  We study the entanglement entropy and the mutual information in coupled
harmonic systems at finite temperature. Interestingly, we find that the mutual
information does not vanish at infinite temperature, but it rather reaches a
specific finite value, which can be attributed to classical correlations
solely. We further obtain high and low temperature expansions for both
quantities. Then, we extend the analysis performed in the seminal paper by
Srednicki (Phys. Rev. Lett. 71, 666 (1993)) for free real scalar field theories
in Minkowski space-time in 3+1 dimensions at a thermal state. We find that the
mutual information obeys an area law, similar to that obeyed by the
entanglement entropy at vanishing temperature. The coefficient of this area law
does not vanish at infinite temperature. Then, we calculate this coefficient
perturbatively in an $1/\mu$ expansion, where $\mu$ is the mass of the scalar
field. Finally, we study the high and low temperature behaviour of the area law
term.
