Classical, quantum, and general probabilistic state-discrimination profiles

Abstract

For a collection of states indexed by $[n]=1,\dots ,n$, let $F(H)$ denote the optimal unnormalized minimum discrimination error for each nonempty subfamily $H\subseteq [n]$. 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 ${\mathfrak{G}}_n$ is a bounded rational polytope, giving a hierarchy ${\mathfrak{C}}_n\subseteq {\mathfrak{Q}}_n\subseteq {\mathfrak{G}}_n$ 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 ${\mathfrak{G}}_3$, the classical region is characterized by $\beta :=F(123)-F(12)-F(13)-F(23)\le 0$, while the GPT maximum is $1$. For qubit triples, the maximal value is attained by the trine, ${\beta }_{\mathrm{tr}}=3\sqrt{3}/2-2$, and we prove the dimension-independent quantum bound ${\beta }_{\mathrm{Q}}\le 2/3$, with a slight further improvement. Hence $0={\beta }_{\mathrm{C}}<3\sqrt{3}/2-2\le {\beta }_{\mathrm{Q}}\le 2/3<1={\beta }_{\mathrm{GPT}}$. We conjecture ${\beta }_{\mathrm{Q}}={\beta }_{\mathrm{tr}}$ 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.