Computing the entanglement entropy in confining gauge theories is often accompanied by puzzles and ambiguities. In this work we show that compactifying the theory on a small circle S1L evades these difficulties. In particular, we study Yang-Mills theory on R3×S1L with double-trace deformations 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 either to an XY-spin model with ℤp symmetry-preserving perturbations or dual Sine-Gordon model. The entanglement entropy of the dual Sine-Gordon model exhibits an extremum at the critical temperature/crossover. We also compute Rényi mutual information (RMI) of the XY-spin model by means of the replica trick and Monte Carlo simulations. These are expensive calculations, since one in general needs to suppress lower winding vortices that do not correspond to physical excitations of the system. We use a T-duality that maps the original XY model to its mirror image, making the extraction of RMI a much efficient process. Our 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. We also discuss the effect of fundamental matter on RMI and the implications of our findings in gauge theories and beyond.
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