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arXiv:2508.00813 · Advanced Quantum Technologies

Entanglement Swapping for Partially Entangled Qudits and the Role of Quantum Complementarity

Diego S. Starke, Marcos L. W. Basso, Lucas C. Céleri, Jonas Maziero

Published Oct 22, 2025• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1002/qute.202500674

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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.
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.
The nonlocal correlations between quantum states in an entangled system are essential to many quantum communications applications. A basic quantum operation, which permits the distribution of entanglement between two initially uncorrelated systems, is entanglement swapping. Here we present a rigorous formulation of entanglement swapping of any two partially mixed two-qubit states without limiting ourselves to any specific type of state or noise. Further, for two important classes of the input states, Bell diagonal and pure states, we describe how the concurrence of the final state is related to the concurrence of the initial states. First, we consider Bell diagonal states, and find bounds on the concurrence of the final state in terms of the concurrences of the initial states. These bounds are important for communications applications because polarization mode dispersion in fibers produces Bell diagonal states up to a local unitary rotation. Second, we show that swapping pure states occasionally results in a state of higher concurrence than either of the initial states. In addition, we find that two pure states are most likely to be capable of swapping to a state of increased concurrence when the two initial states have similar concurrences. Our analysis offers a completely general framework for investigating the behavior of any pair of two-qubit states when used for entanglement swapping.
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.
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.
We study the role of average concurrence in entanglement swapping in quantum networks. We begin with qubit pure states, and there is a very simple rule governing the propagation of average concurrence in multiple swaps. We find a similarly simple rule for average concurrence when creating a Greenberger-Horne-Zeilinger state from three entangled pairs. We look at examples of mixed qubit states and find that the relation for pure states gives an upper bound on what is possible with mixed states. We then move on to qudits, where we make use of the I-concurrence. Here the situation is not as simple as for qubits, but in some cases relatively straightforward results can be obtained.
The basic entanglement-swapping scheme can be seen as a process which allows one to redistribute the Bell states properties between different pairs of a four-qubit system. We analyze a similar scheme by performing a Bell--von Neumann measurement over two local qubits, each one initially correlated through an XXX state with a spatially distant qubit. This process swaps the XXX feature without conditions, whereas the input entanglement is partially distributed in the four possible outcome states under certain conditions. Specifically, we obtain two threshold values for the entanglement of formation of the input XXX states in order for the outcome states to be nonseparable.