  We investigate the decoherence of a qubit coupled to either a quantum
two-level system (TLS) again coupled to an environment, or a classical
fluctuator modeled by random telegraph noise. In order to do this we construct
a model for the quantum TLS where we can adjust the temperature of its
environment, and the decoherence rate independently. The model has a
well-defined classical limit at any temperature and this corresponds to the
appropriate random telegraph process, which is symmetric at high temperatures
and becomes asymmetric at low temperatures. We find that the difference in the
qubit decoherence rates predicted by the two models depends on the ratio
between the qubit-TLS coupling and the decoherence rate in the pointer basis of
the TLS. This is then the relevant parameter which determines whether the TLS
has to be treated quantum mechanically or can be replaced by a classical
telegraph process. We also compare the mutual information between the qubit and
the TLS in the classical and quantum cases.
