Connecting velocity and entanglement in quantum walks

We investigate the relation between transport properties and entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional discrete time quantum walks. We obtain closed-form expressions for the long-time position variance and asymptotic entanglement of quantum walks whose time evolution is given by any balanced quantum coin, starting from any initial qubit and position states following $\delta$-like (local) and Gaussian distributions. We find out that the knowledge of the limit velocity of the walker together with the polar angle of the initial qubit provide the asymptotic entanglement for local states, while this velocity with the quantum coin phases give it for highly delocalized states.

Figure 1

The average long-time position variance averaged over all initial qubits for quantum walks starting from a local state (black) from Eq. (21) and Gaussian states from Eq. (26) with ${\sigma }_0=1$ (red), 2 (blue), 3 (green), and 10 (cyan) for distinct values of $\delta$. Inset: The average long-time variance of Gaussian states (dashed) for $\delta =0$ (black), $\pi /8$ (red), $\pi /4$ (blue), $3\pi /8$ (green), and $\pi /2$ (cyan) over ${\sigma }_0$.

Figure 2

The average long-time position variance (black) from Eq. (26) and asymptotic entanglement (red) obtained from Eq. (32) in $\langle {\overline{S}}_E\rangle$ averaged over all initial qubits for quantum walks starting from a Gaussian state with ${\sigma }_0\gg 1$.

Figure 3

Average (a) variance and (b) entanglement over time for Hadamard (black) and Fourier (red) quantum walks starting from a Gaussian state with ${\sigma }_0=10$. Probability amplitudes for up (black) and down (red) for (c) Hadamard and (d) Fourier quantum walks after $N=3000$ time steps. There is a break region within $j=-2050$ and 2050 in (c). The average was made over a set of 2016 initial qubits [Eq. (1)] from $(\alpha ,\beta )=(0,0)$ to $(\pi ,2\pi )$ with independent increments of 0.1.

Figure 4

Asymptotic entanglement and long-time variance for quantum walks starting from a local state with $\beta =\eta =0$ for $\delta =0$ (black), $\pi /8$ (red), $\pi /4$ (green), $3\pi /8$ (blue), and $\pi /2$ (cyan).

Figure 5

Asymptotic entanglement and long-time variance for quantum walks starting from a Gaussian state (${\sigma }_0\gg 1$) with $\beta =\eta =0$ for $\delta =0$ (black), $\pi /8$ (red), $\pi /4$ (green), $3\pi /8$ (blue), and $0.4995\pi$ (cyan).