Lower bounds on bipartite entanglement in noisy graph states

Aqil Sajjad; Eneet Kaur; Kenneth Goodenough; Don Towsley; Saikat Guha

doi:10.22331/q-2026-05-21-2111

Lower bounds on bipartite entanglement in noisy graph states

Aqil Sajjad¹, Eneet Kaur^1,2, Kenneth Goodenough³, Don Towsley³, and Saikat Guha¹

¹James C. Wyant College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA
²Cisco Quantum Lab, Los Angeles, USA
³College of Information and Computer Sciences, University of Massachusetts Amherst, MA 01002, USA

Published:	2026-05-21, volume 10, page 2111
Editor:	Himadri Shekhar Dhar
Eprint:	arXiv:2404.09014v2
Doi:	https://doi.org/10.22331/q-2026-05-21-2111
Citation:	Quantum 10, 2111 (2026).

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

Abstract

Graph states are a key resource for a number of applications in quantum information theory. Due to the inherent noise in noisy intermediate-scale quantum (NISQ) era devices, it is important to understand the effects noise has on the usefulness of graph states. We consider a noise model where the initial qubits, prepared in $|+\rangle$ states, undergo depolarizing noise before the application of the CZ operations that generate edges between qubits situated at the nodes of the resulting graph state. For this model we develop a method for calculating the coherent information – a lower bound on the rate at which entanglement can be distilled, across a bipartition of the graph state. We also identify some patterns on how adding more nodes or edges affects the bipartite distillable entanglement. As an application, we find a family of graph states that maintain a strictly positive coherent information for any amount of (non-maximal) depolarizing noise.

Featured image: A bipartite graph state shared between Alice (maroon qubits) and Bob (blue qubits).

Popular summary

Graph states are an important resource for distributing entanglement across several parties. A graph state is based on the mathematical notion of a graph, which comprises a collection of vertices, and edges connecting some or all pairs of vertices with each other. In a graph state, each qubit is represented as a node in a graph, with edges representing entangling interactions between two qubits. A graph state thus describes a complex network of many entangled qubits. By manipulating and measuring some of the qubits, the pattern and nature of entanglement between the various qubits can be modified, thus allowing a graph state to serve as a resource for sharing entanglement across different parties. Moreover, graph states also have mathematical properties that can be exploited to protect against errors. Another benefit of working with graph states is that they can be described by the so-called stabilizer formalism, where an $n$-qubit graph state is specified by $n$ operators instead of an exponentially large $2^n$ terms in the quantum state.
In this work, we study the effect of noise on entanglement shared between two parties over a graph state, where the $n$ qubits are divided between Alice and Bob. In the absence of noise, it is straightforward to see how much entanglement is being shared between them by a given graph state. Specifically, both Alice and Bob can carry out a series of operations on the qubits owned by them, to extract a collection of Bell states shared between them. The number of such Bell states is a property of the graph state being employed. However, in the presence of noise, the graph state undergoes degradation, reducing the number of Bell pairs being shared. We study how a particular form of noise, known as depolarizing noise before the application of the “edge” interactions between the various nodes employed to generate the graph state, affects the amount of entanglement being delivered between two parties. We identify some patterns on how adding more nodes or edges affects the amount of entanglement in the presence of noise. We also find a family of graph states that show relatively high robustness to noise, ensuring that at least a certain amount of entanglement is always present.

► BibTeX data

@article{Sajjad2026lowerbounds,
  doi = {10.22331/q-2026-05-21-2111},
  url = {https://doi.org/10.22331/q-2026-05-21-2111},
  title = {Lower bounds on bipartite entanglement in noisy graph states},
  author = {Sajjad, Aqil and Kaur, Eneet and Goodenough, Kenneth and Towsley, Don and Guha, Saikat},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2111},
  month = may,
  year = {2026}
}

► References

[1] Stephanie Wehner, David Elkouss, and Ronald Hanson. Quantum internet: A vision for the road ahead. Science, 362 (6412), October 2018. ISSN 1095-9203. 10.1126/science.aam9288. URL http://dx.doi.org/10.1126/science.aam9288.
https://doi.org/10.1126/science.aam9288

[2] Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A, 52: R2493–R2496, Oct 1995. 10.1103/PhysRevA.52.R2493. URL https://doi.org/10.1103/PhysRevA.52.R2493.
https://doi.org/10.1103/PhysRevA.52.R2493

[3] Sara Bartolucci, Patrick Birchall, Hector Bombín, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, Fernando Pastawski, Terry Rudolph, and Chris Sparrow. Fusion-based quantum computation. Nature Communications, 14 (1), February 2023. ISSN 2041-1723. 10.1038/s41467-023-36493-1. URL http://dx.doi.org/10.1038/s41467-023-36493-1.
https://doi.org/10.1038/s41467-023-36493-1

[4] A.Yu. Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303 (1): 2–30, January 2003. 10.1016/s0003-4916(02)00018-0. URL https://doi.org/10.1016/s0003-4916(02)00018-0.
https://doi.org/10.1016/s0003-4916(02)00018-0

