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Physical Review A

Network nonlocality sharing in the star-shaped scenario using only projective measurements

Hao Sun, Fen-Zhuo Guo, Haifeng Dong, Fei Gao, Su‐Juan Qin

Published Feb 3, 2025• 4 citations
Publisher

Abstract

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 $n$ 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 $n$-local scenario involving $m$ ($1\ensuremath{\le}m\ensuremath{\le}n$) branches with two sequential observers, our optimal strategy shows that non-$n$-locality sharing among all observers can be achieved through the simultaneous violation of ${2}^{m} n$-locality inequality. Our findings provide a distinctive perspective on quantum correlation sharing in networks.

DOI
10.1103/physreva.111.022201

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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≤m\ensuremath≤n1\ensuremath{\le}m\ensuremath{\le}n1\ensuremath≤m\ensuremath≤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.
Quantum network nonlocality sharing provides a unique perspective for constructing large-scale quantum networks and holds promise for numerous potential applications. In this paper we demonstrate the network nonlocality sharing via unsharp measurements in a two-forked nnn-layer tree-shaped network, which features ${2}^{n}\ensuremath{-}2$ independent sources and ${2}^{n}\ensuremath{-}1$ nodes. We investigate the $({2}^{n}\ensuremath{-}2)$-local scenarios via any mmm-sided sequential measurements, and conclude that arbitrarily many independent observers on each sequential side can share the non-$({2}^{n}\ensuremath{-}2)$-locality for a sufficiently large value of nnn. For a finite nnn, we present a method that enables directly obtaining the number of observers on each sequential side, who can share the non-$({2}^{n}\ensuremath{-}2)$-locality simultaneously. To interpret our results more intuitively, we discuss the simplest case of n=3n=3n=3, and find that at most 30, 6, 3, and 2 sequential observers per sequential side can simultaneously share the non-6-locality in mmm-sided cases for m=1,2,3,4m=1,2,3,4m=1,2,3,4, respectively.
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.
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.
Bell's theorem was a cornerstone for our understanding of quantum theory and the establishment of Bell non-locality played a crucial role in the development of quantum information. Recently, its extension to complex networks has been attracting growing attention, but a deep characterization of quantum behavior is still missing for this novel context. In this work we analyze quantum correlations arising in the bilocality scenario, that is a tripartite quantum network where the correlations between the parties are mediated by two independent sources of states. First, we prove that non-bilocal correlations witnessed through a Bell-state measurement in the central node of the network form a subset of those obtainable by means of a local projective measurement. This leads us to derive the maximal violation of the bilocality inequality that can be achieved by arbitrary two-qubit quantum states and arbitrary local projective measurements. We then analyze in details the relation between the violation of the bilocality inequality and the CHSH inequality. Finally, we show how our method can be extended to the n -locality scenario consisting of n two-qubit quantum states distributed among n + 1 nodes of a star-shaped network.
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.
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.