AI 中文总结
该研究揭示奇偶置换问题的超经典优势源于基本系统的线性维度而非纠缠，即使无纠缠，量子理论也具概率优势，部分GPT模型无需纠缠即可确定完成任务。
AI 中文摘要
我们研究对n个粒子施加的未知置换的奇偶性（奇/偶）进行识别的任务。经典情形下，每个粒子使用少于n个不同标签时，成功概率被限制为随机猜测；而量子力学利用制备和测量中的纠缠，每个粒子仅需⌈√n⌉个能级即可完美完成该任务[PRL {\bf 135}, 260603 (2025)]。我们证明，即使没有纠缠制备，量子理论仍比经典策略具有概率优势。此外，这类乘积制备在局域量子理论中可实现完美成功，其中基本系统是量子的，但其组合遵循广义概率理论（GPT）的最小张量积结构。我们进一步识别出GPT模型，这些模型无需在制备阶段或测量阶段使用纠缠即可确定地完成该任务。核心结果表明，基本系统的线性维度而非纠缠是决定置换奇偶性问题中概率优势存在的根本资源，特别是在低于所需维度阈值时，任何纠缠都无法超过随机猜测极限。
英文摘要
We investigate the task of identifying the parity (odd vs even) of an unknown permutation applied to $n$ particles. Classically, using fewer than $n$ distinct labels per particle limits the success probability to random guessing, whereas quantum mechanics, exploiting entanglement in both preparation and measurement, accomplishes the task perfectly with as few as $\big\lceil \sqrt{n}\big\rceil$ levels per particle [\href{https://doi.org/10.1103/yhyv-xnwq}{PRL {\bf 135}, 260603 (2025)}]. We show that even without entangled preparation, quantum theory still offers a probabilistic advantage over classical strategies. Moreover, such product preparations yield perfect success in locally quantum theories, where elementary systems are quantum but their composition follows the minimal tensor product structure of generalized probabilistic theories (GPTs). We further identify GPT models that accomplish the task with certainty without requiring entanglement either at the preparation stage or at the measurement stage. Our central result establishes that the linear dimension of the elementary systems, rather than entanglement, is the fundamental resource governing the existence of probabilistic advantage in the permutation parity problem. In particular, below the required dimension threshold, no amount of entanglement can improve upon the random-guessing limit.
Comments10 pages, 2 figures