[5] Robert Raussendorf and Hans J. Briegel. A one-way quantum computer. Phys. Rev. Lett., 86: 5188–5191, May 2001. 10.1103/PhysRevLett.86.5188. URL https://doi.org/10.1103/PhysRevLett.86.5188.
https://doi.org/10.1103/PhysRevLett.86.5188

[6] Nicolas Sangouard, Christoph Simon, Hugues de Riedmatten, and Nicolas Gisin. Quantum repeaters based on atomic ensembles and linear optics. Rev. Mod. Phys., 83: 33–80, Mar 2011. 10.1103/RevModPhys.83.33. URL https://doi.org/10.1103/RevModPhys.83.33.
https://doi.org/10.1103/RevModPhys.83.33

[7] Koji Azuma, Kiyoshi Tamaki, and Hoi-Kwong Lo. All-photonic quantum repeaters. Nature Communications, 6 (1), April 2015. 10.1038/ncomms7787. URL https://doi.org/10.1038/ncomms7787.
https://doi.org/10.1038/ncomms7787

[8] Nathan Shettell and Damian Markham. Graph states as a resource for quantum metrology. Phys. Rev. Lett., 124: 110502, Mar 2020. 10.1103/PhysRevLett.124.110502. URL https://doi.org/10.1103/PhysRevLett.124.110502.
https://doi.org/10.1103/PhysRevLett.124.110502

[9] Maria Flors Mor-Ruiz and Wolfgang Dür. Noisy stabilizer formalism. Physical Review A, 107 (3), March 2023. ISSN 2469-9934. 10.1103/physreva.107.032424. URL http://dx.doi.org/10.1103/PhysRevA.107.032424.
https://doi.org/10.1103/physreva.107.032424

[10] Clément Meignant, Damian Markham, and Frédéric Grosshans. Distributing graph states over arbitrary quantum networks. Physical Review A, 100 (5), November 2019. ISSN 2469-9934. 10.1103/physreva.100.052333. URL http://dx.doi.org/10.1103/PhysRevA.100.052333.
https://doi.org/10.1103/physreva.100.052333

[11] Alex Fischer and Don Towsley. Distributing graph states across quantum networks. In 2021 IEEE International Conference on Quantum Computing and Engineering (QCE), page 324–333. IEEE, October 2021. 10.1109/qce52317.2021.00049. URL http://dx.doi.org/10.1109/QCE52317.2021.00049.
https://doi.org/10.1109/qce52317.2021.00049

[12] Konrad Szymański, Lina Vandré, and Otfried Gühne. Useful entanglement can be extracted from noisy graph states. Quantum, 10: 1977, January 2026. ISSN 2521-327X. 10.22331/q-2026-01-21-1977. URL http://dx.doi.org/10.22331/q-2026-01-21-1977.
https://doi.org/10.22331/q-2026-01-21-1977

[13] Christopher Vairogs, Samihr Hermes, and Felix Leditzky. Localizing multipartite entanglement with local and global measurements. Quantum, 10: 2007, February 2026. ISSN 2521-327X. 10.22331/q-2026-02-23-2007. URL http://dx.doi.org/10.22331/q-2026-02-23-2007.
https://doi.org/10.22331/q-2026-02-23-2007

[14] Rafail Frantzeskakis, Chenxu Liu, Zahra Raissi, Edwin Barnes, and Sophia E. Economou. Extracting perfect ghz states from imperfect weighted graph states via entanglement concentration. Physical Review Research, 5 (2), May 2023. ISSN 2643-1564. 10.1103/physrevresearch.5.023124. URL http://dx.doi.org/10.1103/PhysRevResearch.5.023124.
https://doi.org/10.1103/physrevresearch.5.023124

[15] Benjamin Schumacher and Michael D. Westmoreland. Quantum mutual information and the one-time pad. Physical Review A, 74 (4), October 2006. ISSN 1094-1622. 10.1103/physreva.74.042305. URL http://dx.doi.org/10.1103/PhysRevA.74.042305.
https://doi.org/10.1103/physreva.74.042305

[16] Mario Berta, Kaushik P. Seshadreesan, and Mark M. Wilde. Rényi generalizations of the conditional quantum mutual information. Journal of Mathematical Physics, 56 (2), February 2015. ISSN 1089-7658. 10.1063/1.4908102. URL http://dx.doi.org/10.1063/1.4908102.
https://doi.org/10.1063/1.4908102

[17] Mark M. Wilde. Quantum Information Theory. Cambridge University Press, November 2016. ISBN 9781316809976. 10.1017/9781316809976. URL http://dx.doi.org/10.1017/9781316809976.
https://doi.org/10.1017/9781316809976

