  We present a characterization of the set of non-signaling correlations in
terms of a two dimensional representation that involves the maximal value of a
Bell functional and the mutual information between the parties. In particular,
we apply this representation to the bipartite Bell scenario with two
measurements and two outcomes. In terms of these physically meaningful
quantities and through numerical optimization methods and some analytical
results, we investigate the frontier between the different subsets of the
non-signaling correlations, focussing on the quantum and post-quantum ones.
Finally, we show that the Tsirelson bound appears as a singular point in this
context without resorting to quantum mechanics.
