Disorder-assisted distribution of entanglement in $XY$ spin chains

We study the creation and distribution of entanglement in disordered $XY$-type spin-$1/2$ chains for the paradigmatic case of a single flipped spin prepared on a fully polarized background. The local magnetic field is set to follow a disordered long-range-correlated sequence with power-law spectrum. Depending on the degree of correlations of the disorder, a set of extended modes emerges in the middle of the band yielding an interplay between localization and delocalization. As a consequence, a rich variety of entanglement distribution patterns arises, which we evaluate here through the concurrence between two spins. We show that, even in the presence of disorder, the entanglement wave can be pushed to spread out, reaching distant sites, and also to enhance pairwise entanglement between the initial site and the rest of the chain. We also study the propagation of an initial maximally entangled state through the chain and show that correlated disorder improves the transmission quite significantly when compared with the uncorrelated counterpart. Our work contributes in designing solid-state devices for quantum information processing in the realistic setting of correlated static disorder.

Figure 1

(a) Sketch of the spin chain featuring $XX$-type exchange interactions with strength $J$ subjected to a random on-site potential landscape (depicted by red bars). The spin chain is initialized in a fully polarized state and a single excitation (a flipped spin) is set, say, in the middle of it. We thus let it evolve via its Hamiltonian dynamics. (b) Single realization of the disorder distribution $\left\{{\varepsilon }_n\right\}$ (in units of $J$) generated from Eq. (3) for $\alpha =0$, 1, and 2, and $N=400$. The entire sequence is normalized satisfying $\langle {\epsilon }_n\rangle =0$ and $\langle {\epsilon }_n^2\rangle =1$. We note that by increasing $\alpha$ the distribution smooths out, resembling the trace of a fractional Brownian motion (see the last panel for $\alpha =2$).

Figure 2

(a) Wave-function amplitude distribution for the ordered case when $tJ=20$ for an excitation initially prepared at site 0. It propagates outwards reaching distant sites at $t\approx m$, with $m$ being the distance from the origin. (b) Corresponding entanglement distribution profile measured by the concurrence $C_{i,j}$. Note that the “entanglement wave” mostly involves pairwise correlations between the sites located within the largest wave amplitude and the rest of the chain. Basically, this checkerboard pattern is maintained while the concurrence evolves and tends to become homogeneously distributed after a long time. Both plots were obtained directly from Eqs. (10) and (15).

Figure 3

Entropy for a block of spins, $S\left[{\rho }_L\right]$, versus its size $L$ when $tJ=40$ for many disorder regimes provided by $\alpha$ including uncorrelated disorder ($\alpha =0$) and the noiseless condition. Each block involves groups of spins ranging from ($x_0+1$)-th to the $L$-th spin, with $x_0$ being the initial site (which is out of the block) located in the middle of the chain. The entanglement saturation threshold sets the extension of the wave-function in a given time. Note it happens quite rapidly for lower values of $\alpha$. The inset provides a zoomed-in view of the saturation region for higher values of $\alpha$. Plots were obtained from exact numerical diagonalization of Hamiltonian (2) for $N=400$ and $S\left[{\rho }_L\right]$ was averaged over ${10}^2$ independent realizations of disorder.

Figure 4

Snapshots of concurrence distribution for (a) $\alpha =0$ (uncorrelated disorder), (b) $\alpha =1$, (c) $\alpha =2$, and (d) $\alpha =3$ averaged over ${10}^2$ independent realizations of disorder. The initial state was set in the middle of a chain with $N=2048$ sites. We set $C_{i,j}=0$ for $i=j$. The first, second, and third columns correspond to times $tJ=20$, 40, and 60, respectively. Here, we clearly see that long-range correlations in the disorder distribution push the entanglement wave to reach distant sites, first extending the localization length and then setting up genuine propagating modes when $\alpha =2$ and higher.

Figure 5

Maximum concurrence $C_{i,j}^{\max }$ for (a) $\alpha =0$ (uncorrelated disorder), (b) $\alpha =1.0$, (c) $\alpha =2.0$, and (d) $\alpha =3.0$ averaged over ${10}^2$ independent realizations of disorder. The scale goes from 0 (dark spots) to 0.1 and above (bright spots). The system's parameters are the same as from Fig. 4 and the time window considered for $C_{i,j}^{\max }$ was $tJ\in [0,200]$, thus making sure that the entanglement wave has passed through the neighborhood of the initial site (which we denote as the zeroth site for ease of use) without reaching the boundaries of the chain ($N=2048$). As we move from uncorrelated disorder towards long-range-correlated disorder, there are many configurations available for entanglement distribution.

Figure 6

Distribution of maximum concurrence $C_{i,j}^{\max }$ for fixed (a) $i=0$ (initial site) and (b) $i=30$ and varying $\alpha$. Those are basically detailed side views of Fig. 5 involving a much smaller region. Again, the time window was set to $tJ\in [0,200]$ and $C_{i,j}^{\max }$ was averaged over ${10}^2$ independent realizations of disorder.

Figure 7

Entanglement transmission and reflection coefficients [Eq. (18)] versus $\alpha$ for (a) $r_0=20$, (b) $r_0=30$, (c) $r_0=40$, and (d) $r_0=50$. The initial state was a maximally entangled pair $|\psi \left(0\right)\rangle =\frac{1}{2}\left(\right|0_s{1}_{s^{\prime }}\rangle +|1_s{0}_{s^{\prime }}\rangle )$ with the sender prepared in the first site ($s=1$) in a chain of size $N=400$ (note that now, for convenience, we are numbering each spin regularly from 1 to $N$). The calculations were performed without letting the wave function reach the other end of the chain and the coefficients were evaluated at times when $C_r^2\left(t\right)$ (averaged over ${10}^2$ samples) achieved a stationary behavior in order to account for the long-time regime.