Routing quantum information in spin chains

Two different models are presented that allow for efficiently performing routing of a quantum state. Both cases involve an $XX$ spin chain working as a data bus and additional spins that play the role of sender and receivers, one of which is selected to be the target of the quantum state transmission protocol via a coherent quantum coupling mechanism making use of local and/or global magnetic fields. Quantum routing is achieved in the first of the models considered by weakly coupling the sender and the receiver to the data bus. On the other hand, in the second model, local magnetic fields acting on additional spins located between the sender and receiver and the data bus allow us to perform high-fidelity routing.

Figure 1

Schematic of the quantum router. The sender and the receivers are coupled to a common chain that acts as a quantum data bus.

Figure 2

Energy-level scheme. Each of the receivers is in resonance with one of the (pairs of) levels of the discrete band. By locally tuning the sender energy, it is possible to select the desired receiver. The spectral separation determines a bound for the maximum number of receivers.

Figure 3

Transition probability as a function of time (scaled with $\overline{\omega }$) and $\overline{k}R$. Integer values of $\overline{k}R/\pi$ guarantee high quality transfer after shorter times [yellow (light) spots in the density plot].

Figure 4

Comparison of the exact transition probability $F$ (dashed line) with the analytic result obtained by applying the weak-coupling approximation (continuous line) for $N=16$ and $g=0.01$. (a) $\overline{k}=0$ and $R=12$; (b) $\overline{k}=7\pi /4$ and $R=10$; (c) $\overline{k}=\pi /2$ and $R=4$.

Figure 5

Modified scheme of levels. The receivers are well separated in energy (${\Delta }_R>2\pi /N$) by tuning the sender and by translating the whole chain spectrum through a uniform external magnetic field.

Figure 6

The sender and one of the receivers are in resonance, whereas, the continuous (as seen from outside) channel is detuned. The effective two-body frequency oscillation associated with this scheme is lower than the other schemes proposed in this section.

Figure 7

The model: A linear $XX$ chain is used as a transmission channel between the sender $S_A$ and one of the receivers $R_{kA}$.

Figure 8

Maximum of the average fidelity in a fixed time interval $Jt<5\times {10}^4$ for a channel of $N=30$ sites with $n=5$ receiver blocks. By tuning the magnetic field $h_S$, respectively, to $h_{R_k},k=1,...,5$, we can perform a routing with high efficiency. The magnetic fields are ordered according to the position of the receivers $h_{R_{(k-1)}}<h_{R_k}$.

Figure 9

Comparison of the exact transition amplitude in Eq. (11) (red solid line) with the result of Eq. (12) (dotted blue line) obtained after a perturbative analysis for $h_X\gg J$ in a chain of $N=20$ sites with $n=3$ receiver blocks. (Time is given in units of $1/J$.)