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arXiv:quant-ph/0307130 · Physical Review A

Multiparty entanglement in graph states

M. Hein, Jens Eisert, Hans J. Briegel

Published Jun 9, 2004• 1,035 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.69.062311

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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.
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 are multipartite entangled states that correspond to mathematical graphs, where the vertices of the graph now play the role of quantum multilevel systems and edges represent interactions of the systems. Graph states are the basis of quantum error correction and one-way quantum computer. We systematically study the entanglement of non-binary graph states. Using iterative algorithm and entanglement bounds, we calculate the entanglement of all the ternary graph states up to nine vertices and parts of quaternary and quinary graph states modulo local unitary transformations and graph isomorphisms. The entanglement measure can be the geometric measure, the measure of relative entropy of entanglement or the measure of logarithmic robustness. We classify the graph states according to the entanglement values obtained. The closest product states obtained in the calculations are studied.
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.
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.
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.
Abstract Characterizing the non-classical correlations of a complex many-body system is an important part of quantum technologies. An ideal tool for this task would scale well with the size of the system, be easily computable and be easily measurable. In this work, we focus on graph states, which are promising platforms for quantum computation, simulation, and metrology. We consider four topologies: star graph states with edges, Turán graphs, r-ary tree graphs, and square grid cluster states. We provide a method to characterize their quantum content: many-body Bell correlations, non-separability and entanglement strength for an arbitrary number of qubits. We also relate the strength of these correlations to the usefulness of graph states for quantum sensing. Finally, we characterize many-body entanglement in graph states with up to eight qubits in 146 classes that are not equivalent under local transformations or graph isomorphisms. This technique is straightforward and does not require any assumptions about the multi-qubit state; therefore it could be applied wherever precise knowledge of many-body quantum correlations is necessary.