Computing entanglement entropy (EE) and other information-theoretic quantities in gauge theories is often accompanied by difficulties and ambiguities. In this talk, I show that one can gain new insights into these computations by considering gauge theories compactified on a small circle. In particular, I study Yang-Mills theory compactified on a circle with a double-trace deformation or adjoint fermions and hold it at temperatures near the deconfinement transition. This theory is dual to a multi-component (electric-magnetic) Coulomb gas that can be mapped to an XY-spin model with Z_p preserving perturbations. With the help of a T-duality, I compute Renyi mutual information (RMI) of the XY-spin model by means of the replica trick and Monte Carlo simulations. The simulations indicate that RMI follows the area law scaling, with subleading corrections, and this quantity can be used as a genuine probe to detect deconfinement transitions. I also discuss the effect of fundamental matter on RMI and the implications of these findings in gauge theories and beyond.