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arXiv:2506.22245 · Physical Review A

End-to-end entanglement of quantum network paths with multiparameter states

Md Sohel Mondal, Aniket Zambare, Siddhartha Santra

Published Nov 7, 2025• 0 citations
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Abstract

Long-range entanglement distribution in a quantum network relies on entanglement swapping at intermediate nodes along a network path to connect short-range entangled states established over the network edges. The end-to-end entanglement of a network path obtained via this process determines the utility of the network path for executing entanglement enabled tasks and for the design of entanglement routing protocols in the quantum network. Here we study the end-to-end entanglement of paths in a quantum network when the edges are characterized by multiparameter quantum states that may be considered to be the output of arbitrary and unknown quantum channels described by the network's edges. We find that over ensembles of multiparameter states with fixed concurrence but varying density matrix elements, the end-to-end entanglement takes a range of values upper bounded by a function of the concurrence of the network-edge states. The scaling behavior of the average end-to-end entanglement reveals that its distribution gets increasingly concentrated around the mean as the paths become longer. For a network path of a given length, the average end-to-end entanglement vanishes for edge concurrence values below a threshold that increases with the path length. Although, for edge concurrence values greater than the threshold the average end-to-end entanglement increases faster with the length of the path. As an implication of our results, we show that the optimal path for entanglement distribution between a pair of end nodes, connected by alternate paths with multiparameter states along the edges, can be indeterminate given only entanglement guarantees along the network edges.

