quantum-computing

Asymptotic robustness of entanglement in noisy quantum networks and graph connectivity

Fernando Lledó, Carlos Palazuelos, and Julio I. de Vicente

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Researchers Lledó, Palazuelos, and de Vicente discovered that quantum network entanglement resilience depends critically on graph connectivity growth rates, not just topology extremes like complete graphs or trees. Their study proves that networks with node degree growing at least linearly (Ω(N)) maintain genuine multipartite entanglement (GME) even with noise, provided it stays below a fixed threshold. Conversely, networks with slowly growing degrees (o(log N)) lose GME entirely as system size increases, becoming asymptotically biseparable regardless of noise levels. The findings bridge graph theory and quantum information, showing that connectivity metrics—particularly degree growth—determine whether entanglement survives scaling in noisy environments. Explicit constructions in the paper demonstrate the tightness of these bounds, offering practical guidelines for designing robust quantum networks.

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AbstractQuantum networks are promising venues for quantum information processing. This motivates the study of the entanglement properties of the particular multipartite quantum states that underpin these structures. In particular, it has been recently shown that when the links are noisy two drastically different behaviors can occur regarding the global entanglement properties of the network. While in certain configurations the network displays genuine multipartite entanglement (GME) for any system size provided the noise level is below a certain threshold, in others GME is washed out if the system size is big enough for any fixed non-zero level of noise. However, this difference has only been established considering the two extreme cases of maximally and minimally connected networks (i.e. complete graphs versus trees, respectively). In this article we investigate this question much more in depth and relate this behavior to the growth of several graph theoretic parameters that measure the connectivity of the graph sequence that codifies the structure of the network as the number of parties increases. The strongest conditions are obtained when considering the degree growth. Our main results are that a sufficiently fast degree growth (i.e. $\Omega(N)$, where $N$ is the size of the network) is sufficient for asymptotic robustness of GME, while if it is sufficiently slow (i.e. $o(\log N)$) then the network becomes asymptotically biseparable. We also present several explicit constructions related to the optimality of these results. ► BibTeX data@article{Lledo2025asymptotic, doi = {10.22331/q-2025-12-31-1958}, url = {https://doi.org/10.22331/q-2025-12-31-1958}, title = {Asymptotic robustness of entanglement in noisy quantum networks and graph connectivity}, author = {Lled{\'{o}}, Fernando and Palazuelos, Carlos and de Vicente, Julio I.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1958}, month = dec, year = {2025} }► References [1] A. Acín, J. I. Cirac, and M. Lewenstein, Entanglement percolation in quantum networks, Nature Physics 3, 256 (2007). https://doi.org/10.1038/nphys549 [2] S. J. Akhtarshenas and M. A. Jafarizadeh, Optimal Lewenstein–Sanpera decomposition for some bipartite systems, J. Phys. A: Math. Gen. 37, 2965 (2004). https://doi.org/10.1088/0305-4470/37/8/008 [3] K. Azuma, S. Bäuml, T. Coopmans, D. Elkouss, and L. Boxi, Tools for quantum network design, AVS Quantum Sci. 3, 014101 (2021). https://doi.org/10.1116/5.0024062 [4] S. Beigi and P. W. Shor, Approximating the set of separable states using the positive partial transpose test, J. Math. Phys. 51, 042202 (2010). https://doi.org/10.1063/1.3364793 [5] J. A. Bondy and U. S. R. Murty, Graph Theory (Springer, New York, 2008). [6] S. Bravyi, Y. Sharma, M. Szegedy, and R. de Wolf, Generating k EPR-pairs from an n-party resource state, Quantum 8, 1348 (2024). https://doi.org/10.22331/q-2024-05-14-1348 [7] M. Caleffi, M. Amoretti, D. Ferrari, J. Illiano, A. Manzalini, and A. S. Cacciapuoti, Distributed quantum computing: A survey, Computer Networks 254, 110672 (2024). https://doi.org/10.1016/j.comnet.2024.110672 [8] G. Chartrand, A graph-theoretic approach to a communications problem, SIAM J. Appl. Math. 14, 778 (1966). [9] E. Chitambar, D. Leung, L. Mancinska, M. Ozols, and A. Winter, Everything you always wanted to know about LOCC (but were afraid to ask), Commun. Math. Phys. 328, 303 (2014). https://doi.org/10.1007/s00220-014-1953-9 [10] F. Chung, Spectral Graph Theory (American Mathematical Society, 1997). [11] J. I. Cirac, A. Ekert, S. F. Huelga, and C. Macchiavello, Distributed Quantum Computation over Noisy Channels, Phys. Rev. A 59, 4249 (1999). https://doi.org/10.1103/PhysRevA.59.4249 [12] P. Contreras-Tejada, C. Palazuelos, and J. I. de Vicente, Genuine multipartite nonlocality is intrinsic to quantum networks, Phys. Rev. Lett. 126, 040501 (2021). https://doi.org/10.1103/PhysRevLett.126.040501 [13] P. Contreras-Tejada, C. Palazuelos, and J. I. de Vicente, Asymptotic survival of genuine multipartite entanglement in noisy quantum networks depends on the topology, Phys. Rev. Lett. 128, 220501 (2022). https://doi.org/10.1103/PhysRevLett.128.220501 [14] S. Das, S. Bäuml, M. Winczewski, and K. Horodecki, Universal limitations on quantum key distribution over a network, Phys. Rev. X 11, 041016 (2021). https://doi.org/10.1103/PhysRevX.11.041016 [15] I. Devetak and A. Winter, Distillation of secret key and entanglement from quantum states, Proc. R. Soc. A 461, 207 (2005). https://doi.org/10.1098/rspa.2004.1372 [16] R. Diestel, Graph Theory (Springer, New York, 2017). [17] P. Erdós, J. Pach, R. Pollack, and Z. Tuza, Radius, diameter, and minimum degree, J. Combin. Theory Ser. B 47, 73 (1989). https://doi.org/10.1016/0095-8956(89)90066-X [18] M. Hamada, Exponential lower bound on the highest fidelity achievable by quantum error-correcting codes, Phys. Rev. A 65, 052305 (2002). https://doi.org/10.1103/PhysRevA.65.052305 [19] M. Horodecki and P. Horodecki, Reduction criterion of separability and limits for a class of distillation protocols, Phys. Rev. A 59, 4206 (1999). https://doi.org/10.1103/PhysRevA.59.4206 [20] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009). https://doi.org/10.1103/RevModPhys.81.865 [21] P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzé, and A. Smerzi, Fisher information and multiparticle entanglement, Phys. Rev. A 85, 022321 (2012). https://doi.org/10.1103/PhysRevA.85.022321 [22] M. Lewenstein and A. Sanpera, Separability and entanglement of composite quantum systems, Phys. Rev. Lett. 80, 2261 (1998). https://doi.org/10.1103/PhysRevLett.80.2261 [23] A. W. Marcus, D. A. Spielman, and N. Srivastava, Interlacing families I: Bipartite Ramanujan graphs of all degrees, Ann. Math. 182, 307 (2015). https://doi.org/10.4007/annals.2015.182.1.7 [24] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010). [25] S. Perseguers, G. J. Lapeyre Jr, D. Cavalcanti, M. Lewenstein, and A. Acín, Distribution of entanglement in large-scale quantum networks, Rep. Prog. Phys. 76, 096001 (2013). https://doi.org/10.1088/0034-4885/76/9/096001 [26] M. B. Plenio and S. Virmani, An introduction to entanglement measures, Quantum Inf. Comput. 7, 1 (2007). https://doi.org/10.26421/QIC7.1-2-1 [27] C. Spee and T. Kraft, Transformations in quantum networks via local operations assisted by finitely many rounds of classical communication, Quantum 8, 1286 (2024). https://doi.org/10.22331/q-2024-03-14-1286 [28] G. Tóth, Multipartite entanglement and high-precision metrology, Phys. Rev. A 85, 022322 (2012). https://doi.org/10.1103/PhysRevA.85.022322 [29] J. Watrous, The Theory of Quantum Information (Cambridge University Press, Cambridge, 2018). [30] S. Wehner, D. Elkouss, and R. Hanson, Quantum internet: A vision for the road ahead, Science 362, 9288 (2018). https://doi.org/10.1126/science.aam9288 [31] H. Yamasaki, A. Soeda, and