[18] Omar Fawzi and Renato Renner. Quantum conditional mutual information and approximate markov chains. Communications in Mathematical Physics, 340 (2): 575–611, September 2015. ISSN 1432-0916. 10.1007/s00220-015-2466-x. URL http://dx.doi.org/10.1007/s00220-015-2466-x.
https://doi.org/10.1007/s00220-015-2466-x

[19] Fernando G. S. L. Brandão, Aram W. Harrow, Jonathan Oppenheim, and Sergii Strelchuk. Quantum conditional mutual information, reconstructed states, and state redistribution. Physical Review Letters, 115 (5), July 2015. ISSN 1079-7114. 10.1103/physrevlett.115.050501. URL http://dx.doi.org/10.1103/PhysRevLett.115.050501.
https://doi.org/10.1103/physrevlett.115.050501

[20] Eneet Kaur, Xiaoting Wang, and Mark M. Wilde. Conditional mutual information and quantum steering. Physical Review A, 96 (2), August 2017. ISSN 2469-9934. 10.1103/physreva.96.022332. URL http://dx.doi.org/10.1103/PhysRevA.96.022332.
https://doi.org/10.1103/physreva.96.022332

[21] Igor Devetak and Andreas Winter. Distillation of secret key and entanglement from quantum states. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461 (2053): 207–235, Jan 2005. 10.1098/rspa.2004.1372. URL https://doi.org/10.1098.
https://doi.org/10.1098/rspa.2004.1372

[22] David Fattal, Toby S Cubitt, Yoshihisa Yamamoto, Sergey Bravyi, and Isaac L Chuang. Entanglement in the stabilizer formalism. arXiv preprint quant-ph/0406168, 2004.
arXiv:quant-ph/0406168

[23] Daniel Gottesman. Stabilizer codes and quantum error correction. arXiv preprint quant-ph/9705052, 1997.
arXiv:quant-ph/9705052

[24] Marc Hein, Wolfgang Dür, Jens Eisert, Robert Raussendorf, M Nest, and H-J Briegel. Entanglement in graph states and its applications. arXiv preprint quant-ph/0602096, 2006.
arXiv:quant-ph/0602096

[25] Kenneth Goodenough, Aqil Sajjad, Eneet Kaur, Saikat Guha, and Don Towsley. Bipartite entanglement of noisy stabilizer states through the lens of stabilizer codes. In 2024 IEEE International Symposium on Information Theory (ISIT), page 545–550. IEEE, July 2024. 10.1109/isit57864.2024.10619666. URL http://dx.doi.org/10.1109/ISIT57864.2024.10619666.
https://doi.org/10.1109/isit57864.2024.10619666

[26] Kovid Goyal, Alex McCauley, and Robert Raussendorf. Purification of large bicolorable graph states. Physical Review A, 74 (3), September 2006. ISSN 1094-1622. 10.1103/physreva.74.032318. URL http://dx.doi.org/10.1103/PhysRevA.74.032318.
https://doi.org/10.1103/physreva.74.032318

[27] Ashlesha Patil and Saikat Guha. A graphical rule book for clifford manipulations of stabilizer states. IEEE Transactions on Quantum Engineering, 7: 1–33, 2026. 10.1109/TQE.2026.3653200.
https://doi.org/10.1109/TQE.2026.3653200

[28] Raúl García-Patrón, Stefano Pirandola, Seth Lloyd, and Jeffrey H. Shapiro. Reverse coherent information. Physical Review Letters, 102 (21), may 2009. 10.1103/physrevlett.102.210501. URL https://doi.org/10.1103.
https://doi.org/10.1103/physrevlett.102.210501

[29] M. Hein, J. Eisert, and H. J. Briegel. Multiparty entanglement in graph states. Physical Review A, 69 (6), June 2004. ISSN 1094-1622. 10.1103/physreva.69.062311. URL http://dx.doi.org/10.1103/PhysRevA.69.062311.
https://doi.org/10.1103/physreva.69.062311

[30] Prajit Dhara, Dirk Englund, and Saikat Guha. Entangling quantum memories via heralded photonic bell measurement. Physical Review Research, 5 (3), September 2023. ISSN 2643-1564. 10.1103/physrevresearch.5.033149. URL http://dx.doi.org/10.1103/PhysRevResearch.5.033149.
https://doi.org/10.1103/physrevresearch.5.033149

Cited by

[1] Konrad Szymański, Lina Vandré, and Otfried Gühne, "Useful entanglement can be extracted from noisy graph states", arXiv:2402.00937, (2024).

[2] Kenneth Goodenough, Aqil Sajjad, Eneet Kaur, Saikat Guha, and Don Towsley, "Bipartite entanglement of noisy stabilizer states through the lens of stabilizer codes", arXiv:2406.02427, (2024).

[3] Konrad Szymański, Lina Vandré, and Otfried Gühne, "Useful entanglement can be extracted from noisy graph states", Quantum 10, 1977 (2026).

The above citations are from SAO/NASA ADS (last updated successfully 2026-08-08 00:07:28). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2026-08-08 00:07:26).

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.