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

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


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.

► BibTeX data

► References

[1] Charles H. Bennett and Gilles Brassard. Quantum cryptography: Public key distribution and coin tossing. Theoretical Computer Science, 560 (P1): 7–11, 2014. ISSN 03043975. 10.1016/​j.tcs.2014.05.025.

[2] Ali Ibnun Nurhadi and Nana Rachmana Syambas. Quantum Key Distribution (QKD) Protocols: A Survey. Proceeding of 2018 4th International Conference on Wireless and Telematics, ICWT 2018, pages 18–22, 2018. 10.1109/​ICWT.2018.8527822.

[3] Anne Broadbent, Joseph Fitzsimons, and Elham Kashefi. Universal blind quantum computation. Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 517–526, 2009. ISSN 02725428. 10.1109/​FOCS.2009.36.

[4] Isaac Chuang. Quantum algorithm for distributed clock synchronization. Physical Review Letters, 85 (9): 2006–2009, May 2000. ISSN 10797114. 10.1103/​PhysRevLett.85.2006.

[5] Daniel Gottesman, Thomas Jennewein, and Sarah Croke. Longer-Baseline Telescopes Using Quantum Repeaters. Physical Review Letters, 109 (7): 070503, July 2011. ISSN 0031-9007. 10.1103/​PhysRevLett.109.070503.

[6] Stephanie Wehner, David Elkouss, and Ronald Hanson. Quantum internet: A vision for the road ahead. Science, 362 (6412): eaam9288, October 2018. ISSN 10959203. 10.1126/​science.aam9288.

[7] Matteo Pompili, Sophie L. N. Hermans, Simon Baier, Hans K. C. Beukers, Peter C. Humphreys, Raymond N. Schouten, Raymond F. L. Vermeulen, Marijn J. Tiggelman, L. dos Santos Martins, Bas Dirkse, Stephanie Wehner, and Ronald Hanson. Realization of a multinode quantum network of remote solid-state qubits. Science, 372 (6539): 259–264, April 2021. ISSN 0036-8075. 10.1126/​science.abg1919.

[8] Muneer Alshowkan, Brian P. Williams, Philip G. Evans, Nageswara S.V. Rao, Emma M. Simmerman, Hsuan-Hao Lu, Navin B. Lingaraju, Andrew M. Weiner, Claire E. Marvinney, Yun-Yi Pai, Benjamin J. Lawrie, Nicholas A. Peters, and Joseph M. Lukens. Reconfigurable Quantum Local Area Network Over Deployed Fiber. PRX Quantum, 2 (4): 040304, October 2021. 10.1103/​PRXQuantum.2.040304.

[9] William J. Munro, Koji Azuma, Kiyoshi Tamaki, and Kae Nemoto. Inside Quantum Repeaters. IEEE Journal of Selected Topics in Quantum Electronics, 21 (3): 78–90, May 2015. ISSN 1077-260X. 10.1109/​JSTQE.2015.2392076.

[10] Marcello Caleffi. Optimal Routing for Quantum Networks. IEEE Access, 5: 22299–22312, 2017. ISSN 21693536. 10.1109/​ACCESS.2017.2763325.

[11] Kaushik Chakraborty, Filip Rozpedek, Axel Dahlberg, and Stephanie Wehner. Distributed Routing in a Quantum Internet, July 2019, arXiv:1907.11630. 10.48550/​arXiv.1907.11630.

[12] Shouqian Shi and Chen Qian. Modeling and Designing Routing Protocols in Quantum Networks, October 2019, arXiv:1909.09329. 10.48550/​arXiv.1909.09329.

[13] Changhao Li, Tianyi Li, Yi-Xiang Xiang Liu, and Paola Cappellaro. Effective routing design for remote entanglement generation on quantum networks. npj Quantum Information, 7 (1): 10, December 2021. ISSN 20566387. 10.1038/​s41534-020-00344-4.

[14] Wenhan Dai, Tianyi Peng, and Moe Z. Win. Optimal Remote Entanglement Distribution. IEEE Journal on Selected Areas in Communications, 38 (3): 540–556, March 2020. ISSN 0733-8716. 10.1109/​JSAC.2020.2969005.

[15] Stefan Bäuml, Koji Azuma, Go Kato, and David Elkouss. Linear programs for entanglement and key distribution in the quantum internet. Communications Physics, 3 (1): 1–12, 2020. ISSN 23993650. 10.1038/​s42005-020-0318-2.

[16] Sara Santos, Francisco A. Monteiro, Bruno C. Coutinho, and Yasser Omar. Shortest path finding in quantum networks with quasi-linear complexity. IEEE Access, 11: 7180–7194, 2023. 10.1109/​ACCESS.2023.3237997.

