  We show that, depending on the ratio between the inter- and the intra-species
interactions, a binary mixture trapped in a three-well potential with periodic
boundary conditions exhibits three macroscopic ground-state configurations
which differ in the degree of mixing. Accordingly, the corresponding quantum
states feature either delocalization or a Schr\"odinger cat-like structure. The
two-step phase separation occurring in the system, which is smoothed by the
activation of tunnelling processes, is confirmed by the analysis of the energy
spectrum that collapses and rearranges at the two critical points. In such
points, we show that also Entanglement Entropy, a quantity borrowed from
quantum-information theory, features singularities, thus demonstrating its
ability to witness the double mixining-demixing phase transition. The developed
analysis, which is of interest to both the experimental and theoretical
communities, opens the door to the study of the demixing mechanism in complex
lattice geometries.
