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.