[17] Changliang Ren and Holger F. Hofmann. Clock synchronization using maximal multipartite entanglement. Physical Review A, 86 (1): 014301, July 2012. ISSN 1050-2947. 10.1103/​PhysRevA.86.014301.

[18] E. T. Khabiboulline, J. Borregaard, K. De Greve, and M. D. Lukin. Quantum-assisted telescope arrays. Physical Review A, 100 (2): 022316, August 2019. ISSN 24699934. 10.1103/​PhysRevA.100.022316.

[19] Zachary Eldredge, Michael Foss-Feig, Jonathan A. Gross, Steven L. Rolston, and Alexey V. Gorshkov. Optimal and secure measurement protocols for quantum sensor networks. Physical Review A, 97 (4): 042337, April 2018. ISSN 2469-9926. 10.1103/​PhysRevA.97.042337.

[20] Timothy Qian, Jacob Bringewatt, Igor Boettcher, Przemyslaw Bienias, and Alexey V. Gorshkov. Optimal measurement of field properties with quantum sensor networks. Physical Review A, 103 (3): L030601, March 2021. ISSN 2469-9926. 10.1103/​PhysRevA.103.L030601.

[21] Mark Hillery, Vladimír Bužek, and André Berthiaume. Quantum secret sharing. Physical Review A - Atomic, Molecular, and Optical Physics, 59 (3): 1829–1834, 1999. ISSN 10502947. 10.1103/​PhysRevA.59.1829.

[22] Changhua Zhu, Feihu Xu, and Changxing Pei. W-state Analyzer and Multi-party Measurement-device-independent Quantum Key Distribution. Scientific Reports, 5 (1): 17449, December 2015. ISSN 2045-2322. 10.1038/​srep17449.

[23] Gláucia Murta, Federico Grasselli, Hermann Kampermann, and Dagmar Bruß. Quantum Conference Key Agreement: A Review. Advanced Quantum Technologies, 3 (11): 2000025, November 2020. ISSN 2511-9044. 10.1002/​qute.202000025.

[24] Ellie D'Hondt and Prakash Panangaden. The Computational Power of the W and GHZ states Quantum Info. Comput., 6 (2): 173–183, Mar 2006. ISSN 1533-7146. arXiv:quant-ph/​0412177. DOI: 10.48550/​arXiv.quant-ph/​0412177.

[25] Robert Raussendorf and Hans J Briegel. A One-Way Quantum Computer. Physical Review Letters, 86 (22): 5188–5191, May 2001. ISSN 0031-9007. 10.1103/​PhysRevLett.86.5188.

[26] Riccardo Laurenza and Stefano Pirandola. General bounds for sender-receiver capacities in multipoint quantum communications. Physical Review A, 96 (3): 032318, September 2017. ISSN 2469-9926. 10.1103/​PhysRevA.96.032318.

[27] Stefano Pirandola. End-to-end capacities of a quantum communication network. Communications Physics, 2 (1): 51, December 2019a. ISSN 2399-3650. 10.1038/​s42005-019-0147-3.

[28] Stefano Pirandola. Bounds for multi-end communication over quantum networks. Quantum Science and Technology, 4 (4): 045006, September 2019b. ISSN 2058-9565. 10.1088/​2058-9565/​ab3f66.

[29] Stefano Pirandola. General upper bound for conferencing keys in arbitrary quantum networks. IET Quantum Communication, 1 (1): 22–25, July 2020. ISSN 2632-8925. 10.1049/​iet-qtc.2020.0006.

[30] Siddhartha Das, Stefan Bäuml, Marek Winczewski, and Karol Horodecki. Universal Limitations on Quantum Key Distribution over a Network. Physical Review X, 11 (4): 041016, October 2021. ISSN 2160-3308. 10.1103/​PhysRevX.11.041016.

[31] Clément Meignant, Damian Markham, and Frédéric Grosshans. Distributing graph states over arbitrary quantum networks. Physical Review A, 100 (5): 052333, November 2019. ISSN 24699934. 10.1103/​PhysRevA.100.052333.

[32] J. Wallnöfer, A. Pirker, M. Zwerger, and W. Dür. Multipartite state generation in quantum networks with optimal scaling. Scientific Reports, 9 (1): 314, December 2019. ISSN 2045-2322. 10.1038/​s41598-018-36543-5.

[33] Kenneth Goodenough, David Elkouss, and Stephanie Wehner. Optimizing repeater schemes for the quantum internet. Physical Review A, 103 (3): 032610, March 2021. ISSN 2469-9926. 10.1103/​PhysRevA.103.032610.

[34] Sergey N. Filippov, Alexey A. Melnikov, and Mário Ziman. Dissociation and annihilation of multipartite entanglement structure in dissipative quantum dynamics. Physical Review A, 88 (6): 062328, December 2013. ISSN 1050-2947. 10.1103/​PhysRevA.88.062328.

