# Qudit hypergraph states and their properties

## TL;DR

This paper introduces qudit hypergraph states, extending hypergraph states to multi-level quantum systems, and explores their entanglement, non-locality, and relationships with other quantum state classes.

## Contribution

It generalizes hypergraph states to qudits, establishing a correspondence with multi-hypergraphs and analyzing their entanglement and non-local properties.

## Key findings

- Multi-hypergraphs uniquely correspond to qudit hypergraph states.
- Entanglement is determined by the connectivity of the multi-hypergraph.
- Qudit hypergraph states exhibit Bell non-locality.

## Abstract

Hypergraph states, a generalization of graph states, constitute a large class of quantum states with intriguing non-local properties and have promising applications in quantum information science and technology. In this paper, we generalize hypergraph states to qudit hypergraph states, i.e., each vertex in the generalized hypergraph (multi-hypergraph) represents a $d$-level quantum system instead of a qubit. It is shown that multi-hypergraphs and $d$-level hypergraph states have a one-to-one correspondence. We prove that if one part of a multi-hypergraph is connected with the other part, the corresponding subsystems are entangled. More generally, the structure of a multi-hypergraph reveals the entanglement property of the corresponding quantum state. Furthermore, we discuss their relationship with some well-known state classes, e.g., real equally weighted states and stabilizer states. These states' responses to the generalized $Z$ ($X$) operations and $Z$ ($X$) measurements are studied. The Bell non-locality, an important resource in fulfilling many quantum information tasks, is also investigated.