The weak limit theorem (WLT), the quantum analogue of the central limit theorem, is foundational to quantum walk (QW) theory. Unlike the universal Gaussian limit of classical walks, deriving analytical forms of the limiting probability density function (PDF) in higher dimensions has remained a challenge since the 1D Konno distribution was established. Previous explicit PDFs for 2D models were limited to specific cases whose fundamental nature was unclear.
This paper resolves this long-standing gap by introducing the notion of maximal speed $v_{\mathrm{max}}$ as a critical parameter. We demonstrate that all previous 2D solutions correspond to a degenerate regime where $v_{\mathrm{max}} = 1$. We then present the first exact analytical representation of the limiting PDF for the physically richer, unexplored regime $v_{\mathrm{max}} < 1$ of a general class of 2D two-state QWs.
Our result reveals 2D Konno functions that govern these dynamics. We establish these as the proper 2D generalization of the 1D Konno distribution by demonstrating their convergence to the 1D form in the appropriate limit. Furthermore, our derivation, based on spectral analysis of the group velocity map, analytically resolves the singular asymptotic structure: we explicitly determine the caustics loci where the PDF diverges and prove they define the boundaries of the distribution's support. By also providing a closed-form expression for the weight functions, this work offers a complete description of the 2D WLT.
Limit Theorems for Decoherent Two Dimensional Quantum Walks
在本文中，我们考虑由[Brun，T.A，et.al，Phys.Rev.A 67（2003）032304]引入的具有退相干算子的模型，该模型最近由[Ampadu，C.，Brun-Type Formalism for Decoherence in Two Dimensional Quantum Walks，Communication in Theoretical Physics To Appear，arXiv：1104.2061（2011）]以获得退相干量子行走的极限。
Central limit theorems for open quantum random walks on the crystal lattices
考虑晶格上的开量子随机游动，研究了其中心极限定理。在整数格上，开量子随机游动满足Attal，{ \it et al}证明的中心极限定理.本文证明了晶格上开量子随机游动的中心极限定理。然后，我们提供了一些例子的Hisononal格。我们还发展了晶格的傅里叶分析。这导致了开量子随机游动的对偶过程的构造。它相当于得到了概率密度的傅里叶变换，在计算游动的特征函数时非常有用。本文构造了晶格上开量子随机游动的对偶过程，并给出了一些例子。
