发表机构
Okegawa City Okegawa West Junior High School; Department of Information and Computer Sciences, Saitama University（大胡市大胡西初中; 埼玉大学信息计算机科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究旋转硬币族中阿达马硬币的正停留时间分布刚性，证明有限时间精确均匀性可刻画阿达马硬币，通过矩阵值生成函数等方法完成相关分析。
AI 中文摘要
我们研究一维两态量子行走在返回原点条件下的正停留时间分布。Konno 证明，对于阿达马（Hadamard）行走，该条件分布在时间为4的倍数时恰好均匀。本文中，我们探究这种有限时间的精确均匀性是否能在旋转硬币族中刻画阿达马硬币。对于固定初态，我们证明旋转硬币满足以下三个条件等价：条件分布在时间8时恰好均匀；条件分布在所有m≥2的时间4m时恰好均匀；该硬币为阿达马硬币。因此，Konno发现的均匀性现象被刻画为旋转硬币族中阿达马硬币的一种刚性现象。证明过程使用了返回原点路径的矩阵值生成函数，分析了半直线上吸收过程产生的代数结构，最后通过比较时间8处的低次系数，证明精确均匀性迫使旋转硬币为阿达马硬币。
英文摘要
We study the distribution of the positive sojourn time for a one-dimensional two state quantum walk, conditioned on return to the origin. Konno showed that, for the Hadamard walk, this conditional distribution is exactly uniform at times divisible by $4$. In this paper, we investigate whether this finite time exact uniformity characterizes the Hadamard coin within the family of rotation coins. For a fixed initial state, we prove that the following three conditions are equivalent for rotation coins: the conditional distribution is exactly uniform at time $8$; the conditional distribution is exactly uniform at every time $4m$ with $m\ge2$; and the coin is the Hadamard coin. Thus, the uniformity phenomenon found by Konno is characterized as a rigidity phenomenon of the Hadamard coin within the rotation coin family. The proof uses a matrix-valued generating function for paths returning to the origin. We analyze the algebraic structure arising from an absorbing process on the half line. Finally, by comparing low degree coefficients at time $8$, we show that exact uniformity forces the rotation coin to be the Hadamard coin.
Comments18 pages, 1 figure
