Hypergraph states form a family of multiparticle quantum states that generalizes the well-known concept of Greenberger-Horne-Zeilinger states, cluster states, and more broadly graph states. We study the nonlocal properties of quantum hypergraph states. We demonstrate that the correlations in hypergraph states can be used to derive various types of nonlocality proofs, including Hardytype arguments and Bell inequalities for genuine multiparticle nonlocality. Moreover, we show that hypergraph states allow for an exponentially increasing violation of local realism which is robust against loss of particles. Our results suggest that certain classes of hypergraph states are novel resources for quantum metrology and measurement-based quantum computation.PACS numbers: 03.65.Ta, 03.65.UdIntroduction.-Multiparticle entanglement is central for discussions about the foundations of quantum mechanics, protocols in quantum information processing, and experiments in quantum optics. Its characterization has, however, turned out to be difficult. One problem hindering the exploration of multiparticle entanglement is the exponentially increasing dimension of the Hilbert space. This implies that making statements about general quantum states is difficult. So, one has to concentrate on families of multiparticle states with an easier-tohandle description. In fact, symmetries and other kinds of simplifications seem to be essential for a state to be a useful resource. Random states can often be shown to be highly entangled, but useless for quantum information processing [1].

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While the circuit model of quantum computation defines its logical depth or "computational time" in terms of temporal gate sequences, the measurement-based model could allow totally different temporal ordering and parallelization of logical gates. By developing techniques to analyze Pauli measurements on multi-qubit hypergraph states generated by the Controlled-Controlled-Z (CCZ) gates, we introduce a deterministic scheme of universal measurement-based computation. In contrast to the cluster-state scheme where the Clifford gates are parallelizable, our scheme enjoys massive parallelization of CCZ and SWAP gates, so that the computational depth grows with the number of global applications of Hadamard gates, or, in other words, with the number of changing computational bases. A logarithmic-depth implementation of an N -times Controlled-Z gate illustrates a novel trade-off between space and time complexity.

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.

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An entanglement witness is an observable with the property that a negative expectation value signals the presence of entanglement. The question arises how a witness can be improved if the expectation value of a second observable is known, and methods for doing this have recently been discussed as so-called ultrafine entanglement witnesses. We present several results on the characterization of entanglement given the expectation values of two observables. First, we explain that this problem can naturally be tackled with the method of the Legendre transformation, leading even to a quantification of entanglement. Second, we present necessary and sufficient conditions that two product observables are able to detect entanglement. Finally, we explain some fallacies in the original construction of ultrafine entanglement witnesses [F. Shahandeh et al., Phys. Rev. Lett. 118, 110502 (2017)]. § Note that the numbering of the Lemmata in the Supplemental Material of Ref.[3] differs from the arxiv version.

In a paper by Popescu and Rohrlich [Phys. Lett. A 166, 293 (1992)] a proof has been presented showing that any pure entangled multiparticle quantum state violates some Bell inequality. We point out a gap in this proof, but we also give a construction to close this gap. It turns out that with some extra effort all the results from the aforementioned publication can be proven. Our construction shows how two-particle entanglement can be generated via performing local projections on a multiparticle state.

Since Bell’s theorem, it is known that local realism fails to explain quantum phenomena. Bell inequality violations manifestly show the incompatibility of quantum theory with classical notions of cause and effect. As recently found, however, the instrumental scenario—a pivotal tool in causal inference—allows for nonclassicality signatures going beyond this paradigm. If we are not limited to observational data and can intervene in our setup, then we can witness quantum violations of classical bounds on the causal influence among the involved variables even when no Bell-like violation is possible. That is, through interventions, the quantum behavior of a system that would seem classical can be demonstrated. Using a photonic setup—faithfully implementing the instrumental causal structure and switching between observation and intervention run by run—we experimentally witness such a nonclassicality. We also test quantum bounds for the causal influence, showing that they provide a reliable tool for quantum causal modeling.