Long-Distance Entanglement in Spin Systems

Most quantum system with short-ranged interactions show a fast decay of entanglement with the distance. In this Letter, we focus on the peculiarity of some systems to distribute entanglement between distant parties. Even in realistic models, like the spin-1 Heisenberg chain, sizable entanglement is present between arbitrarily distant particles. We show that long-distance entanglement appears for values of the microscopic parameters which do not coincide with known quantum critical points, hence signaling a transition detected only by genuine quantum correlations.

Figure 1

A schematic setup for creating entanglement between two selected sites A and B at distance $d$, through the subsystem C.

Figure 2

The finite-size study on the concurrence shows the presence of LDE in model (1) for $\delta >{\delta }_T(\alpha )$. Data were obtained keeping 256 DMRG states, with a truncation error smaller than ${10}^{-10}$.

Figure 3

Concurrence for the two end-qubit state as a function of dimerization $\delta$ for some values of $\alpha$. Exact calculation on a chain of length $L=24$. The LDE increases steeply above a threshold and is enhanced by frustration. Inset: threshold value of dimerization ${\delta }_T(\alpha )$, above which end-to-end concurrence starts to be nonzero for lengths $L=12$, 16, 20. The crosses are the infinite size extrapolations of DMRG data with $L$ up to 100.

Figure 4

Entangled (white) and separable (shaded) states of two globally SU(2)-invariant qutrits are completely determined by means of $⟨S_{\mathrm{A}}^z{S}_{\mathrm{B}}^z⟩$ and $⟨(S_{\mathrm{A}}^z{)}^2(S_{\mathrm{B}}^z{)}^2⟩$.

Figure 5

Concurrence between probes attached to a Heisenberg chain of length $L=26$ as a function of the distance. The calculation was done for various values of $J_p$. A dramatic increase of entanglement between distant probes appears as $J_p$ is lowered.