  We study measures of decoherence and thermalization of a quantum system $S$
in the presence of a quantum environment (bath) $E$. The whole system is
prepared in a canonical thermal state at a finite temperature. 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 the uncoupled system 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 $E$. Numerical
results for both coupled and decoupled systems with up to 40 quantum spins
validate these findings.
