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arXiv:2302.01695 · Journal of Physics A Mathematical and Theoretical

Symmetric hypergraph states: entanglement quantification and robust Bell nonlocality

Jan Nöller, Otfried Gühne, Mariami Gachechiladze

Published Aug 8, 2023• 8 citations
arXivHTMLPDF

Abstract

Abstract Quantum hypergraph states are the natural generalization of graph states. Here we investigate and analytically quantify entanglement and nonlocality for large classes of quantum hypergraph states. More specifically, we connect the geometric measure of entanglement of symmetric hypergraphs to their local Pauli stabilizers. As a result we recognize the resemblance between symmetric graph states and symmetric hypergraph states. This explains both the exponentially increasing violation of local realism for infinitely many classes of hypergraph states and robustness towards particle loss.

DOI
10.1088/1751-8121/acee30

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Abstract Quantum hypergraph states are the natural generalization of graph states. Here we investigate and analytically quantify entanglement and nonlocality for large classes of quantum hypergraph states. More specifically, we connect the geometric measure of entanglement of symmetric hypergraphs to their local Pauli stabilizers. As a result we recognize the resemblance between symmetric graph states and symmetric hypergraph states. This explains both the exponentially increasing violation of local realism for infinitely many classes of hypergraph states and robustness towards particle loss.
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.
In this work, we present a novel method to express the stabilizer of a k-uniform complete hypergraph state as a linear combination of local operators. Quantum hypergraph states generalize graph states and exhibit properties that are not shared by their graph counterparts, most notably, their stabilizers are intrinsically nonlocal, as hyperedges can involve arbitrary subsets of vertices. Our formulation provides an explicit description of the stabilizers for k-uniform complete hypergraphs and may offer new insights for exploring these states within the stabilizer formalism. In particular, this approach could facilitate the construction of new Bell inequalities or find applications in quantum error correction.
The encoding of 3-uniform edge-ordered weighted hypergraphs into multiqubit quantum states is proposed. We derive an analytical expression for the geometric measure of entanglement of hypergraph quantum states and establish its relation to hypergraph properties. The results establish a direct connection between the local structure of hypergraphs, hyperedge weights, and the entanglement of the corresponding quantum states. For equally weighted hypergraphs, it is shown that the entanglement depends explicitly on the vertex degree. The dependence of entanglement on hypergraph parameters is studied analytically and using quantum computing. Hypergraph states associated with chain, star, regular lattice, and binary-tree hypergraphs are examined, and their entanglement is quantified using quantum computing. These results open up the possibility of studying hypergraph properties with quantum programming.
Complex entangled states are the key resources for measurement-based quantum computations, which is realised by performing a sequence of measurements on initially entangled qubits. Executable quantum algorithms in the graph-state quantum computing model are determined by the entanglement structure and the connectivity of entangled qubits. By generalisation from graph-type entanglement in which only the nearest qubits interact to a new type of hypergraph entanglement in which any subset of qubits can be arbitrarily entangled via hyperedges, hypergraph states represent more general resource states that allow arbitrary quantum computation with Pauli universality. Here we report experimental preparation, certification and processing of complete categories of four-qubit hypergraph states under the principle of local unitary equivalence, on a fully reprogrammable silicon-photonic quantum chip. Genuine multipartite entanglement for hypergraph states is certificated by the characterisation of entanglement witness, and the observation of violations of Mermin inequalities without any closure of distance or detection loopholes. A basic measurement-based protocol and an efficient resource state verification by color-encoding stabilizers are implemented with local Pauli measurement to benchmark the building blocks for hypergraph-state quantum computation. Our work prototypes hypergraph entanglement as a general resource for quantum information processing.