For many quantum information protocols such as state transfer, entanglement
transfer and entanglement generation, standard notions of controllability for
quantum systems are too strong. We introduce the weaker notion of accessible
pairs, and prove an upper bound on the achievable fidelity of a transformation
between a pair of states based on the symmetries of the system. A large class
of spin networks is presented for which this bound can be saturated. In this
context, we show how the inaccessible dark states for a given
excitation-preserving evolution can be calculated, and illustrate how some of
these can be accessed using extra catalytic excitations. This emphasises that
it is not sufficient for analyses of state transfer in spin networks to
restrict to the single excitation subspace. One class of symmetries in these
spin networks is exactly characterised in terms of the underlying graph
properties.Comment: 14 pages, 3 figures v3: rewritten for increased clarit
