  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 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.
