Quantum Hypergraph States: A Review

TL;DR

This review comprehensively discusses quantum hypergraph states, highlighting their mathematical structure, entanglement properties, nonclassical features, and applications in quantum information and computation, including error correction and resource theories.

Contribution

It provides a unified overview of hypergraph states, including recent advances in their entanglement classification, nonlocality, and generalizations to higher dimensions.

Findings

Hypergraph states exhibit rich multipartite entanglement.

They serve as resources for measurement-based quantum computation.

Hypergraph states can be generalized to qudits and continuous variables.

Abstract

Quantum hypergraph states extend the well-studied class of graph states by taking into account multi-qubit interactions through hyperedges. They provide a powerful framework to represent a family of quantum states with genuine multipartite entanglement. In this review, we provide a compact overview of the formal structure, entanglement characteristics, and operational relevance of hypergraph states in quantum information theory. We begin by introducing their mathematical foundations and generalizations of the stabilizer formalism. A central focus is placed on their entanglement properties, including the classification under local unitary (LU) and stochastic local operations with classical communication (SLOCC), the quantification of multipartite entanglement, and detection techniques via entanglement witnesses. We also explore other nonclassical features of hypergraph states, such as contextuality and genuine multipartite nonlocality, derived from stabilizer-based Bell-type inequalities. Additional attention is given to the role of hypergraph states in error correction, and as a computational resource in measurement-based quantum computation (MBQC), and to their non-stabilizer character - quantified via resource-theoretic measures of quantum magic. Finally we review their generalization to higher dimensions, i.e. to qudits and continuous variables.