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arXiv:2512.13357 · Physical Review Research

Entanglement thresholds for sequential nonlocality sharing in star networks

Ming-Xiao Li, Yuqi Li, Rui-Bin Xu, Mo-Ran Zhu, Haitao Ma, Chang-Yue Zhang, Zhu-Jun Zheng

quant-phSubmitted Dec 15, 2025• Announced Dec 16, 2025• 0 citations• Revised Oct 9, 2026 (v2)
v1 · Dec 15, 2025v2 · Oct 9, 2026
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Abstract

Quantum networks promise scalable distributed quantum information processing. Realizing this potential depends critically on the efficient use of nonclassical resources. Sequential sharing of quantum nonlocality provides a resource-efficient route for recycling entanglement across multiple observers. In network settings, however, such recycling is typically constrained by a trade-off between the number of active branches and the achievable sharing depth. Here we investigate this problem in star networks using probabilistic projective measurements. We establish an explicit entanglement threshold above which nonlocality can be shared across all branches up to any target depth, and below which we determine the optimal trade-off between the number of active branches and the achievable number of sequential rounds. We further analyze the protocol's robustness to noise and its experimental observability, clarifying its practical feasibility. These results establish quantitative design principles and entanglement requirements for scalable nonlocality recycling in quantum networks.

Journal reference
Phys. Rev. Research 8, 033274 (2026)
DOI
10.1103/26rt-yvpn
Comments
16 pages, 6 figures

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Quantum networks promise scalable distributed quantum information processing. Realizing this potential depends critically on the efficient use of nonclassical resources. Sequential sharing of quantum nonlocality provides a resource-efficient route for recycling entanglement across multiple observers. In network settings, however, such recycling is typically constrained by a trade-off between the number of active branches and the achievable sharing depth. Here we investigate this problem in star networks using probabilistic projective measurements. We establish an explicit entanglement threshold above which nonlocality can be shared across all branches up to any target depth, and below which we determine the optimal trade-off between the number of active branches and the achievable number of sequential rounds. We further analyze the protocol's robustness to noise and its experimental observability, clarifying its practical feasibility. These results establish quantitative design principles and entanglement requirements for scalable nonlocality recycling in quantum networks.
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.
Sequential sharing of quantum nonlocality enables multiparty collaborative information processing, which is typically realized through unsharp measurements in previous studies. By contrast, we experimentally demonstrate that the sequential sharing of quantum nonlocality can be observed using only projective measurements, with the introduction of classical randomness. We achieve sequential violations of the Clauser-Horne-Shimony-Holt inequality in a bipartite quantum system by sharing classical randomness or using a probabilistic projective measurement. Extending the first protocol to a tripartite quantum system, sequential tripartite nonlocality is observed with the Greenberger-Horne-Zeilinger state, where violations of the Mermin inequality are observed in sequential measurements. We experimentally demonstrate these phenomena through the preparation and measurement of a three-photon entangled state. Our work reveals the key role of projective measurements in the sequential sharing of quantum nonlocality, providing a new experimental complement for measuring quantum nonlocality and opening new technological pathways for distributed utilization nonlocality in quantum networks.
The nonlocality of an entangled qubit pair can be recycled for several Bell experiments. Here, we go beyond standard Bell scenarios and investigate the recycling of nonlocal resources in a quantum network. First, we show that the ability to recycle nonlocality can be independent of the size of the network. Then, we realize a photonic quantum three-branch star network in which three sources of entangled pairs independently connect three outer parties with a central node. After measuring, each outer party respectively relays their system to an independent secondary measuring party. We experimentally demonstrate that the outer parties can perform unsharp measurements that are strong enough to violate a network Bell inequality with the central party, but weak enough to maintain sufficient entanglement in the network to allow the three secondary parties to do the same. Moreover, the violations are strong enough to exclude any model based on standard projective measurements on the Einstein-Podolsky-Rosen pairs emitted in the network. Our experiment brings together the research program of recycling quantum resources with that of Bell nonlocality in networks.
Unsharp measurements are widely seen as the key resource for recycling the nonlocality of an entangled state shared between several sequential observers. Contrasting this, we here show that nonlocality can be recycled using only standard, projective, qubit measurements. Focusing on the Clauser-Horne-Shimony-Holt inequality and allowing parties to share classical randomness, we determine the optimal trade-off in the magnitude of Bell violations for a maximally entangled state. We then find that nonmaximally entangled states make possible larger sequential violations, which contrasts the standard Clauser-Horne-Shimony-Holt scenario. Furthermore, we show that nonlocality can be recycled using projective qubit measurements even when no shared classical randomness is available. We discuss the implications of our results for experimental implementations of sequential nonlocality.
Abstract The nonlocality of an entangled qubit pair can be recycled for several Bell experiments. Here, we go beyond standard Bell scenarios and investigate the recycling of nonlocal resources in a quantum network. First, we show that the ability to recycle nonlocality can be independent of the size of the network. Then, we realize a photonic quantum 3-branch star network in which three sources of entangled pairs independently connect three outer parties with a central node. After measuring, each outer party respectively relays their system to an independent secondary measuring party. We experimentally demonstrate that the outer parties can perform unsharp measurements that are strong enough to violate a network Bell inequality with the central party, but weak enough to maintain sufficient entanglement in the network to allow the three secondary parties to do the same. Moreover, the violations are strong enough to exclude any model based on standard projective measurements on the EPR pairs emitted in the network. Our experiment brings together the research program of recycling quantum resources with that of Bell nonlocality in networks.
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.