Controllability of Symmetric Spin Networks

We consider a network of n spin 1/2 systems which are pairwise interacting via Ising interaction and are controlled by the same electro-magnetic control field. Such a system presents symmetries since the Hamiltonian is unchanged if we permute two spins. This prevents full (operator) controllability in that not every unitary evolution can be obtained. We prove however that controllability is verified if we restrict ourselves to unitary evolutions which preserve the above permutation invariance. For low dimensional cases, n=2 and n=3, we provide an analysis of the Lie group of available evolutions and give explicit control laws to transfer between any two permutation invariant states. This class of states includes highly entangled states such as GHZ states and W states, which are of interest in quantum information.