Abstract A detailed investigation of the multiparty entanglement present in the 4 − qubit quantum hypergraph states is presented, following a measurement‐based geometrical approach. Considering a classification of the 4 − party quantum system represented by a mathematical hypergraph based on the connections between its vertices, the genuine 4 − party entanglement present in each bi‐partition of the states have been measured. A strong correlation between the connectivity of the vertices of the hypergraphs and the genuine 4 − party entanglement has been found. The equivalence of the genuine 4 − party entanglement present in each bi‐partition is shown considering similar connectivity of the vertices. This explicates the cyclic permutation symmetry of the multiparty entanglement present in the 4 − qubit hypergraph states. Physically, one may expect the quantum systems with superposition of many states to behave in this symmetric manner while mapped into a network‐type picture, which the authors have quantified, as well as classified in this work.
Abstract A detailed investigation of the multiparty entanglement present in the 4 − qubit quantum hypergraph states is presented, following a measurement‐based geometrical approach. Considering a classification of the 4 − party quantum system represented by a mathematical hypergraph based on the connections between its vertices, the genuine 4 − party entanglement present in each bi‐partition of the states have been measured. A strong correlation between the connectivity of the vertices of the hypergraphs and the genuine 4 − party entanglement has been found. The equivalence of the genuine 4 − party entanglement present in each bi‐partition is shown considering similar connectivity of the vertices. This explicates the cyclic permutation symmetry of the multiparty entanglement present in the 4 − qubit hypergraph states. Physically, one may expect the quantum systems with superposition of many states to behave in this symmetric manner while mapped into a network‐type picture, which the authors have quantified, as well as classified in this work.
Several entanglement measures are used to define equivalence classes in the set of hypergraph states of three qubits. Our classifications reveal that (i) under local unitary transformations, hypergraph states of three qubits are split into six classes and only one of them is not equivalent to any graph state; (ii) under stochastic local operations with classical communication, for the single-copy case hypergraph states of three qubits, partitioned into five classes which cannot be converted into a W state, are equivalent to graph states; and (iii) when bipartite entanglement in three qubits is considered, hypergraph states of three qubits are split into five classes and only one of them has the same entangled graph as the W state.
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 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.
Abstract Quantifying genuine entanglement is a crucial task in quantum information theory. In this work, we give an approach of constituting genuine m -partite entanglement measures from any bipartite entanglement and any k -partite entanglement measure, 3 ⩽ k < m . In addition, as a complement to the three-qubit concurrence triangle proposed in ( Phys. Rev. Lett. 127 040403), we show that the triangle relation is also valid for any continuous entanglement measure and system with any dimension. We also discuss the tetrahedron structure for the four-partite system via the triangle relation associated with tripartite and bipartite entanglement respectively. For multipartite system that contains more than four parties, there is no symmetric geometric structure as that of tri- and four-partite cases.
Abstract Quantifying genuine entanglement is a key task in quantum information theory. We study the quantification of genuine multipartite entanglement for four-qubit systems. Based on the concurrence of nine different classes of four-qubit states, with each class being closed under stochastic local operation and classical communication, we construct a concurrence tetrahedron. Proper genuine four-qubit entanglement measure is presented by using the volume of the concurrence tetrahedron. For non genuine entangled pure states, the four-qubit entanglement measure classifies the bi-separable entanglement. We show that the concurrence tetrahedron based measure of genuine four-qubit entanglement is not equivalent to the genuine four-partite entanglement concurrence. We illustrate the advantages of the concurrence tetrahedron by detailed examples.
Graph states are multiparticle entangled states that correspond to mathematical graphs, where the vertices of the graph take the role of quantum spin systems and edges represent Ising interactions. They are many-body spin states of distributed quantum systems that play a significant role in quantum error correction, multiparty quantum communication, and quantum computation within the framework of the one-way quantum computer. We characterize and quantify the genuine multiparticle entanglement of such graph states in terms of the Schmidt measure, to which we provide upper and lower bounds in graph theoretical terms. Several examples and classes of graphs will be discussed, where these bounds coincide. These examples include trees, cluster states of different dimensions, graphs that occur in quantum error correction, such as the concatenated [7,1,3]-CSS code, and a graph associated with the quantum Fourier transform in the one-way computer. We also present general transformation rules for graphs when local Pauli measurements are applied, and give criteria for the equivalence of two graphs up to local unitary transformations, employing the stabilizer formalism. For graphs of up to seven vertices we provide complete characterization modulo local unitary transformations and graph isomorphisms.