与离散时间量子行走相关的相互作用Fock空间上的Poisson算子
Poisson operator on the interacting Fock space associated with a discrete-time quantum walk
浏览论文内容
中文总结 AI 辅助
该论文研究离散时间量子行走对应的相互作用Fock空间上的QW-Poisson算子，分析其谱性质、密度相变，计算矩生成函数等统计量，还关联了非交换概率论。
中文摘要 AI 辅助
我们研究与离散时间量子行走相关的相互作用Fock空间上的Poisson算子，将其称为QW-Poisson算子。首先，我们研究QW-Poisson分布的谱性质，特别通过尺寸偏置变换建立了一般相互作用Fock空间上Poisson算子与逆Poisson算子谱分布之间的关系。接着，我们研究QW-Poisson分布密度的边界行为，证明在支撑集的左端点处会发生相变：根据参数的不同，密度要么衰减至0，要么爆炸至+∞。此外，该相变与QW-Poisson分布的原子数量的转变（等价于QW-Poisson算子的点谱的转变）以及0是否属于QW-Poisson算子的谱完全一致。然后，我们计算了QW-Poisson算子的矩生成函数和若干统计量，还通过Poisson近似得到了Konno分布的一个极限定理。最后，我们研究了与离散时间量子行走相关的相互作用Fock空间和非交换概率论之间的联系。
英文摘要
We study the Poisson operator on the interacting Fock space associated with a discrete-time quantum walk, which we call the QW-Poisson operator. First, we investigate the spectral properties of the QW-Poisson distribution. In particular, we establish a relation between the spectral distributions of the Poisson operator and the reversed Poisson operator on a general interacting Fock space via a size-biased transform. Next, we study the edge behavior of the density of the QW-Poisson distribution. We show that a phase transition occurs at the left endpoint of the support: depending on the parameter, the density either decays to $0$ or blows up to $+\infty$. Moreover, this phase transition coincides with the transition in the number of atoms of the QW-Poisson distribution, equivalently, in the point spectrum of the QW-Poisson operator, and with whether $0$ belongs to the spectrum of the QW-Poisson operator. Finally, we study a connection between the interacting Fock space associated with a discrete-time quantum walk and noncommutative probability theory. More precisely, we compute the moment-generating function and moments of the QW-Poisson operator, and obtain a limit theorem for the Konno distribution via a Poisson approximation. We also investigate the Boolean self-decomposability of the Konno distribution and the shifted reversed QW-Poisson distribution.
