Classical, quantum, and general probabilistic state-discrimination profiles

For a collection of states indexed by [n]= 1,,n, let F(H) denote the optimal unnormalized minimum discrimination error for each nonempty subfamily H [n]. We call F the discrimination profile and characterize exactly which profiles can arise in a general probabilistic theory (GPT). For fixed n, the GPT region G n is a bounded rational polytope, giving a hierarchy C n Q n G n of classical, quantum, and GPT profiles, analogous to the local--quantum--no-signalling hierarchy in Bell theory. For n=3 we determine the classical and GPT polytopes explicitly. Within G 3, the classical region is characterized by β:=F(123)-F(12)-F(13)-F(23) 0, while the GPT maximum is 1. For qubit triples, the maximal value is attained by the trine, β tr=3 sqrt3/2-2, and we prove the dimension-independent quantum bound β Q 2/3, with a slight further improvement. Hence 0=β C<3 sqrt3/2-2 β Q 2/3<1=β GPT. We conjecture β Q=β tr and prove this for arbitrary pure-state triples, together with further supporting evidence. For arbitrary n, every classical facet admitting a GPT violation already admits a quantum violation in dimension at most 3. A four-state qubit example shows that quantum success profiles need not be submodular, unlike for n=3. Finally, we exhibit a four-dimensional quantum triple whose discrete profile is classical for all subfamilies but becomes nonclassical when unequal priors are allowed.