Network Nonlocality Sharing in Generalized Star Network from Bipartite Bell Inequalities

TL;DR

This work addresses how to realize network nonlocality sharing in generalized star networks using a broad class of bipartite Bell inequalities. It introduces a general extension where a bipartite inequality $I= obreak \sum_{s,t} M_{st} A^s B^t \le C$ is mapped to an $(n,m,k)$-star network inequality $S^{(n,m,k)}$ with quantum violations indicating sharing, under an optimal weak-sharing protocol starting from independent singlet states. The main contributions are (i) a closed-form analytic expression for the bipartite correlator $\langle A_{ij^{(i)}}^{x} B_i^y \rangle$ in terms of measurement vectors, weak-measurement factors, and the matrices $K_{iq}$, valid for arbitrary $n,m,k$; (ii) a practical framework to study network nonlocality sharing beyond CHSH-type inequalities; and (iii) explicit simultaneous violations found for Vértesi inequalities in $(2,2,6)$ and $(2,2,465)$, with the larger $k$ case exhibiting greater robustness. The results show that network nonlocality sharing can persist with many measurement settings, offering a versatile approach for exploring nonlocal correlations in complex quantum networks.

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.

Figures (2)

• Figure 1: Schematic of the generalized star network. A central node $B$ shares $n$ bipartite states $\rho_{A_i B_i}$ with the first observers $A_{i1}$ ($i=1,\dots,n$), thereby generating $n$ branches. Along each branch $i$, the state is passed sequentially through $m$ parties $A_{i1},A_{i2},\dots,A_{im}$: every intermediate Alice performs an optimal weak measurement and forwards the post-measurement state to the next party, while the last Alice $A_{im}$ performs a sharp measurement. Each Alice has $k$ possible measurement settings. Double-headed arrows represent the distribution of the bipartite states $\rho_{A_i B_i}$ between $B$ and $A_{i1}$, and dashed arrows represent the sequential propagation of the post-measurement states along each branch.
• Figure 2: Normalized quantum correlations $S_j$ in the $(2,2,k)$-star network as a function of the precision factor $G$ for varying number of measurement settings $k$. The plots illustrate three correlations—$S_{11}$ (dashed blue line) for Alice$_{11}$ and Alice$_{21}$, $S_{12}$ (dotted gold line) for Alice$_{11}$ and Alice$_{22}$, and $S_{22}$ (solid green line) for Alice$_{12}$ and Alice$_{22}$—plotted against the precision factor, $G$. The vertical axis represents the correlation values $S_{j}$ normalized by the corresponding classical bound $\tilde{n}^2$, with the classical bound indicated by a solid black reference line at $1.0$. Each panel corresponds to a $(2, 2, k)$-star network scenario with a different number of measurement settings: (a) $k = 3$, (b) $k = 6$, and (c) $k = 465$. The plots demonstrate that a simultaneous violation of the classical bound is possible when $k>3$, and the quantum network nonlocality sharing effect becomes more pronounced as the number of measurement settings $k$ increases.

Theorems & Definitions (3)

• Theorem
• Claim
• proof