Multipartite states with restricted unitary symmetry

We investigate the bipartite and tripartite quantum states invariant under the so-called finite-order unitary matrix [Formula: see text]. We show that bipartite states that commute with [Formula: see text] are maximally mixed states. For the unitary matrix [Formula: see text] being the finite-order matrix or a exceptional unitary matrix, the tripartite state commuting with [Formula: see text] adopt the form [Formula: see text] with the restriction of [Formula: see text], where [Formula: see text] are real numbers and [Formula: see text] represents the swap operator. These results can be applied to characterize multipartite states restricted by similar unitary symmetry.