发表机构
Instituto de Física, Universidad Nacional Autónoma de México; Escuela de Ciencias Físicas y Matemáticas, Universidad de San Carlos de Guatemala; Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien（墨西哥国立自治大学物理研究所; 危地马拉圣卡洛斯大学物理与数学院; 维也纳量子科学与技术中心，原子物理研究所，维也纳工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在最小离散时间量子行走中研究帕龙多悖论，提出输运向量几何判据，当组合策略向量超出单个策略锥体时悖论出现，并计算了悖论集合的概率。
AI 中文摘要
我们研究了帕龙多悖论——即组合失败的策略能产生获胜策略的现象——在最小离散时间量子行走中的表现。我们引入了输运向量，它编码在硬币的稳态中，其与初始硬币态的内积给出行走者的渐近速度。更一般地，当组合策略的输运向量落在由单个策略张成的锥体之外时，悖论恰好出现——这一几何判据适用于任何策略组合。例如，它解释了为何在单步内组合两个硬币算子能产生悖论，而简单交替使用它们却不能，因为交替使组合向量保持在锥体内。悖论集合具有非零测度，我们在代表性情形下显式计算了其概率。这将悖论视为量子行走中设计可达输运的一个实例。
英文摘要
We study Parrondo's paradox -- the phenomenon where combining losing strategies yields a winning one -- in a minimal discrete-time quantum walk. We introduce the transport vector, encoded in the coin's steady state, whose inner product with the initial coin state gives the walker's asymptotic velocity. More generally, the paradox emerges exactly when the transport vector of the combined strategy falls outside the cone spanned by the individual ones -- a geometric criterion valid for any combination of strategies. It explains, for instance, why composing two coin operators within a single step can produce the paradox while simple alternation between them cannot, since alternation keeps the combined vector confined to the cone. The paradoxical set has nonzero measure, and we compute its probability explicitly in representative cases. This casts the paradox as one instance of designing reachable transport in quantum walks.
Comments5+12 pages, 4 figures. Comments are welcome
