Device independent security of quantum key distribution from monogamy-of-entanglement games

Enrique Cervero-Martí­n; Marco Tomamichel

doi:10.22331/q-2025-03-05-1652

Device independent security of quantum key distribution from monogamy-of-entanglement games

Enrique Cervero-Martín¹ and Marco Tomamichel^2,3

¹Centre for Quantum Technologies, National University of Singapore
²Centre for Quantum Technologies, National University of Singapore
³Department of Electrical and Computer Engineering, National University of Singapore

Published:	2025-03-05, volume 9, page 1652
Editor:	Máté Farkas
Eprint:	arXiv:2312.04079v2
Doi:	https://doi.org/10.22331/q-2025-03-05-1652
Citation:	Quantum 9, 1652 (2025).

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Abstract

We analyse two party non-local games whose predicate requires Alice and Bob to generate matching bits, and their three party extensions where a third player receives all inputs and is required to output a bit that matches that of the original players. We propose a general device independent quantum key distribution protocol for the subset of such non-local games that satisfy a monogamy-of-entanglement property characterised by a gap in the maximum winning probability between the bipartite and tripartite versions of the game. This gap is due to the optimal strategy for two players requiring entanglement, which due to its monogamy property cannot be shared with any additional players. Based solely on the monogamy-of-entanglement property, we provide a simple proof of information theoretic security of our protocol. Lastly, we numerically optimize the finite and asymptotic secret key rates of our protocol using the magic square game as an example, for which we provide a numerical bound on the maximal tripartite quantum winning probability which closely matches the bipartite classical winning probability. Further, we show that our protocol is robust for depolarizing noise up to about $2.88\%$, providing the first such bound for general attacks for magic square based quantum key distribution.

Featured image: Shaded green area corresponds to the range of bipartite winning probabilities $ω_2$ and tripartite winning probabilities $ω_3$ which yield a positive asymptotic key rate.

Popular summary

We analyse a general framework for device-independent quantum key distribution based on so-called monogamy-of-entanglement games —a type of non-local games requiring the participating parties to generate matching outputs, but whose success probability decreases as more parties are added.
We show that device-independent quantum key distribution using monogamy-of-entanglement games is possible provided a large enough gap in the winning probabilities of the bipartite and tripartite versions of the game.
In particular, we analyse an application of our framework using the magic square game, for which we also show that the tripartite quantum value collapses to the classical value due to the monogamy of entanglement.

► BibTeX data

@article{CerveroMartin2025deviceindependent,
  doi = {10.22331/q-2025-03-05-1652},
  url = {https://doi.org/10.22331/q-2025-03-05-1652},
  title = {Device independent security of quantum key distribution from monogamy-of-entanglement games},
  author = {Cervero-Mart{\'{i}}n, Enrique and Tomamichel, Marco},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1652},
  month = mar,
  year = {2025}
}

► References

[1] Jonathan Barrett, Lucien Hardy, and Adrian Kent. ``No signaling and quantum key distribution''. Phys. Rev. Lett. 95, 010503 (2005).
https://doi.org/10.1103/PhysRevLett.95.010503

[2] Antonio Acín, Serge Massar, and Stefano Pironio. ``Efficient quantum key distribution secure against no-signalling eavesdroppers''. New Journal of Physics 8, 126–126 (2006).
https://doi.org/10.1088/1367-2630/8/8/126

[3] Reinhard F. Werner. ``An application of bell's inequalities to a quantum state extension problem''. Letters in Mathematical Physics 17, 359–363 (1989).
https://doi.org/10.1007/BF00399761

[4] Andrew C. Doherty, Pablo A. Parrilo, and Federico M. Spedalieri. ``Complete family of separability criteria''. Phys. Rev. A 69, 022308 (2004).
https://doi.org/10.1103/PhysRevA.69.022308

[5] Renato Renner. ``Security of quantum key distribution''. PhD thesis. ETH Zurich. (2006). url: arxiv.org/abs/quant-ph/0512258.
arXiv:quant-ph/0512258

[6] Marco Tomamichel and Anthony Leverrier. ``A largely self-contained and complete security proof for quantum key distribution''. Quantum 1, 14 (2017).
https://doi.org/10.22331/q-2017-07-14-14

[7] Marco Tomamichel, Roger Colbeck, and Renato Renner. ``A fully quantum asymptotic equipartition property''. IEEE Transactions on Information Theory 55, 5840–5847 (2009).
https://doi.org/10.1109/TIT.2009.2032797

