Weak limit of the three-state quantum walk on the line

We revisit the one-dimensional discrete time quantum walk with three states and the Grover coin, the simplest model that exhibits localization in a quantum walk. We derive analytic expressions for the localization and a long-time approximation for the entire probability density function (PDF). We find the possibility for asymmetric localization to the extreme that it vanishes completely on one site of the initial conditions. We also connect the time-averaged approximation of the PDF found by Inui et al. [Phys. Rev. E 72, 056112 (2005)] to a spatial average of the walk. We show that this smoothed approximation predicts moments of the real PDF accurately.

Figure 1

The one-dimensional quantum walk with a three-dimensional coin. The matrices $\mathcal{P}$, $\mathcal{Q}$, and $\mathcal{R}$ facilitate a right hop, a left hop, or no change in position, respectively.

Figure 2

Comparison between analytic prediction (lines) and numerical simulation after $T=2^{20}$ time steps (symbols) for the localization around $n=0$. The initial conditions are $|{\psi }_0^0\rangle \propto (1,-1.9,1)$ (red triangles), $|{\psi }_0^0\rangle \propto (10,0,1)$ (blue circles), $|{\psi }_0^0\rangle \propto (1,-3,2+i)$ (black squares), and $|{\psi }_0^0\rangle \propto (410,-800,499)$ (orange diamonds). They have been chosen to show the possible asymmetry of $p_1\left(n\right)$.

Figure 3

The PDF of the walk (blue dots) after $t=4096$ steps with the initial conditions $|{\psi }_0^0\rangle \propto (0,i,1)$. Beyond the shown index range, the probability is essentially zero. Inset: Relative difference between the asymptotic approximation and the numerical values. The prediction is fairly good for a wide range of points.

Figure 4

Top: A short time series of the probability at the initial site. The numerical values (blue circles) and the asymptotic approximation (black line) are indistinguishable on this scale. Bottom: The relative difference for $p(0,t)$ between the simulation and asymptotic approximation. The black line corresponds to $\sim t^{-3/2}$ and is only a guide to the eye.

Figure 5

The smooth approximation $p_{\text{avg}}$ from Eq. (19) (orange line) and the spatial average over 16 sites (blue dots) after 4096 time steps. The initial conditions are the same as in Fig. 3. The actual PDF is shown by gray dots.

Figure 6

Comparison between the numerical values (symbols) and the evaluation of Eq. (C3) for the second moment of the PDF. Inset: The relative difference between the two values.

Figure 7

The numerically observed probabilities (blue dots), the weak limit approximation $p_1\left(n\right)+p_{\text{avg}}$ (orange line), and the corresponding envelopes from the long-time approximation (black lines). For the upper panels, it is $|{\psi }_0^0{\rangle \propto (0,i,1)}^T$; for the lower ones, it is $|{\psi }_0^0{\rangle \propto (1,0,-1)}^T$. The left panels refer to the initial site, whereas the right ones refer to $n=512$, to contrast sites with and without significant localization. Note the difference in scaling with time, depending on the initial conditions. Inset: Zoom for large times on the same $t$ scale. For the data in panel (c), there is no prediction from the method of stationary phase, because for $n=0$ all available orders cancel. However, from the formulas we know that the envelope should scale $\sim t^{-3/2}$, as supported by the numerical data.