Limit distributions of three-state quantum walks: The role of coin eigenstates

We analyze two families of three-state quantum walks which show the localization effect. We focus on the role of the initial coin state and its coherence in controlling the properties of the quantum walk. In particular, we show that the description of the walk simplifies considerably when the initial coin state is decomposed in the basis formed by the eigenvectors of the coin operator. This allows us to express the limit distributions in a much more convenient form. Consequently, striking features which are hidden in the standard basis description are easily identified. Moreover, the dependence of moments of the position distribution on the initial coin state can be analyzed in full detail. In particular, we find that in the eigenvector basis the even moments and the localization probability at the origin depend only on incoherent combination of probabilities. In contrast, odd moments and localization outside the origin are affected by the coherence of the initial coin state.

Figure 1

Second moment of the three-state walk with the coin operator (1) as a function of $|g_1{|}^2$ and $|g_2{|}^2$. We have chosen the coin parameter $\rho =0.5$. The plot indicates that the greatest variance is achieved for the initial state $|{\sigma }_1^{-}\rangle$, while the smallest results from $|{\sigma }^{+}\rangle$. The domain of the plot is restricted to the lower triangle since the probabilities $|g_1{|}^2$ and $|g_2{|}^2$ are limited by the normalization condition (8).

Figure 2

Probability distribution in the vicinity of the origin for the three-state Grover walk starting with the initial state $|{\psi }_C\rangle =\frac{1}{\sqrt{2}}\left(\right|{\sigma }^{+}\rangle +|{\sigma }_2^{-}\rangle )$ after $t=1000$ steps. The localization depicted by the blue dashed line appears only for positive $m$, as predicted by (14).

Figure 3

Probability distribution of the three-state walk with the coin parameter $\rho =0.7$ after $t=100$ steps. As the initial state we have chosen the coin eigenstate $|{\sigma }^{+}\rangle$. For this initial coin state both peaks on the edges of the distribution vanish. This corresponds to the fact that the group-velocity density (15), depicted by the red curve, tends to zero for $v$ approaching $\pm \rho$. To unravel this feature we plot the probability distribution on the logarithmic scale. The blue dashed line depicts the localization probability (13).

Figure 4

Probability distribution of the three-state walk with the coin parameter $\rho =0.8$ after $t=100$ steps. As the initial state we have chosen the coin eigenstate $|{\sigma }_1^{-}\rangle$. For this initial coin state the localization effect disappears. The red curve corresponds to the density (10).

Figure 5

Probability distribution of the three-state walk with the coin parameter $\rho =0.5$ after $t=100$ steps. As the initial state we have chosen the coin eigenstate $|{\sigma }_2^{-}\rangle$. The probability density (16) depicted by the red curve tends to zero near the origin. To unravel this effect we use logarithmic scale on the $y$ axis. The blue dashed line corresponds to the localization probability (13).

Figure 6

Probability distribution of the three-state walk with the coin parameter $\rho =0.6$ after $t=100$ steps. As the initial coin state we have chosen $|{\sigma }_L\rangle$ given by Eq. (17). In this case the peak on the right-hand side of the lattice disappears, as predicted by the density (18) illustrated with the red curve. The blue dashed line depicts the localization probability (13).

Figure 7

The second moment of the three-state walk with the coin (19) as a function of $|g_1{|}^2$ and $|g_2{|}^2$. We have chosen the coin parameter $\phi =\frac{\pi }{4}$. Note the different scale in comparison with Fig. 1. The domain of the plot is restricted due to the normalization condition (8).

Figure 8

Probability distribution of the three-state walk after $t=100$ steps on a logarithmic scale. The red curve denotes the density (22) and the blue dashed line corresponds to the localization probability (26). As the initial state we have chosen the coin eigenstate $|{\gamma }^{+}\rangle$. For small values of the coin parameter, such as $\phi =0.01$ depicted in the upper plot, the density bends to zero for $v$ approaching $\pm \eta$. However, for any $\phi >0$ the density diverges as these points. For increasing values of $\phi$ the density obtains the inverted-bell shape common for quantum walks. This is illustrated in the lower plot where we choose the coin parameter $\phi =\frac{\pi }{4}$.

Figure 9

Probability distribution of the three-state walk with the coin parameter $\phi =\frac{\pi }{6}$ after $t=100$ steps. The red curve corresponds to the density (22). As the initial state we have chosen the coin eigenstate $|{\gamma }_1^{-}\rangle$. For this initial coin state the localization effect disappears for all values of $\phi$.

Figure 10

Probability distribution of the three-state walk after $t=100$ steps on a logarithmic scale. The red curve denotes the density (22) and the blue dashed line corresponds to the localization probability (26). As the initial state we have chosen the coin eigenstate $|{\gamma }_2^{-}\rangle$. For small values of the coin parameter the density has a dip around the origin. This is illustrated in the upper plot, where we choose $\phi =0.01$. With increasing values of the coin parameter the density flattens, as depicted in the lower plot for $\phi =\frac{\pi }{3}$.