Genuine multi-entropy, dihedral invariants and Lifshitz theory

TL;DR

This work develops and exploits two tripartite multi-invariants—the genuine multi-entropy and the dihedral invariant—for quantum states. It provides an analytic continuation of the multi-entropy to noninteger Rényi indices in Lifshitz field theories, yielding a concise relation with the Rényi mutual information at $n=1/2$ and the logarithmic negativity, and demonstrates a universal Lifshitz formula linking $G_n^{(3)}$ to partition functions. It further proves that dihedral invariants equal the Rényi reflected entropies, revealing a deep connection between dihedral replica contractions and reflected-entropy constructions. Collectively, these results offer a unified, tractable framework to quantify genuine multipartite entanglement in Lifshitz RK states and general tripartite pure states, with implications for stabilizer and GHZ-like entanglement structures and potential extensions to broader quantum-many-body contexts.

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 multi-entropy and dihedral invariant. We compute the (genuine) multi-entropy for Lifshitz groundstates, and obtain its analytical continuation to noninteger values of Rényi index. We show that the genuine multi-entropy 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 Rényi 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.

Figures (9)

• Figure 1: Examples of tripartitions, with subsystem sizes $\ell_K$. (a) A system with boundary with $A, B$ disjoint. (b) A periodic system with contiguous $A, B, C$.
• Figure 2: Graphical representation of the calculations leading to the multi-entropy for $A, B$ disconnected in a finite interval (see Fig. \ref{['fig_tri']}). We use different colors to differentiate lengths $\ell_A$ (pink), $\ell_B$ (green), and $\ell_C$ (gray). Dotted lines denote the "indices" of the matrix, while solid lines mean that trace has been taken. Each field value at the interface between the subsystems (black dots), denoted $\phi_i$, has to be integrated over. Hollow circles indicate Dirichlet boundary conditions at the end of the system.
• Figure A1: Disjoint regions $A$ and $B$ in the bulk. Right: Replica graph resulting from the multi-entropy symmetry for $n=3$.
• Figure A2: Adjacent regions $A$ and $B$ in the bulk. Right: Replica graph resulting from the multi-entropy symmetry for $n=3$.
• Figure A3: Disjoint regions $A$ and $B$, with $A$ on the boundary and $B$ in the bulk. Right: Replica graph resulting from the multi-entropy symmetry for $n=3$.