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arXiv:2508.04536 · Annalen der Physik

Entanglement Distribution in Quantum Networks Via Swapping of Partially Entangled Pure States

Henrique Guerra, Tailan S. Sarubi, Rafael Chaves, Jonas Maziero

Published Mar 1, 2026• 0 citations
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Abstract

ABSTRACT The entanglement swapping protocol (ESP) is a fundamental primitive for distributing quantum correlations across distant nodes in a quantum network. Recent studies have demonstrated that even when the involved qubit pairs are only partially entangled, it is still possible to concentrate and transmit entanglement via Bell‐basis measurements. In this work, we extend these ideas to quantum networks with various topologies–including linear, star, and hybrid configurations–by analyzing the application of the ESP to initially partially entangled pure states. We investigate how entanglement evolves under such protocols by considering the transformations of the initial states and evaluating the success probabilities for generating maximally entangled states at the output. Our results offer new insights into the dynamics of the entanglement distribution in quantum networks.

DOI
10.1002/andp.202500475

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ABSTRACT The entanglement swapping protocol (ESP) is a fundamental primitive for distributing quantum correlations across distant nodes in a quantum network. Recent studies have demonstrated that even when the involved qubit pairs are only partially entangled, it is still possible to concentrate and transmit entanglement via Bell‐basis measurements. In this work, we extend these ideas to quantum networks with various topologies–including linear, star, and hybrid configurations–by analyzing the application of the ESP to initially partially entangled pure states. We investigate how entanglement evolves under such protocols by considering the transformations of the initial states and evaluating the success probabilities for generating maximally entangled states at the output. Our results offer new insights into the dynamics of the entanglement distribution in quantum networks.
Abstract The entanglement swapping protocol (ESP) is extended to partially entangled qudit states and the process within the framework of complete complementarity relations (CCRs) is analyzed. Building on previous results for qubits, it is shown that the average distributed entanglement between two parties via ESP is bounded above by the initial entanglement of one of the input pairs, and also by the product of the initial entanglements. Notably, it is found that using initial states with vanishing local quantum coherence is sufficient to capture the essential features of the protocol, simplifying the analysis. By exploring the cases of qubits and qutrits, it is observed that the upper bound on the average distributed entanglement—expressed in terms of the product of the initial entanglements—can be improved, and what this tighter bound might be is conjectured. Finally, the role of quantum complementarity in the ESP is discussed and how local predictability constrains the entanglement that can be operationally distributed via ESP is shown.
We investigate entanglement distribution in pure-state quantum networks. We consider the case when nonmaximally entangled two-qubit pure states are shared by neighboring nodes of the network. For a given pair of nodes, we investigate how to generate the maximal entanglement between them by performing local measurements, assisted by classical communication, on the other nodes. We find optimal measurement protocols for both small and large one-dimensional networks. Quite surprisingly, we prove that Bell measurements are not always the optimal ones to perform in such networks. We generalize then the results to simple small two-dimensional (2D) networks, finding again counterintuitive optimal measurement strategies. Finally, we consider large networks with hierarchical lattice geometries and 2D networks. We prove that perfect entanglement can be established on large distances with probability one in a finite number of steps, provided the initial entanglement shared by neighboring nodes is large enough. We discuss also various protocols of entanglement distribution in 2D networks employing classical and quantum percolation strategies.
We consider entanglement swapping, a key component of quantum network operations and entanglement distribution. Pure entangled states, which are the desired input to the swapping protocol, are typically mixed by environmental interactions, causing a reduction in their degree of entanglement. Thus an understanding of entanglement swapping with partially mixed states is of importance. Here we present a general analytical solution for entanglement swapping of arbitrary two-qubit states. Our result provides a comprehensive method for analyzing entanglement swapping in quantum networks. First, we show that the concurrence of a partially mixed state is conserved when this state is swapped with a Bell state. Then, we find upper and lower bounds on the concurrence of the state resulting from entanglement swapping for various classes of input states. Finally, we determine a general relationship between the ranks of the initial states and the rank of the final state after swapping.
We study quantum entanglement distribution on networks with full-rank bi-partite mixed states linking qubits on nodes. In particular, we use entanglement swapping and purification to partially entangle widely separated nodes. The simplest method consists of performing entanglement swappings along the shortest chain of links connecting the two nodes. However, we show that this method may be improved upon by choosing a protocol with a specific ordering of swappings and purifications. A priori, the design that produces optimal improvement is not clear. However, we parameterize the choices and find that the optimal values depend strongly on the desired measure of improvement. As an initial application, we apply the new improved protocols to the Erd\"os--R\'enyi network and obtain results including low density limits and an exact calculation of the average entanglement gained at the critical point.
We study quantum entanglement distribution on networks with full-rank bi-partite mixed states linking qubits on nodes. In particular, we use entanglement swapping and purification to partially entangle widely separated nodes. The simplest method consists of performing entanglement swappings along the shortest chain of links connecting the two nodes. However, we show that this method may be improved upon by choosing a protocol with a specific ordering of swappings and purifications. A priori, the design that produces optimal improvement is not clear. However, we parametrize the choices and find that the optimal values depend strongly on the desired measure of improvement. As an initial application, we apply the new improved protocols to the Erdös--Rényi network and obtain results including low density limits and an exact calculation of the average entanglement gained at the critical point.
Entanglement swapping is gaining widespread attention due to its application in entanglement distribution among different parts of quantum appliances. We investigate the entanglement swapping for pure and noisy systems, and argue different entanglement quantifiers for quantum states. We explore the relationship between the entanglement of initial states and the average entanglement of final states in terms of concurrence and negativity. We find that if initial quantum states are maximally entangled and we make measurements in the Bell basis, then average concurrence and average negativity of final states give similar results. In this case, we simply obtain the average concurrence (average negativity) of the final states by taking the product of concurrences (negativities) of the initial states. However, the measurement in non-maximally entangled basis during entanglement swapping degrades the average swapped entanglement. Further, the product of the entanglement of the initial mixed states provides an upper bound to the average swapped entanglement of final states obtained after entanglement swapping. The negativity work well for weak entangled noisy states but concurrence gives better results for relatively strong entanglement regimes. We also discuss how successfully the output state can be used as a channel for the teleportation of an unknown qubit.