  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 a 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.
