  We study measures of decoherence and thermalization of a quantum system $S$
in the presence of a quantum environment (bath) $E$. The entirety $S$$+$$E$ is
prepared in a canonical thermal state at a finite temperature, that is the
entirety is in a steady state. Both our numerical results and theoretical
predictions show that measures of the decoherence and the thermalization of $S$
are generally finite, even in the thermodynamic limit, when the entirety
$S$$+$$E$ is at finite temperature. Notably, applying perturbation theory with
respect to the system-environment coupling strength, we find that under common
Hamiltonian symmetries, up to first order in the coupling strength it is
sufficient to consider $S$ uncoupled from $E$, but entangled with $E$, to
predict decoherence and thermalization measures of $S$. This decoupling allows
closed form expressions for perturbative expansions for the measures of
decoherence and thermalization in terms of the free energies of $S$ and of $E$.
Large-scale numerical results for both coupled and uncoupled entireties with up
to 40 quantum spins support these findings.
