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arXiv:1301.3576 · Physical Review A

Multipartite entanglement and hypergraph states of three qubits

Ri Qu, Zong-shang Li, Juan Wang, Yan-ru Bao

Published Mar 25, 2013• 13 citations
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Abstract

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.

DOI
10.1103/physreva.87.032329

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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 WWW 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 WWW state.
Hypergraph states are multi-qubit states that form a subset of the locally maximally entangleable states and a generalization of the well--established notion of graph states. Mathematically, they can conveniently be described by a hypergraph that indicates a possible generation procedure of these states; alternatively, they can also be phrased in terms of a non-local stabilizer formalism. In this paper, we explore the entanglement properties and nonclassical features of hypergraph states. First, we identify the equivalence classes under local unitary transformations for up to four qubits, as well as important classes of five- and six-qubit states, and determine various entanglement properties of these classes. Second, we present general conditions under which the local unitary equivalence of hypergraph states can simply be decided by considering a finite set of transformations with a clear graph-theoretical interpretation. Finally, we consider the question whether hypergraph states and their correlations can be used to reveal contradictions with classical hidden variable theories. We demonstrate that various noncontextuality inequalities and Bell inequalities can be derived for hypergraph states.
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 Hypergraph states form a family of multiparticle quantum states that generalizes cluster states and graph states. We study the action and graphical representation of nonlocal unitary transformations between hypergraph states. This leads to a generalization of local complementation and graphical rules for various gates, such as the CNOT gate and the Toffoli gate. As an application, we show that already for five qubits local Pauli operations are not sufficient to check local equivalence of hypergraph states. Furthermore, we use our rules to construct entanglement witnesses for three-uniform hypergraph states.
Abstract Classification of multipartite pure states via the local unitary (LU) equivalence or the stochastic local operations and classical communication (SLOCC) equivalence is a fundamental task in investigating multipartite entanglement. It is shown in the seminal paper (2000 Phys . Rev . A 62 062314) that pure states of three qubits can be partitioned into six SLOCC classes. We explore here a necessary and sufficient condition for LU equivalence of pure states for any SLOCC class via Acín et al ’s Schmidt coefficients (ASC) of three qubits in (2000 Phys. Rev. Lett. 85 1560). We show that the LU equivalent states in W SLOCC class have the same ASC irrespective of phases while there are generally two different ASCs for the LU equivalent states in GHZ SLOCC class (irrespective of phases for some subclass). Consequently, we obtain a complete LU classification of three qubit pure states in the sense of ASC. Moreover, we discuss the choice among the different ASCs and then propose an approach to find a unique ASC for any given three qubit pure state. At last we analyse the entanglement/separability of the marginal states of the two inequivalent genuine entangled SLOCC classes. We find that any bipartite marginal states of the W SLOCC class are entangled, and that of the GHZ SLOCC class has eight different cases which are depend on the ASCs.
Hypergraph states are multiqubit states whose combinatorial description and entanglement properties generalize the well-studied class of graph states. Graph states are important in applications such as measurement-based quantum computation and quantum error correction. The study of hypergraph states, with their richer multipartite entanglement and other nonlocal properties, has a promising outlook for new insight into multipartite entanglement. We present results analyzing local unitary symmetries of hypergraph states, including both continuous and discrete families of symmetries. In particular, we show how entanglement types can be detected and distinguished by certain configurations in the hypergraphs from which hypergraph states are constructed.