[en] Computing entanglement entropy and its cousins is often challenging even in the simplest continuum and lattice models, partly because such entropies depend nontrivially on all geometric characteristics of the entangling region. Quantum information measures between two or more regions are even more complicated, but contain more, and universal, information. In this paper, we focus on Rényi entropy and information of the order n  =  2. For a free field theory, we show that these quantities are mapped to the change of the thermodynamic free energy by introducing boundaries subject to Dirichlet and Neumann boundary conditions in one higher dimension. This mapping allows us to exploit the powerful tools available in the context of thermal Casimir effect, specifically a multipole expansion suited for computing the Rényi information between arbitrarily-shaped regions. In particular, we compute the Rényi information between two disk-shaped regions at an arbitrary separation distance. We provide an alternative representation of the Rényi information as a sum over closed-loop polymers, which establishes a connection to purely entropic effects, and proves useful in deriving information inequalities. Finally, we discuss extensions of our results beyond free field theories. (paper: quantum statistical physics, condensed matter, integrable systems)