DOI
10.1103/vvw7-chh8

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Long-range entanglement distribution in a quantum network relies on entanglement swapping at intermediate nodes along a network path to connect short-range entangled states established over the network edges. The end-to-end entanglement of a network path obtained via this process determines the utility of the network path for executing entanglement enabled tasks and for the design of entanglement routing protocols in the quantum network. Here we study the end-to-end entanglement of paths in a quantum network when the edges are characterized by multiparameter quantum states that may be considered to be the output of arbitrary and unknown quantum channels described by the network's edges. We find that over ensembles of multiparameter states with fixed concurrence but varying density matrix elements, the end-to-end entanglement takes a range of values upper bounded by a function of the concurrence of the network-edge states. The scaling behavior of the average end-to-end entanglement reveals that its distribution gets increasingly concentrated around the mean as the paths become longer. For a network path of a given length, the average end-to-end entanglement vanishes for edge concurrence values below a threshold that increases with the path length. Although, for edge concurrence values greater than the threshold the average end-to-end entanglement increases faster with the length of the path. As an implication of our results, we show that the optimal path for entanglement distribution between a pair of end nodes, connected by alternate paths with multiparameter states along the edges, can be indeterminate given only entanglement guarantees along the network edges.
Large-scale quantum networks, necessary for distributed quantum information processing, are posited to have quantum entangled systems between distant network nodes. The extent and quality of distributed entanglement in a quantum network, that is its functionality, depends on its topology, edge-parameter distributions and the distribution protocol. We uncover the parametric entanglement topography and introduce the notion of typical and maximal viable regions for entanglement-enabled tasks in a general model of large-scale quantum networks. We show that such a topographical analysis, in terms of viability regions, reveals important functional information about quantum networks, provides experimental targets for the edge parameters and can guide efficient quantum network design. Applied to a photonic quantum network, such a topographical analysis shows that in a network with radius 10310^3103 kms and 1500 nodes, arbitrary pairs of nodes can establish quantum secure keys at a rate of Rsec=1R_{sec}=1Rsec​=1 kHz using 111 MHz entanglement generation sources on the edges and as few as 3 entanglement swappings at intermediate nodes along network paths.
Entanglement routing in near-term quantum networks consists of choosing the optimal sequence of short-range entanglements to combine through swapping operations to establish end-to-end entanglement between two distant nodes. Similar to traditional routing technologies, a quantum routing protocol uses network information to choose the best paths to satisfy a set of end-to-end entanglement requests. However, in addition to network state information, a quantum routing protocol must also take into account the requested entanglement fidelity, the probabilistic nature of swapping operations, and the short lifetime of entangled states. In this work, we formulate a practical entanglement routing problem and analyze and categorize the main approaches to address it, drawing comparisons to, and inspiration from, classical network routing strategies where applicable. We classify and discuss the studied quantum routing schemes into reactive, proactive, and hybrid routing.
In quantum networks, a set of entangled states distributed over multiple, alternative, distinct paths between a pair of source-destination nodes can be purified to obtain a higher fidelity entangled state between the nodes. This multipath entanglement purification (MP-EP) strategy can exploit the network's complex structure to strengthen the entanglement connection between node pairs separated by appropriate graph distances. We investigate the network scenarios in which MP-EP outperforms entanglement distribution over single network paths utilizing a statistical model of a quantum network and find that MP-EP can be an effective entanglement distribution strategy over a range of node separations determined by the average edge fidelities and probabilities of the network. We find that MP-EP can boost the entanglement connection between suitably separated node pairs to reach fidelities sufficient for a given quantum task thereby increasing the functionality of a quantum network. Further, we provide statistical criteria for the feasibility of MP-EP in terms of network parameters that can determine the regions of the network where MP-EP can be a useful entanglement distribution strategy.
We consider the problem of multi-path entanglement distribution to a pair of nodes in a quantum network consisting of devices with non-deterministic entanglement swapping capabilities. Multi-path entanglement distribution enables a network to establish end-to-end entangled links across any number of available paths with pre-established elementary (link-level) entanglement. Probabilistic entanglement swapping, on the other hand, limits the amount of entanglement that is shared between the nodes; this is especially the case when, due to architectural and other practical constraints, swaps must be performed in temporal proximity to each other. Limiting our focus to the case where only bipartite entangled states are generated across the network, we cast the problem as an instance of generalized flow maximization between two quantum end nodes wishing to communicate. We propose a mixed-integer quadratically constrained program (MIQCP) to solve this flow problem for networks with arbitrary topology. We then compute the overall network capacity, defined as the maximum number of EPR states distributed to users per time unit, by solving the flow problem for all possible network states generated by probabilistic entangled link presence and absence, and subsequently by averaging over all network state capacities. The MIQCP can also be applied to networks with multiplexed links – ie, links capable of generating multiple EPR pairs per time unit – by converting the original network state's graph into an equivalent one with unit link capacities. While our approach for computing the overall network capacity has the undesirable property that the total number of states grows exponentially with link multiplexing capability, it nevertheless yields an exact solution that serves as an upper bound comparison basis for the throughput performance of more easily-implementable yet non-optimal entanglement routing algorithms. We apply our capacity computation method to several multiplexed and non-multiplexed networks, including a topology based on SURFnet – a backbone network used for research purposes in the Netherlands – for which real fiber length data is available.
Quantum repeater networks distribute entanglement over lossy links while many users share a limited pool of entangled pairs. Most existing routing schemes either always use a single best path or rely on global optimizations that are hard to run in real time. Here we propose and analyze a simple alternative: a stochastic multipath rule in which each entanglement request is sent at random along one of several edge-disjoint repeater paths, with a single parameter that controls the bias between shorter and longer routes. Using a distance-dependent lossy network model with finite per-link capacities and probabilistic entanglement swapping, we develop an analytic description of the resulting end-to-end entanglement rate as a function of this bias and validate it with large-scale numerical simulations. We find that an intermediate bias consistently outperforms both deterministic extremes across distances, traffic patterns, attenuation, swapping noise, and congestion, bringing the rate close to simple capacity upper bounds and making link usage more even across networks. These results identify stochastic multipath routing as a lightweight classical control strategy for boosting performance and scalability in near-term quantum repeater networks.
How to efficiently distribute entanglement over large-scale quantum networks is still an open problem that greatly depends on the technology considered. In this work, we consider quantum networks where each link is characterized by a trade-off between the entanglement generation rate and fidelity. For such networks, we look at the two following problems: the one of finding the best path to connect any two given nodes, and the problem of finding the best starting node in order to connect three nodes in the network multipartite entanglement routing. Two entanglement distribution models are considered: one where entangled qubits are distributed one at a time, and a flow model where a large number of entangled qubits are distributed simultaneously. The paper proposes of a quite general methodology that uses continuous fidelity curves (i.e., entanglement generation fidelity vs. rate) as the main routing metric. Combined with multi-objective path-finding algorithms, the fidelity curves describing each link allow finding a set of paths that maximize both the end-to-end fidelity and the entanglement generation rate. For the link models and networks considered, it is proven that the algorithm always converges to the optimal solution. It is also shown through simulation that the execution time grows polynomially with the number of network nodes (growing with a power between 111 and 1.41.41.4, depending on the network)