State transfer in highly connected networks and a quantum Babinet principle

The transfer of a quantum state between distant nodes in two-dimensional networks is considered. The fidelity of state transfer is calculated as a function of the number of interactions in networks that are described by regular graphs. It is shown that perfect state transfer is achieved in a network of size $N$, whose structure is that of an $(N∕2)$-cross polytope graph, if $N$ is a multiple of $4$. The result is reminiscent of the Babinet principle of classical optics. A quantum Babinet principle is derived, which allows for the identification of complementary graphs leading to the same fidelity of state transfer, in analogy with complementary screens providing identical diffraction patterns.

Figure 1

(Color online) A circulant spin network can be used to transfer quantum states from $A$ to $B$. In a network of $N=6$ spins there are three possible configurations with connectivity $C=\left(\mathrm{a}\right)$ 1, (b) 2, and (c) 3, as shown. Network (b) is a three-cross polytope graph.

Figure 2

(Color online) (a) Maximum fidelity in the interval $[0,\Delta t=100]$ against connectivity for a network of size $N=200$ and homogeneous interactions. (b) Fidelity against time for the simple ring network. (c) Fidelity against time for the 100-cross polytope graph network.

Figure 3

(Color online) Illustration of Babinet’s principle in an optical setup with Fraunhofer conditions. (a) An unobstructed plane wave is focused by a lens $L$ and produces a diffraction pattern of amplitude $A\left(\vec{r}\right)$ on the screen $S$. (b) Diffraction patterns resulting from complementary screens $s$ and $\overline{s}$, whose opaque and transparent areas are swapped. At any point downstream from $s$ and $\overline{s}$, the sum of the two diffracted amplitudes, $A_s\left(\vec{r}\right)+A_{\overline{s}}\left(\vec{r}\right)$, equals the amplitude diffracted from the unobstructed plane wave, $A\left(\vec{r}\right)$. Away from the central spike, this amplitude is zero and therefore $A_s\left(\vec{r}\right)=-A_{\overline{s}}\left(\vec{r}\right)$, which leads to Babinet’s prediction of identical diffracted light fields for complementary apertures. Complementary apertures play the role of complementary graphs describing quantum spin networks. An increase in $\vec{r}$ corresponds to increasing the number of nodes $N$.