Communication at the quantum speed limit along a spin chain

Spin chains have long been considered as candidates for quantum channels to facilitate quantum communication. We consider the transfer of a single excitation along a spin-1/2 chain governed by Heisenberg-type interactions. We build on the work of Balachandran and Gong [V. Balachandran and J. Gong, Phys. Rev. A 77, 012303 (2008)] and show that by applying optimal control to an external parabolic magnetic field, one can drastically increase the propagation rate by two orders of magnitude. In particular, we show that the theoretical maximum propagation rate can be reached, where the propagation of the excitation takes the form of a dispersed wave. We conclude that optimal control is not only a useful tool for experimental application, but also for theoretical inquiry into the physical limits and dynamics of many-body quantum systems.

Figure 1

The one-dimensional spin chain used for information transfer. The (blue) solid circles represent sites along the chain, with the applied magnetic field depicted. The effective couplings are indicated operating between the sites.

Figure 2

The transfer begins with the state $|\Psi (0)\rangle$ and with the potential minimum centered at $x=0$. (a) The excitation is localized at the first site. (b) The field minimum moves along the chain during the evolution. (c) The spin state has been completely transferred to the final site in the chain.

Figure 3

Excitation probability plotted against $x$ for a spin chain with $N=101$ sites, at times (a) 0, (b) 100, and (c) 200, in units of $J^{-1}$. Here, $d(t)=0.5t$ and $C(t)=1$. The final fidelity is only around 15%.

Figure 4

Excitation probability plotted against $x$ for a spin chain with $N=101$ sites, at times (a) 0, (b) 100, and (c) 200, in units of $J^{-1}$. Here, $d(t)=0.5t$ and $C(t)=0.1$. The final fidelity here is around 5%.

Figure 5

Excitation probability plotted against $x$ for a spin chain with $N=101$ sites, at times (a) 0, (b) 100, and (c) 200, in units of $J^{-1}$. Here, $d(t)$ and $C(t)$ were optimally controlled. The final fidelity is $>99\%$.

Figure 6

The optimal control pulses for (a) $C(t)$ and (b) $d(t)-d_0(t)$, where $d_0(t)=0.5t$. The main features here are large perturbations at the initial and final time due to the boundaries, and slower modulations for the intermediate stage of the transport.

Figure 7

The probability density of the wave function along the chain at different times: (a) $t=0$, (b) $t=T/4$, (c) $t=T/2$, (d) $t=3T/4$, and (e) $t=T$, where $T=T_{\mathrm{QSL}}^{*}=56.50J^{-1}$. Both $d(t)$ and $C(t)$ were found after 100,000 iterations of the optimal control algorithm.

Figure 8

The probability density of the wave function along the chain at different times: (a) $t=0$, (b) $t=T/4$, (c) $t=T/2$, (d) $t=3T/4$, and (e) $t=T$, where $T<T_{\mathrm{QSL}}^{*}=53.30J^{-1}$. Both $d(t)$ and $C(t)$ were found after 100,000 iterations of the optimal control algorithm.

Figure 9

The decrease in infidelity of a transfer across a chain with 101 sites against the iterations of the control algorithm. The solid (red) line is the convergence for a transfer time $T=70.92J^{-1}>T_{\mathrm{QSL}}^{*}$, the dashed (green) line for a transfer time $T=T_{\mathrm{QSL}}^{*}=56.50J^{-1}$, and the dotted (blue) line for a transfer time of $T<T_{\mathrm{QSL}}^{*}=53.33J^{-1}$.

Figure 10

The average speed of the excitation wave $v_d$ versus $v_a$. The solid (red) line shows results for a chain length of 41 sites, the long-dashed (green) line shows the same for 61 sites, the short-dashed (dark blue) line for 81 sites, the dotted (pink) line for 101 sites, and the dashed-dotted (light blue) line for 121 sites. The black dotted line is the line $v_a=v_d$.

Figure 11

The solid (red) line shows the Fourier transform of the pulse $d(t)-d_0(t)$ for a chain length of 101 spins with a total time $T=56.50J^{-1}$.

Figure 12

The infidelity of the transfer related to the maximum frequency component of the controls retained after filtering. For key, see Fig. 10.

Figure 13

The optimal control pulses for (a) $C(t)$ and (b) $d(t)-d_0(t)$, where $d_0(t)=1.77t$. The maximum frequency component is $~4J$.

Figure 14

The infidelity reached after $R=100,000$ iterations for different chain lengths $N$ and effective time-per-site $(T-b)/(N-1)$, where $b=3.65$.

Figure 15

A comparison of the quantum-speed-limit time $T_{\mathrm{QSL}}$ with the optimal control limit $T_{\mathrm{QSL}}^{*}$. The solid (red) line is $T_{\mathrm{QSL}}$ with $\gamma =1$, which is the repeated orthogonal swaps. The (blue) crosses are $T_{\mathrm{QSL}}^{*}$ for different $N$ in the range 21–131 with $\varepsilon =5·{10}^{-5}$. The dashed (green) line is $T_{\mathrm{QSL}}+b$ with $\gamma =0.34$.