Entangling power of disordered quantum walks

We investigate how the introduction of different types of disorder affects the generation of entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional quantum random walks (QRWs). Disorder is modeled by adding another random feature to QRWs, i.e., the quantum coin that drives the system's evolution is randomly chosen at each position and/or at each time step, giving rise to dynamic, fluctuating, or static disorder. The first one is position independent, with every lattice site having the same coin at a given time; the second has time- and position-dependent randomness; and the third one is time independent. We show for several levels of disorder that dynamic disorder is the most powerful entanglement generator, followed closely by fluctuating disorder. Static disorder is the less efficient entangler, being almost always less efficient than the ordered case. Also, dynamic and fluctuating disorder lead to maximally entangled states asymptotically in time for any initial condition, while static disorder has no asymptotic limit and, similarly to the ordered case, has a long-time behavior highly sensitive to the initial conditions.

Figure 1

The dashed lines represent one of many possible realizations of CRWs and the solid curves the probability amplitudes associated with RQRWs. (a)–(c) $\text{The}{\text{RQRW}}_2$ with only two coins $C(j,t)$ (red and blue disks). (d)–(f) $\text{The}{\text{RQRW}}_{\infty }$ with $C(j,t)$ chosen randomly from continuous uniform distributions of quantum coins. For (a) and (d) dynamic disorder we have $C(j,t)=C\left(t\right)$, for (b) and (e) fluctuating disorder $C(j,t)$, and for (c) and (f) the static case $C(j,t)=C\left(j\right)$. Pictorially, for dynamic disorder all coins are the same within a given row (same coin everywhere at time $t$), while for static disorder they are equal within a given column (same coin at a given position $j$ for all time). For fluctuating disorder, at each time step the coin changes randomly at each position and independently of the other sites.

Figure 2

Average entanglement $\langle S_E\rangle$ obtained by averaging over 16 384 initial conditions of the type $|\Psi \left(0\right)\rangle =(cos{\alpha }_s|\uparrow \rangle +e^{i{\beta }_s}sin{\alpha }_s|\downarrow \rangle )\otimes (cos{\alpha }_p|-1\rangle +e^{i{\beta }_p}sin{\alpha }_p|1\rangle )$, where ${\alpha }_{s,p}\in [0,\pi ]$ and ${\beta }_{s,p}\in [0,2\pi ]$. The first numerical experiment was implemented with the initial condition $({\alpha }_s,{\beta }_s,{\alpha }_p,{\beta }_p)=(0,0,0,0)$ and the subsequent ones with quadruples of points in independent increments of $0.4$ until ${\alpha }_{s,p}=\pi$ and ${\beta }_{s,p}=2\pi$. Here we have a 400-step walk. In the upper panel, after many steps and from top to bottom we have $\text{the}{\text{RQRW}}_{\infty }$ built with dynamic disorder (black), fluctuating disorder (red), the ordered QRW with the Hadamard coin (green), and static disorder (blue). In the lower panel, after many steps and from top to bottom we have $\text{the}{\text{RQRW}}_2$ built with dynamic disorder (black), fluctuating disorder (red), static disorder (blue), and the ordered QRW with the Hadamard coin (green).

Figure 3

Average entanglement $\langle S_E\rangle$ computed averaging over 2016 localized initial conditions given as $|\Psi \left(0\right)\rangle =(cos{\alpha }_s|\uparrow \rangle +e^{i{\beta }_s}sin{\alpha }_s|\downarrow \rangle )$ $\otimes |0\rangle$, with ${\alpha }_s\in [0,\pi ]$ and ${\beta }_s\in [0,2\pi ]$. The first realization employed the initial condition $({\alpha }_s,{\beta }_s)=(0,0)$ and the following ones all pairs of points in independent increments of $0.1$ until ${\alpha }_s=\pi$ and ${\beta }_s=2\pi$. Here we work with $\text{the}{\text{RQRW}}_{\infty }$ and a 1000-step walk. After many steps and from top to bottom we have dynamic disorder (black), fluctuating disorder (red), the ordered QRW with the Hadamard coin (green), and static disorder (blue). The inset gives the rate of initial conditions leading to $S_E$ greater than $0.95$, $0.97$, and $0.99$ at step 1000. Note that dynamic disorder outperforms all other types of disorder in its entanglement generation capability (black bars). Fluctuating disorder (right, red bars) ranks second, while static disorder (left, blue bars) is the worst entangler, even worse than the ordered case (green bars).

