Achievable Trade-Off in Network Nonlocality Sharing

TL;DR

This work addresses how entanglement resources bound the ability to recycle nonlocal correlations in quantum networks by introducing a probabilistic projective measurements (PPM) protocol. It identifies a threshold $C(k)$ that allows unbounded, full-network sharing, and derives a depth–breadth trade-off $m+j=n+k-1$ at the threshold when resources are limited. The paper also compares PPM to weak measurements, showing superior detectability, and extends the framework to depolarizing and amplitude-damping noise to render the protocol robust under realistic conditions. Together, these results map the feasible regimes for nonlocality recycling in star-network topologies and guide scalable quantum-network design. The findings offer practical guidance for resource-efficient quantum networks and suggest future directions in network topologies and quantum-resource theory integration.

Abstract

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.

Figures (6)

• Figure 1: The $n$-star network. The central party Bob shares a bipartite entangled state with each of the $n$ peripheral Alice$^{1}$, Alice$^{2}$, $\cdots$, Alice$^{n}$ via independent sources.
• Figure 2: Sequential nonlocality sharing protocol. The central node Bob shares an entangled state with each peripheral Alice$^{i,1}$$(i=1,\cdots,n)$. On $m$ branches, sequential observers (Alice$^{i,j}$) perform the PPM strategy: If the input is $x_{i1}=0$, Alice$^{i,j}$ measures $\sigma_x$. If the input is $x_{ij}=1$, she flips a biased classical coin; on "heads” she measures $\sigma_z$, while on "tails” she leaves the state unchanged. Each observer then passes the post-measurement state to the next observer in the same branch, repeating the PPM up to the final layer. The remaining $n-m$ branches are preserved until the final measurement to detect network nonlocality.
• Figure 3: Achievable sharing rounds. The figure employs a semicircular coordinate system where the angular position directly represents Bob’s measurement parameter $\delta$, ranging from $\delta = 0$ at the right endpoint to $\delta = \frac{\pi}{4}$ at the left endpoint. The contours are labeled from 1 to 5, indicating the value of sharing rounds. We map the nonlocality sharing capability for quantum states with different initial parameter $\theta$. The pentagram markers in the figure denote parameter configurations that satisfy the relation $2\theta + \delta = \frac{\pi}{2}$, which is constructed in the proof of Thm \ref{['theorem1']}. Adherence to this relation guarantees, within our protocol, the achievement of the target sequential sharing round $k$ for a given entanglement resource $C>C(k)$.
• Figure 4: Comparison with unsharp measurements. The orange line corresponds to the PPM protocol, while the gray line represents the unsharp measurement protocol. For the same sharing depth, the PPM protocol yields consistently stronger nonlocality violations across all rounds, indicating superior experimental detectability.
• Figure 5: Sharing capability under noises. The angular coordinate represents the noise strength, with zero noise at the right endpoint and increasing toward the left. Contour lines labeled from 1 to 5 indicate the maximum number of achievable sharing rounds $k$ for different initial states. Panel (a) covers depolarizing noise, with the noise strength ranging from 0 to 0.1 . Panel (b) corresponds to amplitude damping noise, with the damping parameter ranging from 0 to 0.3. The mapping illustrates how the noise level influences the sharing robustness under varying initial entanglement.

Theorems & Definitions (11)

• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4