Multipartite entanglement dynamics in quantum walks

Abstract

Quantum walks constitute a rich area of quantum information science, where multipartite entanglement plays a central role in the dynamics and scalability of quantum advantage over classical simulators. In this work, we study the multipartite entanglement of quantum walks in optical settings. We present methods for computing a geometric measure of entanglement for arbitrary partitions of a single-walker quantum walk and for analyzing the entanglement in multi-walker scenarios. These techniques are used for numerical studies on the entanglement dynamics of quantum walks in large systems and under various initial conditions. For a given bipartition, based on the coin degrees of freedom, we derive exact expressions describing the complete entanglement dynamics for arbitrary localized initial conditions. We use these expressions for analytic statements about the asymptotic behavior of the system. Furthermore, we demonstrate the emergence of entanglement typicality in statistical ensembles of random optical networks.

Figures (9)

• Figure 1: Workflow for solving the separability eigenvalue equations for generalized W states \ref{['eq:WState']}, described by equation \ref{['eq:SEVGeneral']}. For details, see the main text.
• Figure 2: The top panel shows the mean entanglement and the central $68\%$ interval for $5\,000$ uniformly, randomly generated W-states for various mode numbers. The dashed black line marks the maximum attainable entanglement $E_{g,\mathrm{max}}$ for a given mode number, according to \ref{['eq:gmaxMax']}. The bottom panel shows the convergence of the mean value to the minimum value in a log-log plot. The line represents a linear fit for $M\geq 50$ in the log-log plot, which corresponds to a power law fit for the data. The coefficient of determination is $R^2 = 0.999\,987$. Mind the different scales of the horizontal axes in both plots.
• Figure 3: Density plot of the asymptotic entanglement, as given by \ref{['eq:AsymptoticEntCoin']}, for bipartitions with respect to the coin degrees of freedom. The contour follows the form of the great circle, \ref{['eq:GreatCircle']}, when projected onto the Bloch sphere parametrized by $\theta$ (latitude) and $\phi$ (longitude).
• Figure 4: Entanglement dynamics $E_g(n)$ for different number of positions $P$. The gray areas mark regions of $E_g$ that cannot be attained; see \ref{['eq:gmaxMax']}. Note that the entanglement dynamics for $P=3$ and $P=6$ are identical.
• Figure 5: Entanglement dynamics $E_g(n)$ for a QW of size $P=500$. The inset highlights the regular dynamics before the boundaries are reached at time step $n=250$. Local minima are always spaced apart by $6$ steps, except for three cases. The irregular behavior for $n > 250$ is limited to a rather small region of $E_g$.