  Using the Ryu-Takayanagi conjectured formula for entanglement entropy in the
context of gauge-gravity duality, we investigate properties of mutual
information between two disjoint rectangular sub-systems in finite temperature
relativistic conformal field theories in d-spacetime dimensions and
non-relativistic scale-invariant theories in some generic examples. In all
these cases mutual information undergoes a transition beyond which it is
identically zero. We study this transition in detail and find universal
qualitative features for the above class of theories which has holographic dual
descriptions. We also obtain analytical results for mutual information in the
specific regime of the parameter space. This demonstrates that mutual
information contains the quantum entanglement part of the entanglement entropy,
which is otherwise dominated by the thermal entropy at large temperatures.