M. Murao, Graph-associated entanglement cost of a multipartite state in exact and finite-block-length approximate constructions, Phys. Rev. A 96, 032330 (2017). https://doi.org/10.1103/PhysRevA.96.032330Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-31 09:02:05: Could not fetch cited-by data for 10.22331/q-2025-12-31-1958 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-31 09:02:06: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractQuantum networks are promising venues for quantum information processing. This motivates the study of the entanglement properties of the particular multipartite quantum states that underpin these structures. In particular, it has been recently shown that when the links are noisy two drastically different behaviors can occur regarding the global entanglement properties of the network. While in certain configurations the network displays genuine multipartite entanglement (GME) for any system size provided the noise level is below a certain threshold, in others GME is washed out if the system size is big enough for any fixed non-zero level of noise. However, this difference has only been established considering the two extreme cases of maximally and minimally connected networks (i.e. complete graphs versus trees, respectively). In this article we investigate this question much more in depth and relate this behavior to the growth of several graph theoretic parameters that measure the connectivity of the graph sequence that codifies the structure of the network as the number of parties increases. The strongest conditions are obtained when considering the degree growth. Our main results are that a sufficiently fast degree growth (i.e. $\Omega(N)$, where $N$ is the size of the network) is sufficient for asymptotic robustness of GME, while if it is sufficiently slow (i.e. $o(\log N)$) then the network becomes asymptotically biseparable. We also present several explicit constructions related to the optimality of these results. ► BibTeX data@article{Lledo2025asymptotic, doi = {10.22331/q-2025-12-31-1958}, url = {https://doi.org/10.22331/q-2025-12-31-1958}, title = {Asymptotic robustness of entanglement in noisy quantum networks and graph connectivity}, author = {Lled{\'{o}}, Fernando and Palazuelos, Carlos and de Vicente, Julio I.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1958}, month = dec, year = {2025} }► References [1] A. Acín, J. I. Cirac, and M. Lewenstein, Entanglement percolation in quantum networks, Nature Physics 3, 256 (2007). https://doi.org/10.1038/nphys549 [2] S. J. Akhtarshenas and M. A. Jafarizadeh, Optimal Lewenstein–Sanpera decomposition for some bipartite systems, J. Phys. A: Math. Gen. 37, 2965 (2004). https://doi.org/10.1088/0305-4470/37/8/008 [3] K. Azuma, S. Bäuml, T. Coopmans, D. Elkouss, and L. Boxi, Tools for quantum network design, AVS Quantum Sci. 3, 014101 (2021). https://doi.org/10.1116/5.0024062 [4] S. Beigi and P. W. Shor, Approximating the set of separable states using the positive partial transpose test, J. Math. Phys. 51, 042202 (2010). https://doi.org/10.1063/1.3364793 [5] J. A. Bondy and U. S. R. Murty, Graph Theory (Springer, New York, 2008). [6] S. Bravyi, Y. Sharma, M. Szegedy, and R. de Wolf, Generating k EPR-pairs from an n-party resource state, Quantum 8, 1348 (2024). https://doi.org/10.22331/q-2024-05-14-1348 [7] M. Caleffi, M. Amoretti, D. Ferrari, J. Illiano, A. Manzalini, and A. S. Cacciapuoti, Distributed quantum computing: A survey, Computer Networks 254, 110672 (2024). https://doi.org/10.1016/j.comnet.2024.110672 [8] G. Chartrand, A graph-theoretic approach to a communications problem, SIAM J. Appl. Math. 14, 778 (1966). [9] E. Chitambar, D. Leung, L. Mancinska, M. Ozols, and A. Winter, Everything you always wanted to know about LOCC (but were afraid to ask), Commun. Math. Phys. 328, 303 (2014). https://doi.org/10.1007/s00220-014-1953-9 [10] F. Chung, Spectral Graph Theory (American Mathematical Society, 1997). [11] J. I. Cirac, A. Ekert, S. F. Huelga, and C. Macchiavello, Distributed