Dynamically Disordered Quantum Walk as a Maximal Entanglement Generator

We show that the entanglement between the internal (spin) and external (position) degrees of freedom of a qubit in a random (dynamically disordered) one-dimensional discrete time quantum random walk (QRW) achieves its maximal possible value asymptotically in the number of steps, outperforming the entanglement attained by using ordered QRW. The disorder is modeled by introducing an extra random aspect to QRW, a classical coin that randomly dictates which quantum coin drives the system’s time evolution. We also show that maximal entanglement is achieved independently of the initial state of the walker, study the number of steps the system must move to be within a small fixed neighborhood of its asymptotic limit, and propose two experiments where these ideas can be tested.

Figure 1

(a) The dashed line represents a possible realization of CRW, where no superposition occurs. The solid curves represent probability amplitudes for ${\mathrm{RQRW}}_2$, where only two $C(t)$ are allowed (red and blue disks). (b) Schematic view for ${\mathrm{RQRW}}_{\infty }$, where $C(t)$ are chosen randomly from uniform continuous distributions of quantum coins (at each step a different color or coin is used). Note that at each step all coins or colors are the same (dynamical disorder).

Figure 2

$⟨S_E⟩$ was computed by averaging over 16 384 initial conditions of the form $|\Psi (0)⟩=(\cos {\alpha }_s|\uparrow ⟩+e^{i{\beta }_s}\sin {\alpha }_s|\downarrow ⟩)\otimes (\cos {\alpha }_p|-1⟩+e^{i{\beta }_p}\sin {\alpha }_p|1⟩)$, with ${\alpha }_{s,p}\in [0,\pi ]$ and ${\beta }_{s,p}\in [0,2\pi ]$. The first realization used the initial condition $({\alpha }_s,{\beta }_s,{\alpha }_p,{\beta }_p)=(0,0,0,0)$ and the subsequent ones all quadruples of points in independent increments of 0.4 until ${\alpha }_{s,p}=\pi$ and ${\beta }_{s,p}=2\pi$. We worked with a 400-step walk. The blue (square) curve gives ${\mathrm{RQRW}}_{\infty }$, the red (circle) one ${\mathrm{RQRW}}_2$, and the green (solid) one the Hadamard QRW. The left inset shows the average position probability distributions [$⟨P(j)⟩$] after 400 steps and the right one the average dispersions ($⟨\sqrt{{\sigma }^2}⟩$). The black dashed curves represent the expected results for CRW starting at the origin, and they overlap with the curves for ${\mathrm{RQRW}}_{\infty }$, which is more localized than ${\mathrm{RQRW}}_2$. The two spikes on $⟨P(j)⟩$ for ${\mathrm{RQRW}}_2$ are due to the fact that it is built on the Hadamard and Fourier coins, where these spikes are a common trend.

Figure 3

$⟨S_E⟩$ was obtained by averaging over 2016 localized initial conditions given as $|\Psi (0)⟩=(\cos {\alpha }_s|\uparrow ⟩+e^{i{\beta }_s}\sin {\alpha }_s|\downarrow ⟩)\otimes |0⟩$, where ${\alpha }_s\in [0,\pi ]$ and ${\beta }_s\in [0,2\pi ]$. The first realization used the initial condition $({\alpha }_s,{\beta }_s)=(0,0)$ and the next ones all pairs of points in independent increments of 0.1 until ${\alpha }_s=\pi$ and ${\beta }_s=2\pi$. We worked with a 1000-step walk. The inset shows the rate of initial conditions leading to $S_E$ greater than 0.95, 0.97, and 0.99 at step 1000. Note that for RQRW [middle (blue) and right (red) bars] almost 100% of the initial conditions lead to $S_E>0.97$, while for QRW [left (green) bar] this occurs for less than 10%.