99$\%$-fidelity ballistic quantum-state transfer through long uniform channels

Quantum-state transfer with fidelity higher than $0.99$ can be achieved in the ballistic regime of an arbitrarily long one-dimensional chain with uniform nearest-neighbor interaction, except for the two pairs of mirror-symmetric extremal bonds, say $x$ (first and last) and $y$ (second and penultimate). These have to be roughly tuned to suitable values $x\sim 2N^{-1/3}$ and $y\sim 2^{3/4}{N}^{-1/6}$, where $N$ is the chain length. The general framework can describe the end-to-end response in different models, such as fermion or boson hopping models and $XX$ spin chains.

Figure 1

Mirror-symmetric quantum channel consisting of $N$ sites with uniform bonds of unit strength, except for two pairs of weaker bonds at the extrema, $x$ and $y$. State transmission is meant to occur between sites 1 and $N$.

Figure 2

Shifts ${\varphi }_q$ [Eq. (20)] for different values of $x$ and $y$. The dashed curves correspond to $y=1$. The fixed point $q_{{}_{\mathrm{F}}}={\sin }^{-1}\frac{x}{2}$ lies on the dashed straight line.

Figure 3

Normalized large-$N$ mode density ${\mathcal{P}}_q$ [Eq. (21)] for $x=0.6$ and different values of the second bond coupling $y$. The thicker curves are those for $y=1$ and for the threshold value $y=Y\left(x\right)$ (see text and Fig. 4). The broad dash-dotted line is the result for the fully uniform chain, $x=y=1$.

Figure 4

Function $Y\left(x\right)$ [Eq. (22)] separates the occurrence of unimodal and bimodal mode distributions. The optimized pairs $(x,y)$ [i.e., those which maximize the arrival amplitude $u\left(t\right)$], are also reported for selected values of $N$ (see Table ).

Figure 5

End-to-end amplitude $\tilde{u}(x,y)$ at arrival time as a function of $x$ for selected values of $y$ and $N=501$. The thicker curves correspond to $y=1$ and to the optimized value $y=0.5439$.

Figure 6

Contour plot of the average fidelity $\mathcal{F}(x,y)$ at arrival time in $(x,y)$ plane for $N=51$.

Figure 7

Contour plot of the average fidelity $\mathcal{F}(x,y)$ at arrival time in $(x,y)$ plane for $N=501$.

Figure 8

Optimal values for $x$, from Table , reported vs $N$ both for $y=1$ and for the optimized $y=y^{\mathrm{opt}}$. Note that $x\sim N^{-1/6}$ in the former case, while $x\sim N^{-1/3}$ in the latter.

Figure 9

Group velocity $v_q$ and corresponding mode density ${\mathcal{P}}_q$ for $N=501$, $x=x^{\mathrm{opt}}=0.225$, and selected values of $y$. The thicker curves correspond to $y=1$ (dotted) and to the optimized value $y=y^{\mathrm{opt}}=0.544$ (solid).

Figure 10

Space-time perspective views of the propagating wave packet $|u_i\left(t\right)|$ [Eq. (10)], for a chain of length $N=251$, (a) in the fully uniform chain $x=y=1$, (b) in the case $y=1$ and optimal $x=0.422$ [15], (c) in the quasiuniform channel [Eq. (17)], with the optimal $x=0.276$ and $y=0.598$, and (d) in the perfect-transfer channel [24] [Eq. (41)]. In all panels the site variable goes from 1 to 251 and the time variable from 0 to 600, as in Fig. 11.

Figure 11

Space-time contour views of the propagating wave packet $|u_i\left(t\right)|$ [Eq. (10)] for the same cases as Fig. 10.