Figure 4

Same as Fig. 3 but now we work with the balanced ${\text{RQRW}}_2$. After many steps and from top to bottom we have dynamic disorder (black), fluctuating disorder (red), static disorder (blue), and the ordered QRW with the Hadamard coin (green). Note that the last two curves overlap. The inset gives the rate of initial conditions leading to $S_E$ greater than $0.95$, $0.97$, and $0.99$ at step 1000. Again dynamic disorder outperforms all other types of disorder in its entanglement generation capability (black bars). Fluctuating disorder (right, red bars) ranks second, while static disorder (left, blue bars) ranks third, this time being slightly better than the ordered case (green bars).

Figure 5

Same as Figs. 3 and 4 but now we compare the entanglement generation efficiencies between $\text{the}{\text{RQRW}}_{\infty }$ and $\text{the}{\text{RQRW}}_2$. For all panels the upper, black curve represents $\text{the}{\text{RQRW}}_2$ and the lower, red one $\text{the}{\text{RQRW}}_{\infty }$, which shows that $\text{the}{\text{RQRW}}_2$ is more efficient than $\text{the}{\text{RQRW}}_{\infty }$ for all types of disorder. This distinction is more explicit for static disorder (lower right panel).

Figure 6

The curves were obtained averaging over 500 random initial conditions of the form $|\Psi \left(0\right)\rangle =(cos{\alpha }_s|\uparrow \rangle +e^{i{\beta }_s}sin{\alpha }_s|\downarrow \rangle )\otimes (cos{\alpha }_p|-1\rangle +e^{i{\beta }_p}sin{\alpha }_p|1\rangle )$, where ${\alpha }_{s,p}\in [0,\pi ]$ and ${\beta }_{s,p}\in [0,2\pi ]$. We worked with a 1000-step walk and with the ${\text{RQRW}}_{\infty }$, where $\theta (j,t)=\phi (j,t)=0$. The upper panel shows dynamic disorder with $q(j,t)=q\left(t\right)$ picked randomly at each time step from the three different uniform distributions listed in the graph. The lower left panel shows fluctuating disorder with $q(j,t)$ picked randomly at each position and time from the same uniform distributions listed in the upper panel. The lower right panel shows static disorder with $q(j,t)=q\left(j\right)$ chosen at each position from the uniform distributions in the upper panel.

Figure 7

Same as in Fig. 6 but now $\text{the}{\text{RQRW}}_{\infty }$ is such that $q(j,t)=1/2$, $\phi (j,t)=0$, and $\theta (j,t)$ is the random variable.

Figure 8

Same as in Fig. 6 but now $\text{the}{\text{RQRW}}_{\infty }$ is such that $q(j,t)=1/2$, $\theta (j,t)=0$, and $\phi (j,t)$ is the random parameter.

Figure 9

Same as in Fig. 6 but now we implement the biased ${\text{RQRW}}_2$, with $p$ being the probability of picking the Hadamard coin and $1-p$ the Fourier coin. The upper panel shows dynamic disorder where at each time step $t$ the probability $p(j,t)=p\left(t\right)$ to pick the same Hadamard coin for all positions $j$ is given by the three values listed in the graph. All curves tend to overlap, with $p=0.1$ and $0.05$ giving slightly higher entanglement for large $t$. The lower left panel shows fluctuating disorder where for each time $t$ and position $j$ the probability $p(j,t)$ to get a Hadamard coin is given as depicted in the upper panel. Here and in the lower right panel we have from top to bottom the $p=0.05$, $0.1$, and $0.5$ cases. The lower right panel shows static disorder where the probability $p(j,t)=p\left(j\right)$ to get the Hadamard coin at position $j$ is as described in the upper panel. Note that for both the dynamic and fluctuating cases, weak disorder is enough to guarantee $S_E\to 1$ asymptotically.

Figure 10

Discretized Gaussian distributions for the qubit's position. The wider the Gaussian the greater $\sigma$ is.

Figure 11

The solid curves give $\langle S_E\left(t\right)\rangle$ for Gaussian initial conditions with spin state $|{\xi }_1\rangle =\left(\right|\uparrow \rangle$ $+i|\downarrow \rangle )/\sqrt{2}$ and the dashed ones with $|{\xi }_2\rangle =\left(\right|\uparrow \rangle +|\downarrow \rangle )/\sqrt{2}$. The corresponding dispersions are given in the upper left panel, where we show $\langle S_E\left(t\right)\rangle$ for the Hadamard QRW (no averaging needed in this case). Note that in this case the asymptotic entanglement is highly sensitive to the initial conditions. In the other panels we have $\text{the}{\text{RQRW}}_{\infty }$ with $q\in [0,1]$ and $\theta ,\phi \in [0,2\pi ]$.

