Classical, quantum, and general probabilistic state-discrimination profiles
Abstract
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 Gn is a bounded rational polytope, giving a hierarchy Cn⊆Qn⊆Gn 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 G3, 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, βtr=3√3/2−2, and we prove the dimension-independent quantum bound βQ≤2/3, with a slight further improvement. Hence 0=βC<3√3/2−2≤βQ≤2/3<1=βGPT. We conjecture βQ=βtr 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.
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BibTeX
@misc{weiner2026classical,
title = {{Classical, quantum, and general probabilistic state-discrimination profiles}},
author = {Mihály Weiner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17920},
eprint = {2609.17920},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17920},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classical, quantum, and general probabilistic state-discrimination profiles
AU - Mihály Weiner
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17920
UR - https://arxiv.org/abs/2609.17920
AB - 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√3/2−2, and we prove the dimension-independent quantum bound β_Q≤2/3, with a slight further improvement. Hence 0=β_C<3√3/2−2≤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.
ER -