Expediting quantum state transfer through the long-range extended XY model

Abstract

Going beyond short-range interactions, we explore the role of long-range interactions in the extended XY model for transferring quantum states through evolution. In particular, employing a spin-1/2 chain with interactions decaying as a power law, we demonstrate that long-range (LR) interactions significantly enhance the efficiency of a quantum state transfer (QST) protocol, improving the achievable fidelity, mitigating its slow decline as compared with the nearest-neighbor setting, associated with increasing system-size. Our study identifies the LR regime as providing an optimal balance between interaction range and transfer efficiency, outperforming the protocol with the short-range interacting model. Our detailed analysis reveals the impact of system parameters, such as anisotropy, magnetic field strength, and coordination number, on QST dynamics. Specifically, we find that intermediate coordination numbers lead to a faster and more reliable state transfer, while extreme values diminish performance. Furthermore, we exhibit that the presence of LR interactions considerably reduces the minimum time required to achieve fidelity beyond the classical limit.

Figures (8)

• Figure 1: Schematic diagram illustrating quantum state transfer. The first spin in the first line possessed by Alice is the arbitrary state to be transferred while the rest of the sites representing the spin chain involving $k$-body interactions acted as a quantum channel. After a time $t_q$ (second line), the state transfer occurs since the fidelity of the last site crosses the classical limit, $2/3$.
• Figure 2: $f^*$ -- earliest local maximum of fidelity above $2/3$$($vertical axis$)$ as a function of the coordination number $z$$($horizontal axis$)$ for different fall-off rates. From the light to dark shades, $\alpha$ increases. Here $N = 25$. Two combinations of $(\lambda, g)$ are considered -- (a) $\lambda = 0.9$ and $g = 0.7$, and (b) $\lambda = 1$ and $g = 1.7$. $\alpha \geq 5$ mimics $f^*$ with NN interacting evolving Ising Hamiltonian. Hence, low $\alpha$ incorporating distant range interactions can yield higher fidelity as compared to the short-range model subject to the choices of $(\lambda, g)$ values. All the axes are dimensionless.
• Figure 3: Nonmonotonic behavior of $f^*$$($ordinate$)$ with respect to decay strength $\alpha$$($abscissa$)$ of the extended transverse Ising model. Different symbols represent choices of $\lambda$ and $g$. Here $N = 20$ and $z = N-1$. It is observed that the fidelity achieves a maximum value $($even beyond 0.9$)$ for a specific value of $\alpha\ \sim 2$. Again for small $\alpha$ values, $f^*$ fluctuates. All the axes are dimensionless.
• Figure 4: The dependency of fidelity $f^*$$($ordinate$)$ on the system size $N$$($abscissa$)$ is plotted. Stars, circles, and squares represent $\alpha=2.3$, $2.7$ and $10$ respectively. Here $\lambda = 1$ and $g = 1.7$. We fit the decaying curve with $a \exp(-b N^{\eta})$. We find that with $a=1$, $\eta$ decreases along with the increase of $b$. Two observations emerge -- (1) for a given $N$, $f^{*^{\alpha =10}} < f^{*^{\alpha <10}}$; (2) there exists $N$ for which $f^{*^{\alpha =10}}$ cannot beat the classical limit while $f^{*^{\alpha \sim 2}}$ is much higher than the classical fidelity. The coordination number is $z=N-1$. All the axes are dimensionless.
• Figure 5: Variation of $f$ (ordinate) and $E(\rho_{qN})$ (ordinate) with respect to time (abscissa). Here, $N=25$. The other parameters of the long-range Hamiltonian are $\lambda = 0.0$, $g = 0.7$, $\alpha = 0.6$ and $z=16$. Triangles and stars represent the fidelity and the corresponding entanglement calculated via logarithmic negativity. The horizontal dashed-dot line represents the classical limit $2/3$ of fidelity. The graph clearly shows the correspondence between the two, aligning both the maxima(s) at $t=34.6$. All the axes are dimensionless.