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.
Li et al. (Fri,) studied this question.