Classical quantum and probabilistic profiles reveal limits in state discrimination
Classical, quantum, and general probabilistic state-discrimination profiles
Information Theory
Summary
This paper studies how well different types of physical theories—classical, quantum, and more general probabilistic ones—can tell apart collections of states. The authors define a function called the discrimination profile that measures the minimal error when distinguishing subsets of states. They find that these profiles form nested sets, with classical theory inside quantum, which is inside general probabilistic theories. By examining triples and quadruples of states, the authors identify exact boundaries between these sets and uncover surprising cases where quantum behavior differs from classical patterns.
What this means in practice
- •For quantum communication engineers: Design improved protocols for distinguishing quantum states by using refined state-discrimination profiles to optimize error rates.
- •For quantum algorithm developers: Use characterizations of quantum versus classical discrimination limits to guide algorithmic choices in state identification tasks.
A theory result. No direct application yet.
Authors
Mihály Weiner
Abstract
For a collection of states indexed by $[n]={1,\ldots,n}$, 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 $β:=F(123)-F(12)-F(13)-F(23)\le0$, while the GPT maximum is $1$. For qubit triples, the maximal value is attained by the trine, $β_{\mathrm{tr}}=3\sqrt3/2-2$, and we prove the dimension-independent quantum bound $β_{\mathrm Q}\le2/3$, with a slight further improvement. Hence $0=β_{\mathrm C}<3\sqrt3/2-2\leβ_{\mathrm Q}\le2/3<1=β_{\mathrm{GPT}}$. We conjecture $β_{\mathrm Q}=β_{\mathrm{tr}}$ 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.