Quantum Computation over Noisy Channels, Phys. Rev. A 59, 4249 (1999). https://doi.org/10.1103/PhysRevA.59.4249 [12] P. Contreras-Tejada, C. Palazuelos, and J. I. de Vicente, Genuine multipartite nonlocality is intrinsic to quantum networks, Phys. Rev. Lett. 126, 040501 (2021). https://doi.org/10.1103/PhysRevLett.126.040501 [13] P. Contreras-Tejada, C. Palazuelos, and J. I. de Vicente, Asymptotic survival of genuine multipartite entanglement in noisy quantum networks depends on the topology, Phys. Rev. Lett. 128, 220501 (2022). https://doi.org/10.1103/PhysRevLett.128.220501 [14] S. Das, S. Bäuml, M. Winczewski, and K. Horodecki, Universal limitations on quantum key distribution over a network, Phys. Rev. X 11, 041016 (2021). https://doi.org/10.1103/PhysRevX.11.041016 [15] I. Devetak and A. Winter, Distillation of secret key and entanglement from quantum states, Proc. R. Soc. A 461, 207 (2005). https://doi.org/10.1098/rspa.2004.1372 [16] R. Diestel, Graph Theory (Springer, New York, 2017). [17] P. Erdós, J. Pach, R. Pollack, and Z. Tuza, Radius, diameter, and minimum degree, J. Combin. Theory Ser. B 47, 73 (1989). https://doi.org/10.1016/0095-8956(89)90066-X [18] M. Hamada, Exponential lower bound on the highest fidelity achievable by quantum error-correcting codes, Phys. Rev. A 65, 052305 (2002). https://doi.org/10.1103/PhysRevA.65.052305 [19] M. Horodecki and P. Horodecki, Reduction criterion of separability and limits for a class of distillation protocols, Phys. Rev. A 59, 4206 (1999). https://doi.org/10.1103/PhysRevA.59.4206 [20] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009). https://doi.org/10.1103/RevModPhys.81.865 [21] P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzé, and A. Smerzi, Fisher information and multiparticle entanglement, Phys. Rev. A 85, 022321 (2012). https://doi.org/10.1103/PhysRevA.85.022321 [22] M. Lewenstein and A. Sanpera, Separability and entanglement of composite quantum systems, Phys. Rev. Lett. 80, 2261 (1998). https://doi.org/10.1103/PhysRevLett.80.2261 [23] A. W. Marcus, D. A. Spielman, and N. Srivastava, Interlacing families I: Bipartite Ramanujan graphs of all degrees, Ann. Math. 182, 307 (2015). https://doi.org/10.4007/annals.2015.182.1.7 [24] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010). [25] S. Perseguers, G. J. Lapeyre Jr, D. Cavalcanti, M. Lewenstein, and A. Acín, Distribution of entanglement in large-scale quantum networks, Rep. Prog. Phys. 76, 096001 (2013). https://doi.org/10.1088/0034-4885/76/9/096001 [26] M. B. Plenio and S. Virmani, An introduction to entanglement measures, Quantum Inf. Comput. 7, 1 (2007). https://doi.org/10.26421/QIC7.1-2-1 [27] C. Spee and T. Kraft, Transformations in quantum networks via local operations assisted by finitely many rounds of classical communication, Quantum 8, 1286 (2024). https://doi.org/10.22331/q-2024-03-14-1286 [28] G. Tóth, Multipartite entanglement and high-precision metrology, Phys. Rev. A 85, 022322 (2012). https://doi.org/10.1103/PhysRevA.85.022322 [29] J. Watrous, The Theory of Quantum Information (Cambridge University Press, Cambridge, 2018). [30] S. Wehner, D. Elkouss, and R. Hanson, Quantum internet: A vision for the road ahead, Science 362, 9288 (2018). https://doi.org/10.1126/science.aam9288 [31] H. Yamasaki, A. Soeda, and M. Murao, Graph-associated entanglement cost of a multipartite state in exact and finite-block-length approximate constructions, Phys. Rev. A 96, 032330 (2017). https://doi.org/10.1103/PhysRevA.96.032330Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-31 09:02:05: Could not fetch cited-by data for 10.22331/q-2025-12-31-1958 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-31 09:02:06: No response from ADS or unable to decode the received json data when getting the list of citing works.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.

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Asymptotic robustness of entanglement in noisy quantum networks and graph connectivity

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