Sperner state and multipartite entanglement signals

TL;DR

This paper introduces a systematic classification of multipartite entanglement using Sperner states, hypergraph structures, and a novel Multi-entanglement Measure Space to distinguish and quantify complex entanglement patterns.

Contribution

It defines Sperner states and links them to hypergraph structures, creating a unified framework for classifying and measuring multipartite entanglement.

Findings

Established a classification scheme for multipartite entanglement.

Linked Sperner states to hypergraph structures for entanglement analysis.

Introduced MEMS to quantify and distinguish entanglement patterns.

Abstract

We establish a systematic classification scheme for multipartite entanglement structures. We define Sperner states -- a broad class of states where apparent multipartite entanglement decomposes into fewer-partite entanglement among subsystems of each party. Each class of Sperner states is associated with one antichain hypergraph and each hypergraph encodes the maximal entanglement structure permissible under its constraints. We introduce a Multi-entanglement Measure Space (MEMS) where each Sperner class corresponds to a linear subspace defined by the vanishing of specific linear combinations of bipartite and multipartite measures. The nonvanishing of such combinations signals multipartite entanglement beyond the associated hypergraph, thereby distinguishing entanglement structures. We build a two way connection between each hypergraph entanglement structure and a distinct set of combinations, thereby quantifying the entanglement pattern and providing a unified basis for classifying all multipartite entanglement.