Distributing Multipartite Entanglement over Noisy Quantum Networks

Luís Bugalho; Bruno C. Coutinho; Francisco A. Monteiro; Yasser Omar

doi:10.22331/q-2023-02-09-920

Distributing Multipartite Entanglement over Noisy Quantum Networks

Luís Bugalho^1,2,3, Bruno C. Coutinho⁴, Francisco A. Monteiro^4,5, and Yasser Omar^1,2,3

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

Published:	2023-02-09, volume 7, page 920
Eprint:	arXiv:2103.14759v3
Doi:	https://doi.org/10.22331/q-2023-02-09-920
Citation:	Quantum 7, 920 (2023).

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

Abstract

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.

Featured image: Model of a quantum network made up of quantum channels capable of distributing pairwise entanglements, and quantum nodes with quantum memories capable of executing operations over qubits. Each of the constituents of the quantum network is characterized by a set of parameters that depend on the physical implementation of the qubits and the quantum network protocols.

Popular summary

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

@article{Bugalho2023distributing,
  doi = {10.22331/q-2023-02-09-920},
  url = {https://doi.org/10.22331/q-2023-02-09-920},
  title = {Distributing {M}ultipartite {E}ntanglement over {N}oisy {Q}uantum {N}etworks},
  author = {Bugalho, Lu{\'{i}}s and Coutinho, Bruno C. and Monteiro, Francisco A. and Omar, Yasser},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {920},
  month = feb,
  year = {2023}
}

► 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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/10.48550/arXiv.1907.11630
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.
https://doi.org/10.48550/arXiv.1909.09329
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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/10.48550/arXiv.quant-ph/0412177
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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/10.48550/arXiv.2203.10318
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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/10.48550/arXiv.2203.05517
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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/10.1103/PhysRevA.62.062314

Cited by

[1] Yi-Jun Luo, Xuan Leng, and Chengjie Zhang, "Genuine multipartite entanglement verification with convolutional neural networks", Physical Review A 110 4, 042412 (2024).

[2] 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).

[3] Johannes Nokkala, Jyrki Piilo, and Ginestra Bianconi, "Complex quantum networks: a topical review", Journal of Physics A: Mathematical and Theoretical 57 23, 233001 (2024).

[4] 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).

[5] A. Gore and A. Beghelli, Optical Fiber Communication Conference (OFC) 2025 Tu2D.6 (2025).

[6] André Roque, Diogo Cruz, Francisco A. Monteiro, and Bruno C. Coutinho, "Efficient entanglement purification based on noise guessing decoding", Quantum 8, 1476 (2024).

[7] 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.

[8] Logan Patrick, Umar Arshad, Dingyu Guo, and Imran Mirza, "Chirality-assisted enhancement of tripartite entanglement in waveguide QED", Scientific Reports 14 1, 11175 (2024).

[9] Yuan Lee, Wenhan Dai, Don Towsley, and Dirk Englund, "Quantum network utility: A framework for benchmarking quantum networks", Proceedings of the National Academy of Sciences 121 17, e2314103121 (2024).

[10] Maxwell Tang, Garrett Hinkley, Kenneth Goodenough, Stefan Krastanov, and Guus Avis, "Routing in Nonisotonic Quantum Networks", IEEE Transactions on Quantum Engineering 7, 4101522 (2026).

[11] Evan Sutcliffe and Alejandra Beghelli, "Fidelity-Aware Multipath Routing for Multipartite State Distribution in Quantum Networks", IEEE Transactions on Quantum Engineering 6, 1 (2025).

[12] Saheli Mukherjee, Bivas Mallick, Sahil Gopalkrishna Naik, Ananda G. Maity, and A. S. Majumdar, "Detecting genuine multipartite entanglement using moments of positive maps", Physical Review A 112 6, 062428 (2025).

[13] Annu Paul, Jobin Varghese P, Rejina P V, Paulin Paul, Anat Suman Jose, and Jobin Jose, "Research Trends and Collaboration Patterns in Quantum Internet: A Bibliometric Mapping Using Biblioshiny, VOSviewer, and CiteSpace", LatIA 3, 365 (2025).

[14] Lisa T. Weinbrenner, Lina Vandré, Tim Coopmans, and Otfried Gühne, "Aging and reliability of quantum networks", Physical Review A 109 5, 052611 (2024).

[15] Maria Flors Mor-Ruiz and Wolfgang Dür, "Influence of Noise in Entanglement-Based Quantum Networks", IEEE Journal on Selected Areas in Communications 42 7, 1793 (2024).

[16] 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).

