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arXiv:2601.20403

Network Nonlocality Sharing in Generalized Star Network from Bipartite Bell Inequalities

Hao-Miao Jiang, Xiang-Jiang Chen, Liu-Jun Wang, Qing Chen

quant-phSubmitted Jan 28, 2026• Announced Jan 29, 2026• 0 citations
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Abstract

This work investigates network nonlocality sharing for a broad class of bipartite Bell inequalities in a generalized star network with an $(n,m,k)$ configuration, comprising $n$ independent branches, $m$ sequential Alices per branch, and $k$ measurement settings per party. On each branch, the intermediate Alices implement optimal weak measurements, whereas the final Alice and the central Bob perform sharp projective measurements. Network nonlocality sharing is witnessed when the quantum values of the network correlations associated with relevant parties simultaneously violate a star-network Bell inequality generated from the given class of bipartite Bell inequalities. We streamline the calculation of the quantum values of the network correlations and derive an analytical expression for the bipartite quantum correlator, valid for arbitrary measurement settings and weak-measurement strengths. The network nonlocality sharing for Vértesi inequalities has been studied within the framework, and simultaneous violations are found in $(2,2,6)$ and $(2,2,465)$ cases, with the latter exhibiting greater robustness. Our approach suggests a practical route to studying network nonlocality sharing by utilizing diverse bipartite Bell inequalities beyond the commonly used CHSH-type constructions.

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12 pages, 2 figures

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This work investigates network nonlocality sharing for a broad class of bipartite Bell inequalities in a generalized star network with an (n,m,k)(n,m,k)(n,m,k) configuration, comprising nnn independent branches, mmm sequential Alices per branch, and kkk measurement settings per party. On each branch, the intermediate Alices implement optimal weak measurements, whereas the final Alice and the central Bob perform sharp projective measurements. Network nonlocality sharing is witnessed when the quantum values of the network correlations associated with relevant parties simultaneously violate a star-network Bell inequality generated from the given class of bipartite Bell inequalities. We streamline the calculation of the quantum values of the network correlations and derive an analytical expression for the bipartite quantum correlator, valid for arbitrary measurement settings and weak-measurement strengths. The network nonlocality sharing for Vértesi inequalities has been studied within the framework, and simultaneous violations are found in (2,2,6)(2,2,6)(2,2,6) and (2,2,465)(2,2,465)(2,2,465) cases, with the latter exhibiting greater robustness. Our approach suggests a practical route to studying network nonlocality sharing by utilizing diverse bipartite Bell inequalities beyond the commonly used CHSH-type constructions.
Abstract This work investigates network nonlocality sharing for a broad class of bipartite Bell inequalities in generalized star networks with an ( n , m , k ) configuration, comprising n independent branches, m sequential Alices per branch, and k measurement settings per party. On each branch, the intermediate Alices implement weak measurements, whereas the final Alice and the central Bob perform sharp projective measurements. Network nonlocality sharing is witnessed when the quantum values of the network correlations associated with relevant parties simultaneously violate a star-network Bell inequality generated from the given class of bipartite Bell inequalities. We streamline the calculation of the quantum values of the network correlations and derive an analytical expression for the bipartite quantum correlator, valid for arbitrary two-qubit source states, measurement settings and weak-measurement strengths. Network nonlocality sharing for Vértesi inequalities has been studied within the framework, and simultaneous violations are found in the ( 2 , 2 , 6 ) and ( 2 , 2 , 465 ) cases, with the latter exhibiting greater robustness. Our approach suggests a practical route to studying network nonlocality sharing by utilizing diverse bipartite Bell inequalities beyond the commonly used CHSH-type constructions.
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Network nonlocality exhibits completely novel quantum correlations compared to standard quantum nonlocality. It has been shown that network nonlocality can be shared in a generalized bilocal scenario via weak measurements [Hou, Liu, and Ren, Phys. Rev. A 105, 042436 (2022)]. In this paper, we investigate network nonlocality sharing via weak measurements in a generalized star-shaped network configuration with arbitrary numbers of unbiased dichotomic input kkk, which includes nnn branches and adds $(m\ensuremath{-}1)$ more parties in each branch to the original star network (n,m=1,k=2)(n,m=1,k=2)(n,m=1,k=2) scenario. It is shown that network nonlocality sharing among all observers can be revealed from simultaneous violation of 2n{2}^{n}2n inequalities in the (n,m=2,k=2)(n,m=2,k=2)(n,m=2,k=2) and (n,m=2,k=3)(n,m=2,k=3)(n,m=2,k=3) scenarios for any nnn branches. The noise resistance of network nonlocality sharing with a precise noise model is also analyzed.
Sequential measurements present a novel framework for constructing large-scale quantum networks, particularly in quantum correlation sharing. In this paper, we design projective measurements and combination strategies to achieve nonlocality sharing in star-shaped networks by leveraging shared classical randomness. This network consists of nnn branches, each containing one or two sequential observers. By investigating two-sided nonlocality sharing within a 2-local network, we identify the optimal projective measurement combination strategy based on the double violation of the 2-locality inequality. Utilizing this optimal strategy, we demonstrate that four inequalities can be violated simultaneously in the 2-local scenario. Furthermore, in an nnn-local scenario involving mmm ($1\ensuremath{\le}m\ensuremath{\le}n$) branches with two sequential observers, our optimal strategy shows that non-nnn-locality sharing among all observers can be achieved through the simultaneous violation of 2mn{2}^{m} n2mn-locality inequality. Our findings provide a distinctive perspective on quantum correlation sharing in networks.
The concept of bilocality was introduced to study the correlations which arise in an entanglement-swapping scenario, where one has two sources which can naturally be taken to be independent. This additional constraint leads to stricter requirements than simply imposing locality, in the form of bilocality inequalities. In this work we consider a natural generalization of the bilocality scenario; namely, the star network consisting of a single central party surrounded by nnn edge parties, each of which shares an independent source with the center. We derive inequalities which are satisfied by all local correlations in this scenario, for the cases when the central party performs (i) two dichotomic measurements (ii) a single Bell-state measurement. We demonstrate quantum violations of these inequalities and study both the robustness to noise and to losses.
The multipartite quantum networks feature multiple independent sources, in contrast to the conventional multipartite Bell experiment involving a single source. Despite the initial independence of resources, the multiple observers in the network can suitably choose measurements on their local subsystems and generate a form of quantum nonlocality across the network. So far, network nonlocality has been explored when each source produces a two-qubit entangled state. In this work we demonstrate the network nonlocality when each party performs a black-box measurement, and the dimension of the system remains unspecified. In an interesting work, by considering each source produces two-qubit entangled states in the conventional bilocal scenario, Gisin et al. [Phys. Rev. A 96, 020304 (2017)] demonstrated a correspondence between the violations of bipartite Clauser-Horne-Shimony-Halt inequality and the bilocality inequality. We introduce a variant of the sum-of-squares approach to reproduce their results without assuming the dimension of the system. We then generalize the argument for network nonlocality in star-network topology. Furthermore, we propose a set of nnn-locality inequalities in a star-network configuration where each of the nnn parties performs an arbitrary number of dichotomic measurements and demonstrates the above correspondence between the quantum violations of the nnn-locality inequalities and the chained Bell inequalities. A similar correspondence is demonstrated based on a recently formulated family of nnn-locality inequalities whose optimal quantum violation cannot be obtained when each source emits a two-qubit entangled state and requires multiple copies of two-qubit entangled states. Throughout this paper, each party in the network performs black-box measurements, and the dimension of the system remains unspecified.