[8] Frédéric Dupuis, Omar Fawzi, and Renato Renner. ``Entropy accumulation''. Communications in Mathematical Physics 379, 867–913 (2020).
https://doi.org/10.1007/s00220-020-03839-5

[9] Frédéric Dupuis and Omar Fawzi. ``Entropy accumulation with improved second-order term''. IEEE Transactions on Information Theory 65, 7596–7612 (2019).
https://doi.org/10.1109/TIT.2019.2929564

[10] Tony Metger, Omar Fawzi, David Sutter, and Renato Renner. ``Generalised entropy accumulation''. 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)Pages 844–850 (2022).
https://doi.org/10.1109/FOCS54457.2022.00085

[11] Ernest Y. Z. Tan. ``Prospects for device-independent quantum key distribution''. PhD thesis. ETH Zurich. (2021). url: arxiv.org/abs/2111.11769.
arXiv:2111.11769

[12] Valerio Scarani and Renato Renner. ``Security bounds for quantum cryptography with finite resources''. In Theory of Quantum Computation, Communication, and Cryptography. Pages 83–95. Berlin, Heidelberg (2008). Springer Berlin Heidelberg.
https://doi.org/10.1007/978-3-540-89304-2_8

[13] Lana Sheridan, Thinh Phuc Le, and Valerio Scarani. ``Finite-key security against coherent attacks in quantum key distribution''. New Journal of Physics 12, 123019 (2010).
https://doi.org/10.1088/1367-2630/12/12/123019

[14] Marco Tomamichel, Charles C.-W. Lim, Nicolas Gisin, and Renato Renner. ``Tight finite-key analysis for quantum cryptography''. Nature Communications 3 (2012).
https://doi.org/10.1038/ncomms1631

[15] Marco Tomamichel, Serge Fehr, Jędrzej Kaniewski, and Stephanie Wehner. ``A monogamy-of-entanglement game with applications to device-independent quantum cryptography''. New Journal of Physics 15, 103002 (2013).
https://doi.org/10.1088/1367-2630/15/10/103002

[16] Umesh Vazirani and Thomas Vidick. ``Fully device-independent quantum key distribution''. Physical Review Letters 113 (2014).
https://doi.org/10.1103/physrevlett.113.140501

[17] Rotem Arnon-Friedman, Frédéric Dupuis, Omar Fawzi, Renato Renner, and Thomas Vidick. ``Practical device-independent quantum cryptography via entropy accumulation''. Nature Communications 9, 459 (2018).
https://doi.org/10.1038/s41467-017-02307-4

[18] Rotem Arnon-Friedman, Renato Renner, and Thomas Vidick. ``Simple and tight device-independent security proofs''. SIAM Journal on Computing 48, 181–225 (2019).
https://doi.org/10.1137/18M1174726

[19] Ernest Y.-Z. Tan, René Schwonnek, Koon Tong Goh, Ignatius William Primaatmaja, and Charles C.-W. Lim. ``Computing secure key rates for quantum cryptography with untrusted devices''. npj Quantum Information 7 (2021).
https://doi.org/10.1038/s41534-021-00494-z

[20] René Schwonnek, Koon Tong Goh, Ignatius W. Primaatmaja, Ernest Y.-Z. Tan, Ramona Wolf, Valerio Scarani, and Charles C.-W. Lim. ``Device-independent quantum key distribution with random key basis''. Nature Communications 12 (2021).
https://doi.org/10.1038/s41467-021-23147-3

[21] Ernest Y.-Z. Tan, Pavel Sekatski, Jean-Daniel Bancal, René Schwonnek, Renato Renner, Nicolas Sangouard, and Charles C.-W. Lim. ``Improved DIQKD protocols with finite-size analysis''. Quantum 6, 880 (2022).
https://doi.org/10.22331/q-2022-12-22-880

[22] D. P. Nadlinger, P. Drmota, B. C. Nichol, G. Araneda, D. Main, R. Srinivas, D. M. Lucas, C. J. Ballance, K. Ivanov, E. Y.-Z. Tan, P. Sekatski, R. L. Urbanke, R. Renner, N. Sangouard, and J.-D. Bancal. ``Experimental quantum key distribution certified by bell's theorem''. Nature 607, 682–686 (2022).
https://doi.org/10.1038/s41586-022-04941-5

[23] Tony Metger and Renato Renner. ``Security of quantum key distribution from generalised entropy accumulation''. Nature Communications 14, 5272 (2023).
https://doi.org/10.1038/s41467-023-40920-8

