Long-distance entanglement and quantum teleportation in $XX$ spin chains

Isotropic $XX$ models of one-dimensional spin-1/2 chains are investigated with the aim to elucidate the formal structure and the physical properties that allow these systems to act as channels for long-distance, high-fidelity quantum teleportation. We introduce two types of models: (i) open, dimerized $XX$ chains, and (ii) open $XX$ chains with small end bonds. For both models we obtain the exact expressions for the end-to-end correlations and the scaling of the energy gap with the length of the chain. We determine the end-to-end concurrence and show that model (i) supports true long-distance entanglement at zero temperature, while model (ii) supports “quasi-long-distance” entanglement that slowly falls off with the size of the chain. Due to the different scalings of the gaps, respectively exponential for model (i) and algebraic in model (ii), we demonstrate that the latter allows for efficient qubit teleportation with high fidelity in sufficiently long chains even at moderately low temperatures.

Figure 1

(Color online) Absolute value of the zero-temperature correlation $x_L=\left|⟨c_1^{†}{c}_L⟩\right|$ as a function of the size $L$ of the chain according to the scaling $x_L=x_{\infty }-A_0{L}^2\text{exp}\left(-L∕\zeta \right)$. The different curves reproduce $\text{ln}\left[\left(x_{\infty }-x_L\right)∕L^2\right]$ as a function of $L$ for different values of the dimerization parameter $\delta$. Symbols denote the exact numerical values. The values of $\zeta$ are obtained from Eq. (19).

Figure 2

End-to-end concurrence $C_{1,L}$, for long chains of length $L⪢{\zeta }_0$ in the dimer model (12), as a function of the dimerization parameter $\delta$. Nonvanishing LDE is achieved and is rapidly growing as soon as $\delta >{\delta }_0=0.132$.

Figure 3

(Color online) End-to-end concurrence $C_{1,L}$ at zero temperature as a function of $\lambda$ for different lengths $L$ of the chain. Curves from right to left are, respectively, for $L=26$, $L=50$, $L=100$, $L=200$, and $L=400$.

Figure 4

Comparison between the exact numerical evaluation and the analytic approximations for the end-to-end correlation function $x=⟨c_1^{+}{c}_L⟩$ for a chain of $L=100$ sites. The interpolating curve is given by $x_{\text{int}}={\left(-1\right)}^{L∕2}∕\left(2+4cL{\lambda }^2\right)$. It is exact up to $O\left({\lambda }^2\right)$ when $\lambda \to 0$, but retains its validity also for higher values of the coupling.

Figure 5

(Color online) Behavior of the energy gap ${\Delta }_L$ as a function of the length $L$ of the chain for different values of $\lambda$. Diamonds denote the exact numerical data. Continuous curves display the approximate expression for the gap [Eq. (33) in the text].

Figure 6

(Color online) Teleportation fidelity for different values of $\lambda$ as a function of the rescaled temperature $T∕J$ for an $XX$ chain with small end bonds, supporting QLDE. The six different curves refer to increasing values of $\lambda$ from 0.02 to 0.22 in steps of 0.04. From top to bottom, left to right: solid line $\left(\lambda =0.02\right)$; dotted line $\left(\lambda =0.06\right)$; dashed line $\left(\lambda =0.1\right)$; dot-dashed line $\left(\lambda =0.14\right)$; double-dot-dashed line $\left(\lambda =0.18\right)$; double-dash-dotted line $\left(\lambda =0.22\right)$. The length of the chain is fixed at $L=50$ sites. The horizontal dashed line $f=2∕3$ is the maximum attainable fidelity using a classical teleportation channel (classical threshold).