Achievable Trade-Off in Network Nonlocality Sharing

Quantum networks are essential for advancing scalable quantum information processing. Quantum nonlocality sharing provides a crucial strategy for the resource-efficient recycling of quantum correlations, offering a promising pathway toward scaling quantum networks. Despite its potential, the limited availability of resources introduces a fundamental trade-off between the number of sharable network branches and the achievable sequential sharing rounds. The relationship between available entanglement and the sharing capacity remains largely unexplored, which constrains the efficient design and scalability of quantum networks. Here, we establish the entanglement threshold required to support unbounded sharing across an entire network by introducing a protocol based on probabilistic projective measurements. When resources fall below this threshold, we derive an achievable trade-off between the number of sharable branches and sharing rounds. To assess practical feasibility, we compare the detectability of our protocol with weak-measurement schemes and extend the sharing protocol to realistic noise models, providing a robust framework for nonlocality recycling in quantum networks.

💡 Research Summary

This paper addresses a fundamental problem in the scalability of quantum networks: how to recycle nonlocal quantum correlations across many users while contending with limited entanglement resources. The authors focus on a star‑shaped network in which a central node (Bob) shares independent bipartite entangled states with n peripheral nodes (Alice i). Nonlocality is certified by the standard star‑network Bell inequality Sₙ≤2, which must be violated to demonstrate genuine network‑wide quantum correlations.

The core contribution is the introduction of a probabilistic projective measurement (PPM) protocol. For each observer, the input x=0 triggers a σₓ projective measurement, while x=1 triggers a biased classical coin flip: “heads” leads to a σ_z measurement, “tails” leaves the state untouched. This simple stochastic element allows the post‑measurement state to be passed to the next observer in the same branch, enabling sequential extraction of nonlocality.

The authors derive an explicit entanglement threshold C(k)=2^{1−k}√(4k−1)−1 (expressed in terms of the concurrence C of the shared pure state). Theorem 1 proves that if every source has concurrence C∈