[17] Yiming Zeng, Jiarui Zhang, Ji Liu, Zhenhua Liu, and Yuanyuan Yang, "Multi-Entanglement Routing Design Over Quantum Networks Using Greenberger–Horne–Zeilinger Measurements", IEEE Transactions on Networking 34, 783 (2026).

[18] Junghoon Park, Jiook Cha, Samuel Yen-Chi Chen, Shinjae Yoo, and Huan-Hsin Tseng, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 1583 (2024) ISBN:979-8-3315-4137-8.

[19] Zhonghui Li, Jian Li, Kaiping Xue, and Hyundong Shin, "MixED: A Mixed Entanglement Distribution Design for Efficient Quantum Teleportation in Quantum Communication Networks", IEEE Transactions on Communications 74, 9376 (2026).

[20] Yiming Zeng, Jiarui Zhang, Xiaojun Shang, Ji Liu, Zhenhua Liu, and Yuanyuan Yang, 2024 IEEE 44th International Conference on Distributed Computing Systems (ICDCS) 266 (2024) ISBN:979-8-3503-8605-9.

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

[22] Samuel Oslovich, Md Zakir Hossain, Trevor Thomas, Bing Wang, Walter O. Krawec, and Kenneth Goodenough, 2025 International Conference on Quantum Communications, Networking, and Computing (QCNC) 315 (2025) ISBN:979-8-3315-3159-1.

[23] 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).

[24] Diogo Cruz, Francisco A. Monteiro, André Roque, and Bruno C. Coutinho, "Fault-Tolerant Noise Guessing Decoding of Quantum Random Codes", IEEE Transactions on Quantum Engineering 6, 1 (2025).

[25] Luís Bugalho, Majid Hassani, Yasser Omar, and Damian Markham, "Private and Robust States for Distributed Quantum Sensing", Quantum 9, 1596 (2025).

[26] Mandeep Kumar and Bhaskar Mondal, "A brief review on Quantum Key Distribution Protocols", Multimedia Tools and Applications 84 27, 33267 (2025).

[27] Walter O. Krawec, "Quantum conference key agreement with classical advantage distillation", Discover Networks 1 1, 5 (2025).

[28] Pankaj Kumar, Binayak Kar, and Shan-Hsiang Shen, "Trace-distance based end-to-end entanglement fidelity with information preservation in quantum networks", Journal of Network and Computer Applications 244, 104366 (2025).

[29] 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).

[30] Luise Prielinger, Kenneth Goodenough, Guus Avis, Stefan Krastanov, Don Towsley, and Gayane Vardoyan, "Piecemaker: A Resource-Efficient Entanglement Distribution Protocol", IEEE Journal on Selected Areas in Communications 44, 5341 (2026).

[31] Daniel Bhatti and Kenneth Goodenough, "Distributing graph states with a photon-weaving quantum server", New Journal of Physics 28 7, 074507 (2026).

[32] Jorge Miguel-Ramiro and Wolfgang Dür, "Graph state fission", Physical Review A 111 5, 052624 (2025).

[33] Sajede Harraz, Yueyan Wang, and Shuang Cong, "Perfect entanglement swapping of W states", Communications in Theoretical Physics 77 10, 105102 (2025).

[34] 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).

[35] Si-Yi Chen, Jessica Illiano, Angela Sara Cacciapuoti, and Marcello Caleffi, "Entanglement-Based Artificial Topology: Neighboring Remote Network Nodes", IEEE Open Journal of the Communications Society 6, 2220 (2025).

[36] Raja Yehia, Matteo Schiavon, Valentina Marulanda Acosta, Tim Coopmans, Iordanis Kerenidis, David Elkouss, and Eleni Diamanti, "Connecting quantum cities: simulation of a satellite-based quantum network", New Journal of Physics 26 7, 073015 (2024).

[37] Suman Kumar, Ved Dharkar, Awinash Kumar, and Reshmi Mitra, 2026 International Conference on Computing, Networking and Communications (ICNC) 1040 (2026) ISBN:979-8-3315-7111-5.

[38] Hahn Lam, Evan Sutcliffe, and Alejandra Beghelli, Quantum 2.0 Conference and Exhibition QTh3A.35 (2024) ISBN:978-1-55752-518-5.

[39] Allen Zang, Alexander Kolar, Alvin Gonzales, Joaquin Chung, Stephen K. Gray, Rajkumar Kettimuthu, Tian Zhong, and Zain H. Saleem, "Quantum advantage in distributed sensing with noisy quantum networks", Physical Review Research 8 3, 033040 (2026).

[40] Carlo Marconi, Guillem Müller-Rigat, Jordi Romero-Pallejà, Jordi Tura, and Anna Sanpera, "Symmetric quantum states: a review of recent progress", Reports on Progress in Physics 89 2, 024001 (2026).

