经典、量子与一般概率态判别轮廓
Classical, quantum, and general probabilistic state-discrimination profiles
- Institute of Mathematics, Budapest University of Technology and Economics（布达佩斯技术与经济大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文刻画了经典、量子与一般概率理论中态判别轮廓的可行区域，建立了层级关系，并给出了三态情形下的精确边界与量子界猜想。
AI中文摘要：
对于由$[n]={1,\ldots,n}$索引的一组态，令$F(H)$表示每个非空子族$H\subseteq[n]$的最优非归一化最小判别误差。我们将$F$称为判别轮廓，并精确刻画了在一般概率理论（GPT）中可能出现哪些轮廓。对于固定的$n$，GPT区域$\mathfrak{G}_n$是一个有界有理多胞形，从而给出经典、量子与GPT轮廓的层级关系$\mathfrak{C}_n\subseteq\mathfrak{Q}_n\subseteq\mathfrak{G}_n$，类似于Bell理论中的局域-量子-无信号层级。对于$n=3$，我们显式确定了经典与GPT多胞形。在$\mathfrak{G}_3$内，经典区域由$\beta:=F(123)-F(12)-F(13)-F(23)\le0$刻画，而GPT最大值为$1$。对于量子比特三元组，最大值由trine态达到，$\beta_{\mathrm{tr}}=3\sqrt3/2-2$，并且我们证明了与维度无关的量子界$\beta_{\mathrm Q}\le2/3$，并略有进一步改进。因此$0=\beta_{\mathrm C}<3\sqrt3/2-2\le\beta_{\mathrm Q}\le2/3<1=\beta_{\mathrm{GPT}}$。我们猜想$\beta_{\mathrm Q}=\beta_{\mathrm{tr}}$，并针对任意纯态三元组证明了这一猜想，同时提供了进一步的支持证据。对于任意$n$，每个允许GPT违背的经典面在至多3维中已经允许量子违背。一个四态量子比特例子表明，量子成功轮廓不必是子模的，这与$n=3$的情况不同。最后，我们展示了一个四维量子三元组，其离散轮廓对于所有子族都是经典的，但当允许不等先验时变为非经典的。
英文摘要：
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.
