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.
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.
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.
Abstract Computationally feasible multipartite entanglement measures are essential for advancing our understanding of complex quantum systems. Entanglement distance (ED), introduced by Cocchiarella et al (2020 Phys. Rev. A 101 042129), based on the Fubini–Study metric, offers several advantages over existing methods, including ease of computation, a profound geometrical interpretation, and applicability to multipartite entanglement. Although ED has been successfully applied to systems of qudits, an explicit formulation for continuous quantum variables, particularly for pure Gaussian states, remains unexplored. In this work, we address this limitation by deriving the analytical expression for the Gaussian entanglement measure (GEM), a multipartite entanglement monotone for multimode pure Gaussian states based on the purity of fragments of the whole system, through a generalization of ED to group-theoretic coherent states. We show the efficacy of GEM across various scenarios, including the analysis of a two-mode Gaussian state under beam-splitter and squeezing transformations, and exploring graph states involving three and four modes. Notably, comparing GEM values for states with different graph topologies reveals insights into the connectivity of the underlying graphs. Additionally, we illustrate how GEM provides insights into free bosonic field theory on R t × S 1 beyond standard bipartite entanglement entropy, paving the way for quantum information-theoretic tools to probe the topological properties of quantum field theories.
Computationally feasible multipartite entanglement measures are needed to advance our understanding of complex quantum systems. An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers, showing several advantages over existing methods, including ease of computation, a deep geometrical interpretation, and applicability to multipartite entanglement. Here, we present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states, based on the purity of fragments of the whole systems. Our analysis includes the application of GEM to a two-mode Gaussian state coupled through a combined beamsplitter and a squeezing transformation. Additionally, we explore 3-mode and 4-mode graph states, where each vertex represents a bosonic mode, and each edge represents a quadratic transformation for various graph topologies. Interestingly, the ratio of the geometric entanglement measures for graph states with different topologies naturally captures properties related to the connectivity of the underlying graphs. Finally, by providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory on Rt×S1, going beyond the standard bipartite entanglement entropy approach between different regions of spacetime. The results presented herein suggest how the GEM paves the way for using quantum information-theoretical tools to study the topological properties of the space on which a quantum field theory is defined.
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.
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.
Quantum State Graphs (QSGs) are a fundamental framework in quantum information theory, representing the entangled states of qubits. Analyzing the topology of QSGs is crucial for understanding their properties and for developing quantum algorithms. This paper introduces a novel approach to QSG topology analysis, leveraging graph theory to effectively characterize and manipulate the structure of QSGs. We propose a method that utilizes graph traversal and connectivity analysis to identify and quantify key topological features, allowing for the optimization of quantum algorithms. The core mechanism focuses on establishing a robust and efficient framework for studying the topology of QSGs, ultimately contributing to advancements in quantum computation and information processing. This research explores how the inherent graph structure of QSGs can be harnessed to simplify complex computations and improve overall performance.