Genuine multientropy, dihedral invariants and Lifshitz theory

Multi-invariants are local-unitary invariants of state replicas introduced as potential probes of multipartite entanglement and correlations in quantum many-body systems. In this paper, we investigate two multi-invariants for tripartite pure states, namely multientropy and dihedral invariant. We compute the (genuine) multientropy for Lifshitz groundstates, and obtain its analytical continuation to noninteger values of R\'enyi index. We show that the genuine multientropy can be expressed in terms of mutual information and logarithmic negativity, a relation that also holds for stabilizer states. For general tripartite pure states, we demonstrate that dihedral invariants are related to R\'enyi reflected entropies. In particular, we show that the dihedral permutations of replicas are equivalent to the reflected construction, or alternatively to the realignment of density matrices.