Distributing Multipartite Entanglement over Noisy Quantum Networks

Luís Bugalho1,2,3, Bruno C. Coutinho4, Francisco A. Monteiro4,5, and Yasser Omar1,2,3

1Instituto Superior Técnico, Universidade de Lisboa, Portugal
2Physics of Information and Quantum Technologies Group, Centro de Física e Engenharia de Materiais Avançados (CeFEMA), Portugal
3PQI – Portuguese Quantum Institute, Portugal
4Instituto de Telecomunicações, Portugal
5ISCTE - Instituto Universitário de Lisboa, Portugal

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A quantum internet aims at harnessing networked quantum technologies, namely by distributing bipartite entanglement between distant nodes. However, multipartite entanglement between the nodes may empower the quantum internet for additional or better applications for communications, sensing, and computation. In this work, we present an algorithm for generating multipartite entanglement between different nodes of a quantum network with noisy quantum repeaters and imperfect quantum memories, where the links are entangled pairs. Our algorithm is optimal for GHZ states with 3 qubits, maximising simultaneously the final state fidelity and the rate of entanglement distribution. Furthermore, we determine the conditions yielding this simultaneous optimality for GHZ states with a higher number of qubits, and for other types of multipartite entanglement. Our algorithm is general also in the sense that it can optimise simultaneously arbitrary parameters. This work opens the way to optimally generate multipartite quantum correlations over noisy quantum networks, an important resource for distributed quantum technologies.

Quantum technologies hold the promise of faster computing, securer private communications, and more precise sensing and metrology. In particular, quantum networks open the possibility to explore these applications in distributed scenarios, allowing for increased performance and/or tasks involving multiple parties. However, to realize some applications between multiple parties multipartite entanglement is often required.
In this work we aim at finding the optimal way to distribute multipartite entanglement between different nodes of a quantum network with noisy quantum repeaters and imperfect quantum memories, where the links are entangled pairs. This has particular relevance for applications where noise and the distribution of the state impacts the application itself. To that end, we introduce a new methodology that allows to maximise two different objectives – the rate of distribution and the fidelity of the distributed state – even though our approach is easily generalizable to include more. We develop an algorithm with tools from classical routing theory that finds the optimal way of distributing a 3-qubit GHZ state, in a way that is adaptable to different underlying physical implementations and protocols of distribution. We also provide results both for higher number of qubits and another class of multipartite entangled states, namely W-states.

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[2] Kiara Hansenne, Zhen-Peng Xu, Tristan Kraft, and Otfried Gühne, "Symmetries in quantum networks lead to no-go theorems for entanglement distribution and to verification techniques", Nature Communications 13, 496 (2022).

[3] Jian Li, Mingjun Wang, Qidong Jia, Kaiping Xue, Nenghai Yu, Qibin Sun, and Jun Lu, "Fidelity-Guarantee Entanglement Routing in Quantum Networks", arXiv:2111.07764, (2021).

[4] Guus Avis, Filip Rozpedek, and Stephanie Wehner, "Analysis of multipartite entanglement distribution using a central quantum-network node", Physical Review A 107 1, 012609 (2023).

[5] Seid Koudia, "The Quantum Internet: an Efficient Stabilizer states Distribution Scheme", arXiv:2305.02656, (2023).

[6] Stav Haldar, Pratik J. Barge, Sumeet Khatri, and Hwang Lee, "Fast and reliable entanglement distribution with quantum repeaters: principles for improving protocols using reinforcement learning", arXiv:2303.00777, (2023).

[7] Paolo Fittipaldi, Anastasios Giovanidis, and Frédéric Grosshans, "A Linear Algebraic Framework for Quantum Internet Dynamic Scheduling", arXiv:2205.10000, (2022).

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