummings type
interaction. We investigate the situation for which the two-level system is
initially in mixed state taken from a one-parameter set, whereas the field has
been prepared in an arbitrary thermal state. Depending on the particular choice
for the initial state and the initial temperature of the quantised field mode,
three cases can be distinguished: (i) free entanglement will be created
immediately, (ii) free entanglement will be generated, but only at a later time
different from zero, (iii) the partial transpose of the joint state remains
positive at all times. It will be demonstrated that increasing the initial
temperature of the field mode may cause the joint state to become distillable
during the time evolution, in contrast to a non-distillable state at lower
initial temperatures. We further assess the generated entanglement
quantitatively, by evaluating the logarithmic negativity numerically, and by
providing an analytical upper bound.
