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arXiv 2605.13645math.COmath.SP
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量子分数复兴与单位ary Cayley 图中的纠缠熵

Quantum Fractional Revival and Entanglement Entropy in Unitary Cayley Graphs

Moscow Institute of Physics and Technology（莫斯科物理技术学院）

Duaa Abdullah

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16 pages, 3 figures and 3 tables
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AI总结：

本文扩展了单位ary Cayley 图上量子分数复兴理论，探讨拉普拉斯矩阵哈密顿量下的复兴现象，推导了最小复兴时间公式，并研究了强共谱顶点对与纠缠熵的关系。

摘要：

本文扩展了单位ary Cayley 图上量子分数复兴（QFR）理论，从多个未解决的方向进行扩展。首先，我们研究了拉普拉斯矩阵哈密顿量下的QFR，证明了正则图中两种模型仅相差一个全局相位因子，并确定了拉普拉斯框架独立允许QFR的条件。其次，对于阶数为n=2p的单位ary Cayley图（p为奇素数），我们推导出最小复兴时间t*=2π/p，并证明相应的复兴振幅由α=cos(2π/p)和β=-i sin(2π/p)给出。第三，我们通过Z_n的算术结构完全表征了强共谱顶点对，证明当n是两次素数时，强共谱性等价于反向性。最后，我们计算了由QFR生成的von Neumann纠缠熵，从而获得一组量子信息度量，并证明熵仅依赖于复兴振幅|α|和|β|。

英文摘要：

This paper extends the theory of quantum fractional revival (QFR) on unitary Cayley graphs $X=(V(\mathbb{Z}_n),E(S))$ in several directions that remained unresolved in previous work. First, we investigate QFR with respect to the Laplacian matrix Hamiltonian in addition to the adjacency matrix Hamiltonian. In particular, we prove that for regular graphs the two models differ only by a global phase factor, and we determine the conditions under which the Laplacian framework independently admits QFR. Second, for unitary Cayley graphs of order $n=2p$, where $p$ is an odd prime, we derive an explicit closed-form expression for the minimum revival time, $t^{*}=\frac{2π}{p},$ and show that the associated revival amplitudes are given by \[ α=\cos\!\left(\frac{2π}{p}\right), \qquad β=-i\sin\!\left(\frac{2π}{p}\right). \] Third, we provide a complete characterization of strongly cospectral vertex pairs in $X=(V(\mathbb{Z}_n),E(S))$ through the arithmetic structure of $\mathbb{Z}_n$, establishing that strong cospectrality is equivalent to antipodality whenever $n$ is twice a prime. Finally, we compute the von Neumann entanglement entropy generated by QFR for all admissible graphs, thereby obtaining a collection of quantum information measures and proving that the entropy depends solely on the revival amplitudes $|α|$ and $|β|$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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