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Physical Review A

Robust coincidence Bell inequalities for the noisy n -qutrit Greenberger-Horne-Zeilinger state

Hui-Xian Meng, Zhongyan Li, Xing-Yan Fan, Jia-Le Miao, Hong-Ye Liu, Yi-Jia Liu, Wei-Min Shang, Jie Zhou, Jing‐Ling Chen

Published Jun 27, 2022• 2 citations
Publisher

Abstract

The general form of the coincidence Bell inequalities for an arbitrary $(n,2,3)$ scenario [i.e., an $n$-party, two-setting, and three-dimensional system (qutrit) scenario] is presented. To detect the nonlocal properties of the noisy $n$-qutrit Greenberger-Horne-Zeilinger states, in this work we investigate the most robust (4,2,3)-scenario and (5,2,3)-scenario coincidence Bell inequalities. By deriving the most robust $(n\ensuremath{-}1)$-party coincidence inequalities, we have established two tight and inequivalent $(n,2,3)$ coincidence Bell inequalities with a visibility of 0.5 for $n=4$ and, similarly, two tight and inequivalent $(n,2,3)$ coincidence Bell inequalities with a visibility of 0.488785 for $n=5$. These inequalities are all tight. To our knowledge, up to now these inequalities have been the most robust Bell inequalities for the corresponding scenarios. Our results are useful for building the iteration formula of $(n,2,3)$-scenario coincidence Bell inequalities with the lowest critical visibility.

DOI
10.1103/physreva.105.062215

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The general form of the coincidence Bell inequalities for an arbitrary (n,2,3)(n,2,3)(n,2,3) scenario [i.e., an nnn-party, two-setting, and three-dimensional system (qutrit) scenario] is presented. To detect the nonlocal properties of the noisy nnn-qutrit Greenberger-Horne-Zeilinger states, in this work we investigate the most robust (4,2,3)-scenario and (5,2,3)-scenario coincidence Bell inequalities. By deriving the most robust (n\ensuremath−1)(n\ensuremath{-}1)(n\ensuremath−1)-party coincidence inequalities, we have established two tight and inequivalent (n,2,3)(n,2,3)(n,2,3) coincidence Bell inequalities with a visibility of 0.5 for n=4n=4n=4 and, similarly, two tight and inequivalent (n,2,3)(n,2,3)(n,2,3) coincidence Bell inequalities with a visibility of 0.488785 for n=5n=5n=5. These inequalities are all tight. To our knowledge, up to now these inequalities have been the most robust Bell inequalities for the corresponding scenarios. Our results are useful for building the iteration formula of (n,2,3)(n,2,3)(n,2,3)-scenario coincidence Bell inequalities with the lowest critical visibility.
Abstract In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in Salavrakos et al (2017 Phys. Rev. Lett. 119 040402) a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger–Horne–Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled Greenberger–Horne–Zeilinger-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.
Abstract In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in Salavrakos et al (2017 Phys. Rev. Lett. 119 040402) a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger–Horne–Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled Greenberger–Horne–Zeilinger-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.
We derive tight bipartite Bell inequalities for various scenarios. A bipartite Bell scenario (X,Y,A,B)(X,Y,A,B)(X,Y,A,B) is defined by the numbers of settings and outcomes per party, X,AX,AX,A, and Y,BY,BY,B for Alice and Bob, respectively. We derive the complete set of facets of the local polytopes of (6,3,2,2), (3,3,3,2), (3,2,3,3), and (2,2,3,5). We provide extensive lists of facets for (2,2,4,4), (3,3,4,2), and (4,3,3,2). For each inequality we compute the maximum quantum violation, the resistance to noise, and the minimal symmetric detection efficiency required to close the detection loophole, for qubits, qutrits, and ququarts. Based on these results, we identify scenarios which perform better in terms of visibility, resistance to noise, or both, when compared to the Clauser-Horne-Shimony-Holt inequality. Such scenarios could find important applications in quantum communication.
Ardehali (Phys. Rev. A, 46, 5375 (1992). 10.1103/PhysRevA.46.5375 ) derived a Bell-type inequality for an n-particle system in a Greenberger–Horne–Zeilinger (GHZ) state. This inequality helps relax the detection efficiencies required for closing the detection loophole in Bell tests. Coffman et al. (Phys. Rev. A, 61, 052306 (2000). doi: 10.1103/PhysRevA.61.052306 ) gave a simplified form of tripartite entanglement of three-qubit pure states. We use theory and experiment to investigate both results using three-qubit GHZ states with different intensities of bit-flip noise. The purpose of our work is to relate the Ardehali inequality and the tripartite concurrence of the tripartite GHZ states in a noisy environment. Our results show that the experimental values are quite consistent with theoretical predictions for three-qubit GHZ states in a bit-flip noisy environment.
In multipartite Bell scenarios, we study the nonlocality robustness of the Greenberger-Horne-Zeilinger (GHZ) state. When each party performs planar measurements forming a regular polygon, we exploit the symmetry of the resulting correlation tensor to drastically accelerate the computation of (i) a Bell inequality via Frank-Wolfe algorithms and (ii) the corresponding local bound. The Bell inequalities obtained are facets of the symmetrized local polytope and they give the best-known upper bounds on the nonlocality robustness of the GHZ state for three to ten parties. Moreover, for four measurements per party, we generalize our facets and hence show, for any number of parties, an improvement on Mermin's inequality in terms of noise robustness. We also compute the detection efficiency of our inequalities and show that some give rise to the activation of nonlocality in star networks, a property that was only shown with an infinite number of measurements.
Bell inequality is a vital tool to detect the nonlocal correlations, but the construction of it for multipartite systems is still a complicated problem. In this work, inspired via a decomposition of (n+1)(n+1)(n+1)-partite Bell inequalities into nnn-partite ones, we present a generalized iterative formula to construct nontrivial (n+1)(n+1)(n+1)-partite ones from the nnn-partite ones. Our iterative formulas recover the well-known Mermin-Ardehali-Belinsk\ifmmode \breve{\imath}\else \u{\i}\fi{}-Klyshko (MABK) and other families in the literature as special cases. Moreover, a family of ``dual-use'' Bell inequalities is proposed, in the sense that for the generalized Greenberger-Horne-Zeilinger states these inequalities lead to the same quantum violation as the MABK family and, at the same time, the inequalities are able to detect the nonlocality in the entire entangled region. Furthermore, we present a generalization of the I3322 inequality to any nnn-partite case which is still tight, and of the 46 \ifmmode \acute{S}\else \'{S}\fi{}liwa's inequalities to the four-partite tight ones, by applying our iteration method to each inequality and its equivalence class.