Graph states represent a significant class of multi-partite entangled quantum states with applications in quantum error correction, quantum communication, and quantum computation. In this work, we introduce a novel formalism called the Entanglement Matrix for quantifying and classifying entanglement in n-qubit graph states. Leveraging concepts from graph theory and quantum information, we develop a systematic approach to analyze entanglement by identifying primary and secondary midpoints in graph representations, where midpoints correspond to controlled-Z gate operations between qubits. Using Von Neumann entropy as our measure, we derive precise mathematical relationships for maximum entanglement in graph states as a function of qubit number. Our analysis reveals that entanglement follows a quadratic relationship with the number of qubits, but with distinct behaviors for odd versus even qubit systems. For odd n-qubit graph states, maximum entanglement follows $E_{\max} = n^2 - n$, while even n-qubit states exhibit higher entanglement with varying formulae depending on specific configurations. Notably, systems with qubit counts that are multiples of 12 demonstrate enhanced entanglement properties. This comprehensive classification framework provides valuable insights into the structure of multi-qubit entanglement, establishing an analytical foundation for understanding entanglement distribution in complex quantum systems that may inform future quantum technologies.
Entanglement Classification in the Graph States: The generalization to n-Qubits States using the Entanglement Matrix
Graph states represent a significant class of multi-partite entangled quantum states with applications in quantum error correction, quantum communication, and quantum computation. In this work, we introduce a novel formalism called the Entanglement Matrix for quantifying and classifying entanglement in n-qubit graph states. Leveraging concepts from graph theory and quantum information, we develop a systematic approach to analyze entanglement by identifying primary and secondary midpoints in graph representations, where midpoints correspond to controlled-Z gate operations between qubits. Using Von Neumann entropy as our measure, we derive precise mathematical relationships for maximum entanglement in graph states as a function of qubit number. Our analysis reveals that entanglement follows a quadratic relationship with the number of qubits, but with distinct behaviors for odd versus even qubit systems. For odd n-qubit graph states, maximum entanglement follows Emax=n2−n, while even n-qubit states exhibit higher entanglement with varying formulae depending on specific configurations. Notably, systems with qubit counts that are multiples of 12 demonstrate enhanced entanglement properties. This comprehensive classification framework provides valuable insights into the structure of multi-qubit entanglement, establishing an analytical foundation for understanding entanglement distribution in complex quantum systems that may inform future quantum technologies.
Graph states form a rich class of entangled states that exhibit important aspects of multi-partite entanglement. At the same time, they can be described by a number of parameters that grows only moderately with the system size. They have a variety of applications in quantum information theory, most prominently as algorithmic resources in the context of the one-way quantum computer, but also in other fields such as quantum error correction and multi-partite quantum communication, as well as in the study of foundational issues such as non-locality and decoherence. In this review, we give a tutorial introduction into the theory of graph states. We introduce various equivalent ways how to define graph states, and discuss the basic notions and properties of these states. The focus of this review is on their entanglement properties. These include aspects of non-locality, bi-partite and multi-partite entanglement and its classification in terms of the Schmidt measure, the distillability properties of mixed entangled states close to a pure graph state, as well as the robustness of their entanglement under decoherence. We review some of the known applications of graph states, as well as proposals for their experimental implementation.
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.
Graph states form a rich class of entangled states that exhibit important aspects of multi-partite entanglement. At the same time, they can be described by a number of parameters that grows only moderately with the system size. They have a variety of applications in quantum information theory, most prominently as algorithmic resources in the context of the one-way quantum computer, but also in other fields such as quantum error correction and multi-partite quantum communication, as well as in the study of foundational issues such as non-locality and decoherence. In this review, we give a tutorial introduction into the theory of graph states. We introduce various equivalent ways how to define graph states, and discuss the basic notions and properties of these states. The focus of this review is on their entanglement properties. These include aspects of non-locality, bi-partite and multi-partite entanglement and its classification in terms of the Schmidt measure, the distillability properties of mixed entangled states close to a pure graph state, as well as the robustness of their entanglement under decoherence. We review some of the known applications of graph states, as well as proposals for their experimental implementation.
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 and stochastic local operations with classical communication, 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, 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.
We propose a method for constructing multi-qubit entangled quantum states that represent weighted tripartite graphs, and develop approaches for investigating their structural properties using quantum computing. In the general case of multi-qubit states corresponding to arbitrary tripartite graph structures, we derive an expression for the entanglement distance. We establish a connection between entanglement and properties of the corresponding tripartite graphs. Namely, we show that the entanglement of a qubit with the rest of the system in a quantum tripartite graph state depends on the weights of the arcs in the neighborhood of the corresponding vertex, as well as on its degree with respect to the vertex sets of the tripartite graph. As an illustrative example, we consider a tripartite graph forming a triangle and evaluate the entanglement distance with quantum computing. We also compute quantum correlators for the general case of tripartite quantum graph states and relate these quantities to structural features of the underlying graphs, including the number of non-overlapping neighbors, the number of common neighbors of the corresponding vertices, and the number of 4-cycles. Obtained relationships between the quantum properties of multi-qubit states and the structural features of tripartite graphs opens up the possibility of investigating classical systems, such as tripartite graphs, using quantum computing. It is worth emphasizing that tripartite graphs have applications in practical problems, including resource allocation, scheduling, and database and hypergraph modeling.
Abstract This work presents a comprehensive exploration of the entanglement and graph connectivity properties of Graph States (GSs). Qubit entanglement in Pseudo Graph States (PGSs) is quantified using the Entanglement Distance (ED), a recently introduced measure of bipartite entanglement. In addition, a new approach is proposed for probing the underlying graph connectivity of genuine GSs, using Pauli matrix quantum correlators. These findings also reveal interesting implications for measurement processes, demonstrating the equivalence of some projective measurements. Finally, the emphasis is placed on the simplicity of data analysis in this framework. This work contributes to a deeper understanding of the entanglement and connectivity properties of GSs, offering valuable information for quantum information processing and quantum computing applications. The famous stabiliser formalism, which is the typically preferred framework for the study of this type of states, is not used in this work; on the contrary, this approach is based exclusively on the concepts of expectation values, quantum correlations, and projective measurement, which have the advantage of being very intuitive and fundamental tools of quantum theory.