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arXiv:1608.03084 · Physical Review A

Hierarchy of multipartite nonlocality in the nonsignaling scenario

Xiaoxu Wang, Chengjie Zhang, Qing Chen, Sixia Yu, Haidong Yuan, C. H. Oh

Published Aug 9, 2016• 13 citations
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Abstract

We propose a hierarchy of Bell-type inequalities for arbitrary $n$-partite systems that identifies the different degrees of nonlocality ranging from standard to genuine multipartite nonlocality. After introducing the definition of nonsignaling $m$ locality, we show that the observed joint probabilities in any nonsignaling $m$-local realistic models should satisfy the $(m\ensuremath{-}1)\mathrm{th}$ Bell-type inequality. When $m=2$, the corresponding inequality reduces to the one shown earlier [Q. Chen et al., Phys. Rev. Lett. 112, 140404 (2014)] whose violation indicates genuine multipartite nonlocality, and when $m=n$, the corresponding inequality is just Hardy's inequality whose violation indicates standard multipartite nonlocality. Furthermore, several examples are provided to demonstrate their hierarchy of multipartite nonlocality.

DOI
10.1103/physreva.94.022110

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We propose a hierarchy of Bell-type inequalities for arbitrary nnn-partite systems that identifies the different degrees of nonlocality ranging from standard to genuine multipartite nonlocality. After introducing the definition of nonsignaling mmm locality, we show that the observed joint probabilities in any nonsignaling mmm-local realistic models should satisfy the (m\ensuremath−1)th(m\ensuremath{-}1)\mathrm{th}(m\ensuremath−1)th Bell-type inequality. When m=2m=2m=2, the corresponding inequality reduces to the one shown earlier [Q. Chen et al., Phys. Rev. Lett. 112, 140404 (2014)] whose violation indicates genuine multipartite nonlocality, and when m=nm=nm=n, the corresponding inequality is just Hardy's inequality whose violation indicates standard multipartite nonlocality. Furthermore, several examples are provided to demonstrate their hierarchy of multipartite nonlocality.
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)].
In this Letter we propose a set of conditions on the joint probabilities as a test of genuine multipartite nonlocality without inequality. Our test is failed by all nonsignaling local models in which even nonlocal correlations among some observables (not all) are allowed as long as these correlations respect the nonsignaling principle. A pass of our test by a state therefore indicates that this state cannot be simulated by any nonsignaling local models; i.e., the state exhibits genuine multipartite nonlocality. It turns out that all entangled symmetric n-qubit (n≥3) states pass our test and therefore are n-way nonlocal. Also we construct two Bell-type inequalities from our proposed test whose violations indicate genuine multipartite nonlocal correlations.
The structure of Bell-type inequalities detecting genuine multipartite nonlocality, and hence detecting genuine multipartite entanglement, is investigated. We first present a simple and intuitive approach to Svetlichny's original inequality, which provides a clear understanding of its structure and of its violation in quantum mechanics. Based on this approach, we then derive a family of Bell-type inequalities for detecting genuine multipartite nonlocality in scenarios involving an arbitrary number of parties and systems of arbitrary dimension. Finally, we discuss the tightness and quantum mechanical violations of these inequalities.
Genuine multipartite nonlocality constitutes the strongest form of nonlocality, which is a key resource for many quantum communication and computational tasks. Standard methods for certifying nonlocality often require accessing contradictions in paradoxes or violations of Bell inequalities for composite quantum systems. However, the existing approaches become infeasible and cannot be adopted straightforwardly for a quantum network scenario, which usually possesses complex topological structure and many independent quantum sources. We have developed a simple and customizable strategy to achieve scalable, efficient, and flexible characterizations for various typical quantum network scenarios. The subtle something-versus-nothing contradiction in Hardy-type paradoxes without inequality allows us to substantiate genuine multipartite nonlocality in general quantum networks. Examples range from chain-type network to star-type network, with source connections of any pure entangled permutation symmetric multiqubit states, entangled two-qubit states, etc. The tools extend significantly our ability to certify genuine multipartite nonlocality in state-of-the-art network situations, and make a prominent step forward toward deeper understanding and practical depiction of quantum networks.
While Bell nonlocality of a bipartite system is counterintuitive, multipartite nonlocality in our many-body world turns out to be even more so. Recent theoretical study reveals in a theory-agnostic manner that genuine multipartite nonlocal correlations cannot be explained by any causal theory involving fewer-partite nonclassical resources and global shared randomness. Here, we provide a Bell-type inequality as a test for genuine multipartite nonlocality in network by exploiting a matrix representation of the causal structure of a multipartite system. We further present experimental demonstrations that both four-photon GHZ state and generalized four-photon GHZ state significantly violate the inequality, i.e., the observed four-partite correlations resist explanations involving three-way nonlocal resources subject to local operations and common shared randomness, hence confirming that nature is boundless multipartite nonlocal.
In this work, we study a particular class of Bell inequalities involving only direct equality-comparisons of outcomes. In the bipartite case, the scenario can be interpreted as a natural generalization of full-correlator in- equalities (XOR games) beyond binary outputs. We define the sub-polytope of the local polytope corresponding to this scenario and solve it for several bipartite and multipartite scenarios by leveraging some structural prop- erties. In doing so, we obtain thousands of new tight inequalities, many of which are also facets of the standard local polytope. We also define unanimous Bell inequalities, a particular case of the previous class applied to the multipartite setting in which only full-equality events (all outcomes equal) are considered. We show that such inequalities can always be written as deterministic nonlocal games, and we give a simple multipartite unanimous family and prove its local bound. We show that most of these inequalities admit quantum violations, and we also display aspects of their importance for nonlocality. For instance, we identify examples where such inequalities can act as dimension witnesses, outcome witnesses, witnesses of genuine multipartite nonlocality, as well as being relevant to CHSH. These results show that these simple and elegant inequalities by themselves provide a powerful tool for discovering new Bell inequalities and device-independent witnesses.