[35] J.L. Sobrinho. An algebraic theory of dynamic network routing. IEEE/​ACM Transactions on Networking, 13 (5): 1160–1173, October 2005. ISSN 1063-6692. 10.1109/​TNET.2005.857111.

[36] Sofie Demeyer, Jan Goedgebeur, Pieter Audenaert, Mario Pickavet, and Piet Demeester. Speeding up Martins' algorithm for multiple objective shortest path problems. 4or, 11 (4): 323–348, 2013. ISSN 16142411. 10.1007/​s10288-013-0232-5.

[37] Sebastiaan Brand, Tim Coopmans, and David Elkouss. Efficient Computation of the Waiting Time and Fidelity in Quantum Repeater Chains. IEEE Journal on Selected Areas in Communications, 38 (3): 619–639, March 2020. ISSN 0733-8716. 10.1109/​JSAC.2020.2969037.

[38] Reinhard F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Physical Review A, 40 (8): 4277–4281, 1989. ISSN 10502947. 10.1103/​PhysRevA.40.4277.

[39] M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest, and H. J. Briegel. Entanglement in Graph States and its Applications. Proceedings of the International School of Physics "Enrico Fermi", 162: 115–218, February 2006. ISSN 0074784X. 10.3254/​978-1-61499-018-5-115.

[40] W. Dür and H. J. Briegel. Entanglement purification and quantum error correction. Reports on Progress in Physics, 70 (8): 1381–1424, 2007. ISSN 00344885. 10.1088/​0034-4885/​70/​8/​R03.

[41] You neng Guo, Qing long Tian, Ke Zeng, and Zheng da Li. Quantum coherence of two-qubit over quantum channels with memory. Quantum Information Processing, 16 (12): 1–18, 2017. ISSN 15700755. 10.1007/​s11128-017-1749-x.

[42] Lars Kamin, Evgeny Shchukin, Frank Schmidt, and Peter van Loock. Exact rate analysis for quantum repeaters with imperfect memories and entanglement swapping as soon as possible, March 2022, arXiv:2203.10318. 10.48550/​arXiv.2203.10318.

[43] Ernesto Queirós Vieira Martins. On a multicriteria shortest path problem. European Journal of Operational Research, 16 (2): 236–245, 1984. ISSN 03772217. 10.1016/​0377-2217(84)90077-8.

[44] João Luís Sobrinho. Network Routing with Path Vector Protocols: Theory and Applications. Computer Communication Review, 33 (4): 49–60, 2003. ISSN 01464833. 10.1145/​863955.863963.

[45] Albert-László Barabási and Márton Pósfai. Network Science. Cambridge University Press, Cambridge, 2016. ISBN 978-1-107-07626-6 1-107-07626-9.

[46] S. N. Dorogovtsev, A. V. Goltsev, and J. F.F. Mendes. Critical phenomena in complex networks. Reviews of Modern Physics, 80 (4): 1275–1335, 2008. ISSN 00346861. 10.1103/​RevModPhys.80.1275.

[47] Robert B. Ellis, Jeremy L. Martin, and Catherine Yan. Random geometric graph diameter in the unit ball. Algorithmica (New York), 47 (4): 421–438, 2007. ISSN 01784617. 10.1007/​s00453-006-0172-y.

[48] Jesper Dall and Michael Christensen. Random geometric graphs. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 66 (1), 2002. ISSN 1063651X. 10.1103/​PhysRevE.66.016121.

[49] Takahiro Inagaki, Nobuyuki Matsuda, Osamu Tadanaga, Masaki Asobe, and Hiroki Takesue. Entanglement distribution over 300 km of fiber. Optics Express, 21 (20): 23241, 2013. ISSN 1094-4087. 10.1364/​oe.21.023241.

[50] Bruno Coelho Coutinho, William John Munro, Kae Nemoto, and Yasser Omar. Robustness of noisy quantum networks. Communications Physics, 5 (1): 1–9, April 2022. ISSN 2399-3650. 10.1038/​s42005-022-00866-7.

[51] Guus Avis, Filip Rozpędek, and Stephanie Wehner. Analysis of Multipartite Entanglement Distribution using a Central Quantum-Network Node, March 2022, arXiv:2203.05517. 10.48550/​arXiv.2203.05517.

[52] J. Wallnöfer, M. Zwerger, C. Muschik, N. Sangouard, and W. Dür. Two-dimensional quantum repeaters. Physical Review A, 94 (5): 1–12, 2016. ISSN 24699934. 10.1103/​PhysRevA.94.052307.

[53] Takahiko Satoh, Kaori Ishizaki, Shota Nagayama, and Rodney Van Meter. Analysis of quantum network coding for realistic repeater networks. Physical Review A, 93 (3): 1–10, 2016. ISSN 24699934. 10.1103/​PhysRevA.93.032302.

