Two-dimensional quantum central limit theorem by quantum walks
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.