Quantum correlations between distant qubits conveyed by large-$S$ spin chains

We consider two distant spin-$\frac{1}{2}$ particles (or qubits) and a number of interacting objects, all with the same value $S\gg 1$ of their respective spin, distributed on a one-dimensional lattice (or large-$S$ spin chain). The quantum states of the chain are constructed by linearly combining tensor products of single-spin coherent states, whose evolution is determined accordingly, i.e., via classical-like equations of motions. We show that the quantum superposition of the above product states resulting from a local interaction between the first qubit and one spin of the chain evolves so that the second qubit, after having itself interacted with another spin of the chain, can be entangled with the first qubit. Obtaining such an outcome does not imply imposing constraints on the length of the chain or the distance between the qubits, which demonstrates the possibility of generating quantum correlations at a distance by means of a macroscopic system, as far as local interactions with just a few of its components are feasible.

Figure 1

Schematic representation of the system: The connected blue spheres represent the spins of a Heisenberg chain, while the orange ones are the two qubits.

Figure 2

TW soliton: $1-cos{\theta }_{\beta }\left(\xi \right)$ for $tan\beta =2$.

Figure 3

Schematic representation of the three stages of the system dynamics (1), (2), and (3) from top to bottom, respectively (see text).

Figure 4

$E_{\mathrm{A},{\mathit{S}}_{\mathrm{A}}}\left(t\right)$ for $t\in [t_0,t_1], S=5, g_{\mathrm{A}}=1, h_{\mathrm{A}}=0.25$. The $\mathrm{A}$ initial state is $|1\rangle$ while the chain is initially in the state corresponding to a propagating Heisenberg soliton (see text) with ${\lambda }_{\beta }=10$ and $\beta =\pi /4$ centered in $n_A$ (specifically meaning ${\mathrm{\Omega }}_{n_{\mathrm{A}}}^0=\{{\theta }_{n_{\mathrm{A}}}^0=\pi /2,{\phi }_{n_{\mathrm{A}}}^0=0\}$). The lower panel shows a zoom of the small-time part of the upper one.

Figure 5

Configurations ${\mathrm{\Omega }}_n(t,\mathrm{\Omega })$ for a starting soliton with ${\lambda }_{\beta }=10$, visualized via $(1-cos{\theta }_n(t,\mathrm{\Omega }))$ as functions of $n$ and $\mathrm{\Omega }$, after an integration time $t-t_1$ equal to 400 (800) in the upper (lower) panel. Colored curves are for $\mathrm{\Omega }$'s that define correspondingly colored points on the small sphere in the upper-left corner.

Figure 6

Same as in Fig. 5, for a starting soliton with ${\lambda }_{\beta }=2.5$ and integration times equal to 200 (upper panel) and 400 (lower panel).

Figure 7

$E_{{\mathit{S}}_n}\left(t\right)$ as a function of $n$ at $t-t_1=800$ with a starting soliton of width ${\lambda }_{\beta }=10$ (upper panel) and at $t-t_1=400$ with a starting soliton of width ${\lambda }_{\beta }=2.5$ (lower panel). Curves for different values of $S$, as indicated.

Figure 8

Upper panel: $\mathcal{C}\left[{\rho }_{\mathrm{A},\mathrm{B}}(t-t_2)\right]$ for $g=1, h_{\mathrm{A}}=h_{\mathrm{B}}=0.25, S=5$, and a starting soliton with ${\lambda }_{\beta }=10$. Lower panel: eigenvalues $\{{\mu }_1,{\mu }_2,{\mu }_3,{\mu }_4\}$ for the same parameters values as above. Cusps in the upper panel originate from the eigenvalues crossings seen in the lower panel.

Figure 9

Same as in Fig. 8 for a starting soliton with ${\lambda }_{\beta }=2.5$.