Two-dimensional quantum central limit theorem by quantum walks

Abstract

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.

Figures (11)

• Figure 1: The walker in $\mathbb{Z}^2$ that follows \ref{['state_evol']}.
• Figure 2: Parameter region for $(a, b) \in \Delta$. The parameter pair can take any value within the interior of the gray-shaded triangle and its boundary where $a + b = 1$.
• Figure 3: The support of 2D Konno functions $f_{\pm}$ in \ref{['eq:fpm']} is $E_1\cap E_2,$ which is the intersection of two ellipses (whose boundaries are shown as dashed lines) and is colored gray. The region is plotted for $(a_1,a_2)=(1/4, 3/4),$ i.e., $(a, b) = (3/16, \sqrt{105}/16).$ In general, the domain $\Sigma(\hat{\bm{v}}) = E_1\cap E_2$ is included in $\sigma(\hat{v}_1)\times \sigma(\hat{v}_2) = [-(a + b), a + b]^2$, which is also confirmed in this figure. In particular, $E_1 \cap E_2$ is contained in $[-1, 1]^2$, as visually confirmed in this figure, and therefore Lemma \ref{['lem:5']}(3) holds.
• Figure 4: (a) and (c) show 3D plots of 2D Konno functions $f_+({\bm{v}})$ and $f_-({\bm{v}})$, respectively, for $(a_1, a_2)=(1/4, 3/4),$ i.e., $(a, b) = (3/16, \sqrt{105}/16)$, similar to FIG. \ref{['fig:E1_cap_E2']}. (b) and (d) show the corresponding heat maps. In the heat maps, the color scale represents the function's value, which is zero (dark blue) outside the domain and increases toward the boundary (light blue to white/red), as indicated by the color bars. The set in (b) and (d) is identical to that in FIG. \ref{['fig:E1_cap_E2']}, and it can be seen that $f_{\pm}({\bm{v}})$ diverges on this boundary. This fact corresponds to the divergence of 1D Konno function $f_{\mathrm{K}}$ at the points $v = \pm v_{\mathrm{max}}$.
• Figure 5: Domain deformation towards the 1D limit. The support of the 2D Konno functions $f_\pm$ is depicted in the rotated coordinate system $(v,u)$, which is decomposed into $\tilde{D}_1$ and $\tilde{D}_2$. The region $\tilde{D}_1$ collapses to points $(v, u) = (\pm a_1, 0)$, while $\tilde{D}_2$ shrinks to the 1D interval $[-a_1, a_1]$, which coincides with the support of $f_{\mathrm{K}}$.

Theorems & Definitions (26)

• Lemma 1
• Lemma 2
• Proposition 3: Maximal speed
• Theorem 4: 2D Konno Functions
• Lemma 5
• Definition 6: 1D Limit
• Theorem 7: 1D Limit
• Lemma 8
• proof
• Theorem 9