[54] Pavel Sekatski, Sabine Wölk, and Wolfgang Dür. Optimal distributed sensing in noisy environments. Physical Review Research, 2 (2): 1–8, May 2019. 10.1103/​PhysRevResearch.2.023052.

[55] Nathan Shettell, William J. Munro, Damian Markham, and Kae Nemoto. Practical limits of error correction for quantum metrology. New Journal of Physics, 23 (4): 043038, April 2021. ISSN 1367-2630. 10.1088/​1367-2630/​abf533.

[56] X. Wang. Exact Algorithms for Steiner Tree Problem. 2008. ISBN 978-90-365-2660-9. 10.3990/​1.9789036526609.

[57] Gabriel Robins and Alexander Zelikovsky. Tighter Bounds for Graph Steiner Tree Approximation. SIAM Journal on Discrete Mathematics, 19 (1): 122–134, January 2005. ISSN 0895-4801. 10.1137/​S0895480101393155.

[58] W. Dür, G Vidal, and J I Cirac. Three qubits can be entangled in two inequivalent ways. Physical Review A, 62 (6): 062314, November 2000. ISSN 1050-2947. 10.1103/​PhysRevA.62.062314.

Cited by

[1] Michelle Victora, Spyros Tserkis, Stefan Krastanov, Alexander Sanchez de la Cerda, Steven Willis, and Prineha Narang, "Entanglement purification on quantum networks", Physical Review Research 5 3, 033171 (2023).

[2] Paolo Fittipaldi, Anastasios Giovanidis, and Frédéric Grosshans, "A Linear Algebraic Framework for Dynamic Scheduling Over Memory-Equipped Quantum Networks", IEEE Transactions on Quantum Engineering 5, 1 (2024).

[3] 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", Physical Review Applied 21 2, 024041 (2024).

[4] Nitish K. Panigrahy, Matheus Guedes De Andrade, Shahrooz Pouryousef, Don Towsley, and Leandros Tassiulas, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 391 (2023) ISBN:979-8-3503-4323-6.

[5] Evan Sutcliffe and Alejandra Beghelli, "Multiuser Entanglement Distribution in Quantum Networks Using Multipath Routing", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[6] Jing-Chen Hao, Pei-Lin Du, Heng-Xin Sun, Kui Liu, Jing Zhang, Rong-Guo Yang, and Jiang-Rui Gao, "Quadripartite entanglement from two-port resonator with second-order harmonic generation", Acta Physica Sinica 73 7, 074203 (2024).

[7] Mohammad Ghaderibaneh, Caitao Zhan, Himanshu Gupta, and C. R. Ramakrishnan, "Efficient Quantum Network Communication Using Optimized Entanglement Swapping Trees", IEEE Transactions on Quantum Engineering 3, 1 (2022).

[8] Diogo Cruz, Francisco A. Monteiro, and Bruno C. Coutinho, "Quantum Error Correction Via Noise Guessing Decoding", IEEE Access 11, 119446 (2023).

[9] Áron Rozgonyi, Gábor Széchenyi, Orsolya Kálmán, and Tamás Kiss, "Training iterated protocols for distillation of GHZ states with variational quantum algorithms", Physics Letters A 499, 129349 (2024).

[10] Álvaro G. Iñesta and Stephanie Wehner, "Performance metrics for the continuous distribution of entanglement in multiuser quantum networks", Physical Review A 108 5, 052615 (2023).

[11] Mohammad Ghaderibaneh, Himanshu Gupta, and C.R. Ramakrishnan, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 1120 (2023) ISBN:979-8-3503-4323-6.

[12] Álvaro G. Iñesta, Gayane Vardoyan, Lara Scavuzzo, and Stephanie Wehner, "Optimal entanglement distribution policies in homogeneous repeater chains with cutoffs", npj Quantum Information 9 1, 46 (2023).

[13] Marzieh Bathaee and Jawad A. Salehi, "Entangled-Based Quantum Wavelength-Division-Multiplexing and Multiple-Access Networks", Entropy 25 12, 1658 (2023).

[14] Rute Oliveira, Raabe Oliveira, Nadja K. Bernardes, and Rafael Chaves, "Statistical properties and repetition rates for a quantum network with geographical distribution of nodes", Physics Letters A 506, 129458 (2024).

[15] 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).

[16] 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).

[17] 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).

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

[19] Mohammad Ghaderibaneh, Caitao Zhan, Himanshu Gupta, and C. R. Ramakrishnan, "Efficient Quantum Network Communication using Optimized Entanglement-Swapping Trees", arXiv:2112.11002, (2021).

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

The above citations are from Crossref's cited-by service (last updated successfully 2024-04-15 12:42:05) and SAO/NASA ADS (last updated successfully 2024-04-15 12:42:06). The list may be incomplete as not all publishers provide suitable and complete citation data.