  We study quantum correlations and complexity of simulation, characterized by
quantum mutual information and entanglement entropy in operator space
respectively, for thermal states in critical, non-critical and quantum chaotic
spin chains. A simple general relation between the two quantities is proposed.
We show that in all cases mutual information and entanglement entropy saturate
with the system size, whereas as a function of the inverse temperature, we find
logarithmic divergences for critical cases and uniform bounds in non-critical
cases. A simple efficient quasi-exact method for computation of arbitrary
entropy related quantities in thermalized XY spin chains is proposed.
