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arXiv:2001.02143 · Physical Review A

Testing multipartite entanglement with Hardy's nonlocality

Lijinzhi Lin, Zhaohui Wei

Published May 26, 2020• 2 citations
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Abstract

Multipartite quantum states may exhibit different types of quantum entanglement in that they cannot be converted into each other by local quantum operations only, and fully understanding mathematical structures of different types of multipartite entanglement is a very challenging task. In this paper, from the viewpoint of Hardy's nonlocality, we compare W and GHZ (Greenberger-Horne-Zeilinger) states and show a couple of crucial different behaviors between them. Particularly, by developing a geometric model for Hardy's nonlocality problem of W states, we derive an upper bound for its maximal violation probability, which turns out to be strictly smaller than the corresponding probability of GHZ states. This gives us a new comparison between these two quantum states, and the result is also consistent with our intuition that GHZ states are more entangled. Furthermore, we generalize our approach to obtain an asymptotic characterization for general $N$-qubit W states, revealing that when $N$ goes up, the speed that the maximum violation probabilities decay is exponentially slower than that of general $N$-qubit GHZ states. We provide some numerical simulations to verify our theoretical results.

DOI
10.1103/physreva.101.052118

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Multipartite quantum states may exhibit different types of quantum entanglement in that they cannot be converted into each other by local quantum operations only, and fully understanding mathematical structures of different types of multipartite entanglement is a very challenging task. In this paper, from the viewpoint of Hardy's nonlocality, we compare W and GHZ (Greenberger-Horne-Zeilinger) states and show a couple of crucial different behaviors between them. Particularly, by developing a geometric model for Hardy's nonlocality problem of W states, we derive an upper bound for its maximal violation probability, which turns out to be strictly smaller than the corresponding probability of GHZ states. This gives us a new comparison between these two quantum states, and the result is also consistent with our intuition that GHZ states are more entangled. Furthermore, we generalize our approach to obtain an asymptotic characterization for general NNN-qubit W states, revealing that when NNN goes up, the speed that the maximum violation probabilities decay is exponentially slower than that of general NNN-qubit GHZ states. We provide some numerical simulations to verify our theoretical results.
Entangled states of multiple qubits can violate Bell-type inequalities indicating nonlocal behavior of multiqubit quantum correlations. We analyze the relation between multipartite entanglement and genuine multipartite nonlocality, characterized by Svetlichny inequality violations, for two families of N − qubit states. We show that for the generalized Greenberger-Horne-Zeilinger family of states, Svetlichny inequality is not violated when the n − tangle is less than 1 / 2 for any even number of qubits. On the other hand, the maximal slice states always violate the Svetlichny inequality when n − tangle is nonzero, and the violation increases monotonically with tangle. Our work generalizes the relations between tangle and Svetlichny inequality violations previously derived for three qubits.
We propose a family of Hardy-type tests for an arbitrary n-partite system, which can detect different degrees of non-locality ranging from standard to genuine multipartite non-locality. For any non-signaling m-local hidden variable model, the corresponding tests fail, whereas a pass of this type of test indicates that this state is m non-local. We show that any entangled generalized GHZ state exhibits Hardy’s non-locality for each rank of multipartite non-locality. Furthermore, for the detection of m non-localities, a family of Bell-type inequalities based on our test is constructed. Numerical results show that it is more efficient than the inequalities proposed in [Phys. Rev. A 94 022110 (2016)].
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides three-qubit states into separable, biseparable, WWW, and Greenberger-Horne-Zeilinger (GHZ) classes. The WWW and GHZ are two subclasses of genuine tripartite entanglement. We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both the WWW and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.
Practical robust self-testing of multipartite entanglement has so far been restricted to small-scale systems due to error bounds that degrade severely with system size. In this work, we establish multipartite self-testing with robustness independent of the size of the quantum network. We derive a fully analytic, device-independent self-testing bound for nnn-qubit Greenberger-Horne-Zeilinger (GHZ) states. The bound scales linearly with the observed violation error and lies universally within a constant factor of two from a theoretical upper bound. Furthermore, the operator-inequality framework reduces the verification of the conjectured optimal bound to a highly efficient numerical check, which we perform up to n=100n=100n=100. Consequently, GHZ entanglement can be certified under a fixed noise level in arbitrarily large systems, enabling scalable device-independent verification.
Nonlocality can be witnessed either through Bell-inequality violations or through logical contradictions such as Hardy's paradox. In the bipartite two input two outcome scenario, these two routes have distinct geometric behavior: CHSH-maximal correlations are exposed points of the quantum set, whereas known Hardy-type self-testing correlations on the no-signaling boundary are non-exposed. Here we show that this bipartite intuition fails in the tripartite two input two outcome scenario. We study the tripartite instance of a multipartite Hardy-type paradox and prove that the correlation attaining the maximal Hardy success probability self-tests the Greenberger--Horne--Zeilinger state and the associated measurements. Although this correlation lies on the no-signaling boundary, we show that it is an extremal and exposed point of the quantum correlation set. Moreover, it coincides with the correlation attaining the maximal violation of the Mermin inequality. Thus, in the tripartite GHZ scenario, the logical-paradox and Bell-inequality routes to nonlocality select the same exposed quantum boundary point. We also establish a robust version of the self-test, showing that small deviations from the ideal Hardy constraints imply quantitative closeness to the target state and measurements. Our results reveal a qualitative geometric difference between bipartite and tripartite Hardy-type nonlocality and suggest a broader investigation of exposedness for multipartite Hardy correlations in the multiparty setting.
Characterizing entanglement of systems composed of multiple particles is a very complex problem that is attracting increasing attention across different disciplines related to quantum physics. The task becomes even more complex when the particles have many accessible levels, i.e., they are of high dimension, which leads to a potentially high-dimensional multipartite entangled state. These are important resources for an ever-increasing number of tasks, especially when a network of parties needs to share highly entangled states, e.g., for communicating more efficiently and securely. For these applications, as well as for purely theoretical arguments, it is important to be able to certify both the high-dimensional and the genuine multipartite nature of entangled states, possibly based on simple measurements. Here we derive a novel method that achieves this and improves over typical entanglement witnesses like the fidelity with respect to states of a Greenberger-Horne-Zeilinger (GHZ) form, without needing more complex measurements. We test our condition on paradigmatic classes of high-dimensional multipartite entangled states like imperfect GHZ states with random noise, as well as on purely randomly chosen ones and find that, in comparison with other available criteria our method provides a significant advantage and is often also simpler to evaluate.