# Genuine multientropy, dihedral invariants and Lifshitz theory

## TL;DR

This paper explores multi-invariants like multientropy and dihedral invariants in tripartite quantum states, linking them to entanglement measures and analyzing their behavior in Lifshitz groundstates.

## Contribution

It introduces analytical expressions for multientropy in Lifshitz states and relates dihedral invariants to Re9nyi reflected entropies, advancing understanding of quantum correlations.

## Key findings

- Multientropy for Lifshitz groundstates expressed in terms of mutual information and logarithmic negativity.
- Dihedral invariants related to Re9nyi reflected entropies and realignment of density matrices.
- Analytical continuation of multientropy to noninteger Re9nyi indices.

## Abstract

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.