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Chinese Physics B

Generalized Hardy-type tests for hierarchy of multipartite non-locality*

Fei Yang, Yuan Yu, Wen-Lu Lin, Shu-Ao Liao, Chengjie Zhang, Qing Chen

Published Nov 6, 2019• 0 citations
Publisher

Abstract

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)].

DOI
10.1088/1674-1056/ab54b8

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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)].
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 $(m\ensuremath{-}1)\mathrm{th}$ 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.
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.
Nonlocal correlations as revealed by the violations to Bell inequalities are incompatible with local models without any nonlocal correlations. However some tripartite entangled states, e.g., symmetric pure states, exhibit a stronger nonlocality even incompatible with hybrid local/nonlocal models allowing nonlocal correlations between any two parties. Here we propose a three-particle Hardy-type test without inequality, as illustrated by a gedanken experiment, that is failed by all non-signaling local models while is passed by all tripartite entangled asymmetric pure states. Our result implies that every tripartite entangled state, regardless of the dimensions of the underlying Hilbert spaces, passes a Hardy-type test and therefore is genuine tripartite nonlocal. Maximal success probability and a generalization to multipartite cases of our test are also presented.
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.
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.
Genuine multipartite nonlocality is a salient feature of quantum systems, empowering the security of multi-party device independent cryptographic protocols. Given a correlation, characterizing and detecting genuineness have been subjected to recent studies. In this regard, we propose a Hardy-type argument which is able to detect genuine nnn-way nonlocality of arbitrary quantum systems. To understand the strength of this argument we also study the optimal success probability of the argument in a minimally constrained theory, namely the generalized no-signaling theory.