Figure 12

The curves were obtained averaging over 1000 random initial conditions given as $|\Psi \left(0\right)\rangle =(cos{\alpha }_s|\uparrow \rangle +e^{i{\beta }_s}sin{\alpha }_s|\downarrow \rangle )\otimes (cos{\alpha }_p|-1\rangle +e^{i{\beta }_p}sin{\alpha }_p|1\rangle )$, where ${\alpha }_{s,p}\in [0,\pi ]$ and ${\beta }_{s,p}\in [0,2\pi ]$. We work with the ${\text{RQRW}}_{\infty }$ with $q$, $\theta$, and $\phi$ random and we go up to 1000 steps. The solid curves were obtained by a $95\%$ confidence level linear fitting of the simulated data, where we have dropped the first 100 points to better highlight the asymptotic behavior. For dynamic and fluctuating disorder the angular coefficients are ($-0.2429\pm 0.0009$) and ($-0.2500\pm 0.0008$), respectively, while for the ordered Hadamard QRW it is ($0.5005\pm 0.0001$). The static case gives a parallel line to the $x$ axis with angular coefficient ($-0.0006\pm 0.0006$). These results corroborate the fact that whenever we have randomness in time (dynamic and fluctuating cases) the distance between successive matrices ${\rho }_C\left(t\right)$ tend to zero as $D\left(t\right)\sim t^{-1/4}$, while for the ordered case $D\left(t\right)\sim t^{-1/2}$. For static disorder ${\rho }_C\left(t\right)$ does not approach a particular state and we do not have an asymptotic limit (${lim}_{t\to \infty }D\left(t\right)\ne 0$ or, equivalently, ${lim}_{t\to \infty }ln\left[D\left(t\right)\right]\nrightarrow -\infty$).

Figure 13

Same as Fig. 12 but now we have the balanced ${\text{RQRW}}_2$, with Fourier and Hadamard coins. For dynamic and fluctuating disorder the angular coefficients are ($-0.2533\pm 0.0009$) and ($-0.2604\pm 0.0008$), respectively. The static case gives a parallel line to the $x$ axis with angular coefficient ($-0.0002\pm 0.0007$). The simulated data once again corroborate the fact that whenever we have randomness in time (dynamic and fluctuating cases) the distance between successive matrices ${\rho }_C\left(t\right)$ tend to zero as $D\left(t\right)\sim t^{-1/4}$, while for the ordered case $D\left(t\right)\sim t^{-1/2}$.

Figure 14

Same configuration as given in Figs. 12 and 13. This time we compare within a given type of disorder the behavior of $\langle D\left(t\right)\rangle$ for the ${\text{RQRW}}_{\infty }$ and $\text{the}{\text{RQRW}}_2$.

Figure 15

Average probability distributions $\langle P\left(j\right)\rangle$ for all RQRWs described in Figs. 12 (upper panel) and 13 (lower panel) after 1000 steps. The black, dashed curves represent the expected probability distribution for the balanced CRW starting at the origin and subjected to the same number of steps of the quantum processes. In the upper panel the dynamic (black, solid) and fluctuating (red, solid) curves overlap with the classical one (black, dashed). In the lower panel, for the ${\text{RQRW}}_2$, dynamic disorder still has spikes away from the origin, while this feature is no longer present for fluctuating disorder. In both panels static disorder (blue, solid) gives distributions that are narrower than the classical one (Anderson localization).

Figure 16

Average dispersions $\langle \sqrt{{\sigma }^2}\rangle$ for all RQRWs described in Figs. 12 (upper panel) and 13 (lower panel). The black, dashed curves represent the expected dispersion for the balanced CRW. In the upper panel the dynamic (black, solid) and fluctuating (red, solid) curves overlap with the classical dispersion (black, dashed) and are given as $\sigma \sim \sqrt{t}$ (diffusive behavior). The Hadamard QRW curve (green, solid) is such that $\sigma \sim t$ (ballistic behavior). The static case (blue, solid) has $\sigma \sim \text{const}$ (see inset) and is far below all curves, clearly indicating Anderson localization. In the lower panel, for $\text{the}{\text{RQRW}}_2$, all curves are distinct with dynamic and fluctuating disorder lying between the ordered and the classical case. Static disorder dispersion is no longer a constant, but still lies below all curves.