[24] Stefano Pirandola, Ulrik L. Andersen, Leonardo Banchi, Mario Berta, Darius Bunandar, Roger Colbeck, Dirk Englund, Tobias Gehring, Cosmo Lupo, Carlo Ottaviani, Jason L. Pereira, Mohsen Razavi, Jesni Shamsul Shaari, Marco Tomamichel, Vladyslav C. Usenko, Giuseppe Vallone, Paolo Villoresi, and Petros Wallden. ``Advances in quantum cryptography''. Advances in Optics and Photonics 12, 1012 (2020).
https://doi.org/10.1364/aop.361502

[25] Víctor Zapatero, Tim van Leent, Rotem Arnon-Friedman, Wen-Zhao Liu, Qiang Zhang, Harald Weinfurter, and Marcos Curty. ``Advances in device-independent quantum key distribution''. npj Quantum Information 9 (2023).
https://doi.org/10.1038/s41534-023-00684-x

[26] Ignatius W. Primaatmaja, Koon Tong Goh, Ernest Y.-Z. Tan, John T.-F. Khoo, Shouvik Ghorai, and Charles C.-W. Lim. ``Security of device-independent quantum key distribution protocols: a review''. Quantum 7, 932 (2023).
https://doi.org/10.22331/q-2023-03-02-932

[27] Tim Zhang, Weiand van Leent, Kai Redeker, Robert Garthoff, René Schwonnek, Florian Fertig, Sebastian Eppelt, Wenjamin Rosenfeld, Valerio Scarani, Charles C.-W. Lim, and Harald Weinfurter. ``A device-independent quantum key distribution system for distant users''. Nature 607, 687–691 (2022).
https://doi.org/10.1038/s41586-022-04891-y

[28] Thomas Vidick. ``Parallel diqkd from parallel repetition'' (2017). arXiv:1703.08508.
arXiv:1703.08508

[29] Rahul Jain, Carl A. Miller, and Yaoyun Shi. ``Parallel device-independent quantum key distribution''. IEEE Transactions on Information Theory 66, 5567–5584 (2020).
https://doi.org/10.1109/tit.2020.2986740

[30] Rahul Jain and Srijita Kundu. ``A direct product theorem for quantum communication complexity with applications to device-independent cryptography'' (2023). arXiv:2106.04299.
arXiv:2106.04299

[31] Xingyao Wu, Jean-Daniel Bancal, Matthew McKague, and Valerio Scarani. ``Device-independent parallel self-testing of two singlets''. Physical Review A 93 (2016).
https://doi.org/10.1103/physreva.93.062121

[32] Eric Culf and Thomas Vidick. ``A monogamy-of-entanglement game for subspace coset states''. Quantum 6, 791 (2022).
https://doi.org/10.22331/q-2022-09-01-791

[33] Eric Culf, Thomas Vidick, and Victor V. Albert. ``Group coset monogamy games and an application to device-independent continuous-variable qkd'' (2022). arXiv:2212.03935v15.
arXiv:2212.0393

[34] Damián Pitalúa-García and Iordanis Kerenidis. ``Practical and unconditionally secure spacetime-constrained oblivious transfer''. Physical Review A 98 (2018).
https://doi.org/10.1103/physreva.98.032327

[35] Andrea Coladangelo, Jiahui Liu, Qipeng Liu, and Mark Zhandry. ``Hidden cosets and applications to unclonable cryptography''. In Advances in Cryptology – CRYPTO 2021. Pages 556–584. Springer International Publishing (2021).
https://doi.org/10.1007/978-3-030-84242-0_20

[36] Srijita Kundu and Ernest Y. Z. Tan. ``Device-independent uncloneable encryption'' (2023). arXiv:2210.01058.
https://doi.org/10.22331/q-2025-01-08-1582
arXiv:2210.01058

[37] Stefano Pironio, Antonio Acín, Nicolas Brunner, Nicolas Gisin, Serge Massar, and Valerio Scarani. ``Device-independent quantum key distribution secure against collective attacks''. New Journal of Physics 11, 045021 (2009).
https://doi.org/10.1088/1367-2630/11/4/045021

[38] Pavel Sekatski, Jean-Daniel Bancal, Xavier Valcarce, Ernest Y.-Z. Tan, Renato Renner, and Nicolas Sangouard. ``Device-independent quantum key distribution from generalized CHSH inequalities''. Quantum 5, 444 (2021).
https://doi.org/10.22331/q-2021-04-26-444

