  We study an extension of Bernstein's theorem to the setting of quantum
groups. For a d-tuple of free, identically distributed random variables we
consider a problem of preservation of freeness under the action of a quantum
subset of the free orthogonal quantum group. For a subset not contained in the
hyperoctahedral quantum group we prove that preservation of freeness
characterizes Wigner's semicircle law. We show that freeness is always
preserved if the quantum subset is contained in the hyperoctahedral quantum
group. We provide examples of quantum subsets which show that our result is an
extension of results known in the literature.
