Quantum fidelity of symmetric multipartite states

For two symmetric quantum states one may be interested in maximizing
the overlap under local operations applied to one of them. The
question arises whether the maximal overlap can be obtained by applying
the same local operation to each party. We show [1] that for two
symmetric multiqubit states and local unitary transformations this is
the case; the maximal overlap can be reached by applying the same
unitary matrix everywhere. For local invertible operations (stochastic
local operations assisted by classical communication equivalence),
however, we present counterexamples, demonstrating that considering
the same operation everywhere is not enough.