[39] Carl A. Miller and Yaoyun Shi. ``Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices''. J. ACM 63 (2016).
https://doi.org/10.1145/2885493

[40] N. David Mermin. ``Simple unified form for the major no-hidden-variables theorems''. Phys. Rev. Lett. 65, 3373–3376 (1990).
https://doi.org/10.1103/PhysRevLett.65.3373

[41] Asher Peres. ``Incompatible results of quantum measurements''. Physics Letters A 151, 107–108 (1990).
https://doi.org/10.1016/0375-9601(90)90172-K

[42] Miguel Navascués, Stefano Pironio, and Antonio Acín. ``A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations''. New Journal of Physics 10, 073013 (2008).
https://doi.org/10.1088/1367-2630/10/7/073013

[43] Stefano Pironio, Miguel Navascués, and Antonio Acín. ``Convergent relaxations of polynomial optimization problems with noncommuting variables''. SIAM Journal on Optimization 20, 2157–2180 (2010).
https://doi.org/10.1137/090760155

[44] Peter Wittek. ``Algorithm 950''. ACM Transactions on Mathematical Software 41, 1–12 (2015).
https://doi.org/10.1145/2699464

[45] Peter Brown, Hamza Fawzi, and Omar Fawzi. ``Device-independent lower bounds on the conditional von neumann entropy''. Quantum 8, 1445 (2024).
https://doi.org/10.22331/q-2024-08-27-1445

[46] Yi-Zheng Zhen, Yingqiu Mao, Yu-Zhe Zhang, Feihu Xu, and Barry C. Sanders. ``Device-independent quantum key distribution based on the mermin-peres magic square game''. Phys. Rev. Lett. 131, 080801 (2023).
https://doi.org/10.1103/PhysRevLett.131.080801

[47] Carlos Palazuelos and Thomas Vidick. ``Survey on nonlocal games and operator space theory''. Journal of Mathematical Physics 57, 015220 (2016).
https://doi.org/10.1063/1.4938052

[48] Lluís Masanes, Renato Renner, Matthias Christandl, Andreas Winter, and Jonathan Barrett. ``Full security of quantum key distribution from no-signaling constraints''. IEEE Transactions on Information Theory 60, 4973–4986 (2014).
https://doi.org/10.1109/TIT.2014.2329417

[49] Renato Renner and Stefan Wolf. ``The exact price for unconditionally secure asymmetric cryptography''. In Advances in Cryptology - EUROCRYPT 2004. Pages 109–125. Berlin, Heidelberg (2004). Springer Berlin Heidelberg.
https://doi.org/10.1007/978-3-540-24676-3_7

[50] Dodis, Yevgeniy and Wichs, Daniel. ``Non-malleable extractors and symmetric key cryptography from weak secrets''. In Proceedings forty-first annual acm symposium on theory computing. Page 601–610. New York, NY, USA (2009). Association for Computing Machinery.
https://doi.org/10.1145/1536414.1536496

[51] Christopher Portmann and Renato Renner. ``Cryptographic security of quantum key distribution'' (2014). arXiv:1409.3525.
arXiv:1409.3525

[52] Joseph M. Renes and Renato Renner. ``One-shot classical data compression with quantum side information and the distillation of common randomness or secret keys''. IEEE Transactions on Information Theory 58, 1985–1991 (2012).
https://doi.org/10.1109/TIT.2011.2177589

[53] Marco Tomamichel, Jesus Martinez-Mateo, Christoph Pacher, and David Elkouss. ``Fundamental finite key limits for one-way information reconciliation in quantum key distribution''. Quantum Information Processing 16 (2017).
https://doi.org/10.1007/s11128-017-1709-5

[54] 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, 207–235 (2005).
https://doi.org/10.1098/rspa.2004.1372

[55] Enrique Cervero-Martín. ``Python script for obtaining tripartite winning probabilities, and finite and asymptotic keyrates''. https://github.com/EnriqueCerv/NLG_DI_QKD.git (2023).
https://github.com/EnriqueCerv/NLG_DI_QKD.git

[56] Ben Toner. ``Monogamy of non-local quantum correlations''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 59–69 (2008).
https://doi.org/10.1098/rspa.2008.0149

[57] Marco Tomamichel. ``Quantum information processing with finite resources''. Springer International Publishing. (2016).
https://doi.org/10.1007/978-3-319-21891-5

[58] Michael Mitzenmacher and Eli Upfal. ``Probability and computing: Randomized algorithms and probabilistic analysis''. Cambridge University Press. USA (2005).

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