[41] 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).

[42] 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.

[43] Markku Hahto, Jyrki Piilo, and Johannes Nokkala, "State Transfer in Noisy Modular Quantum Networks", Advanced Quantum Technologies 8 1, 2400316 (2025).

[44] Yuan Li, Hao Zhang, Chen Zhang, Tao Huang, and F. Richard Yu, "A Survey of Quantum Internet Protocols From a Layered Perspective", IEEE Communications Surveys & Tutorials 26 3, 1606 (2024).

[45] Angela Sara Cacciapuoti, Jessica Illiano, Michele Viscardi, and Marcello Caleffi, "Multipartite Entanglement Distribution in the Quantum Internet: Knowing When to Stop!", IEEE Transactions on Network and Service Management 21 6, 6041 (2024).

[46] Jessica Illiano, Marcello Caleffi, Michele Viscardi, and Angela Sara Cacciapuoti, "Quantum MAC: Genuine Entanglement Access Control via Many-Body Dicke States", IEEE Transactions on Communications 72 4, 2090 (2024).

[47] 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).

[48] Xinan Chen, Stefano Chessa, Ian George, Felix Leditzky, and Eric Chitambar, "Capacities of Entanglement Distribution From a Central Source", IEEE Transactions on Information Theory 72 6, 4001 (2026).

[49] Lisa T. Weinbrenner, Nidhin Prasannan, Kiara Hansenne, Sophia Denker, Jan Sperling, Benjamin Brecht, Christine Silberhorn, and Otfried Gühne, "Certifying the Topology of Quantum Networks: Theory and Experiment", Physical Review Letters 132 24, 240802 (2024).

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

[51] Maria Flors Mor-Ruiz, Julius Wallnöfer, and Wolfgang Dür, "Imperfect quantum networks with tailored resource states", Quantum 9, 1605 (2025).

[52] Si-Yi Chen, Jessica Illiano, Angela Sara Cacciapuoti, and Marcello Caleffi, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 1980 (2024) ISBN:979-8-3315-4137-8.

[53] Xiaojie Fan, Caitao Zhan, Himanshu Gupta, and C. R. Ramakrishnan, "Optimized Distribution of Entanglement Graph States in Quantum Networks", IEEE Transactions on Quantum Engineering 6, 1 (2025).

[54] Aniruddha Sen, Kenneth Goodenough, and Don Towsley, "Multipartite Entanglement Distribution in Quantum Networks using Subgraph Complementations", Quantum 9, 1911 (2025).

[55] Zhonghui Li, Jian Li, Kaiping Xue, Lutong Chen, Nenghai Yu, Qibin Sun, and Jun Lu, "NarrowGap: Reducing Bottlenecks for End-to-End Entanglement Distribution in Quantum Networks", IEEE Transactions on Networking 33 1, 162 (2025).

[56] Pankaj Kumar, Binayak Kar, and Shan-Hsiang Shen, "Trace-Distance Based End-to-End Entanglement Fidelity with Information Preservation in Quantum Networks", (2025).

[57] 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).

[58] Bruno F. E. Matarèse and Arnie Purushotham, "Quantum Oncology", Quantum Reports 7 1, 9 (2025).

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

[60] Julia Freund, Alexander Pirker, Lina Vandré, and Wolfgang Dür, "Graph state extraction from two-dimensional cluster states", New Journal of Physics 27 9, 094505 (2025).

[61] 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).

[62] 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).

[63] Marco Canteri, James Bate, Ida Mishra, Nicolai Friis, Victor Krutyanskiy, and Benjamin P. Lanyon, "Generation of multipartite photonic entanglement using a trapped-ion quantum processing node", arXiv:2510.15693, (2025).

[64] Naphan Benchasattabuse, "Resource Management in Heterogeneous Quantum Repeater Networks", arXiv:2605.25132, (2026).

[65] Daniel Bhatti and Kenneth Goodenough, "Distributing graph states with a photon-weaving quantum server", arXiv:2504.07410, (2025).

[66] 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).

[67] 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 (2023).

[68] Dibakar Das, Shiva Kumar Malapaka, Jyotsna Bapat, and Debabrata Das, "Network Centralities in Quantum Entanglement Distribution due to User Preferences", arXiv:2308.08170, (2023).

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

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

[71] Seid Koudia, "The quantum internet: an efficient stabilizer states distribution scheme", Physica Scripta 99 1, 015115 (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2026-07-17 06:41:02) and SAO/NASA ADS (last updated successfully 2026-07-16 18:26:37). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch ADS cited-by data during last attempt 2026-07-17 06:41:02: Cannot retrieve data from ADS due to rate limitations.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.