Device-independent lower bounds on the conditional von Neumann entropy

Peter Brown; Hamza Fawzi; Omar Fawzi

doi:10.22331/q-2024-08-27-1445

Device-independent lower bounds on the conditional von Neumann entropy

Peter Brown^1,3, Hamza Fawzi², and Omar Fawzi¹

¹Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France
²DAMTP, University of Cambridge, United Kingdom
³Télécom Paris, LTCI, Institut Polytechnique de Paris, 91120 Palaiseau, France

Published:	2024-08-27, volume 8, page 1445
Eprint:	arXiv:2106.13692v3
Doi:	https://doi.org/10.22331/q-2024-08-27-1445
Citation:	Quantum 8, 1445 (2024).

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Abstract

The rates of several device-independent (DI) protocols, including quantum key-distribution (QKD) and randomness expansion (RE), can be computed via an optimization of the conditional von Neumann entropy over a particular class of quantum states. In this work we introduce a numerical method to compute lower bounds on such rates. We derive a sequence of optimization problems that converge to the conditional von Neumann entropy of systems defined on general separable Hilbert spaces. Using the Navascués-Pironio-Acín hierarchy we can then relax these problems to semidefinite programs, giving a computationally tractable method to compute lower bounds on the rates of DI protocols. Applying our method to compute the rates of DI-RE and DI-QKD protocols we find substantial improvements over all previous numerical techniques, demonstrating significantly higher rates for both DI-RE and DI-QKD. In particular, for DI-QKD we show a minimal detection efficiency threshold which is within the realm of current capabilities. Moreover, we demonstrate that our method is capable of converging rapidly by recovering all known tight analytical bounds up to several decimal places. Finally, we note that our method is compatible with the entropy accumulation theorem and can thus be used to compute rates of finite round protocols and subsequently prove their security.

Popular summary

Device-independent protocols offer the pinnacle of cryptographic security: the generation of randomness and secret key on untrusted quantum hardware. However the security analysis of such protocols can be challenging in practice. In principle, we must understand the power of an adversary's best attack when that adversary has the freedom to choose the unknown quantum hardware on which we execute the protocol. In turn, to compute the key-rate of such a protocol we are required to optimize entropic quantities over potentially unbounded quantum systems. In this work we demonstrate a semidefinite programming technique to compute lower bounds on the key-rates of such protocols and demonstrate its practicality on numerous examples.

► BibTeX data

@article{Brown2024deviceindependent,
  doi = {10.22331/q-2024-08-27-1445},
  url = {https://doi.org/10.22331/q-2024-08-27-1445},
  title = {Device-independent lower bounds on the conditional von {N}eumann entropy},
  author = {Brown, Peter and Fawzi, Hamza and Fawzi, Omar},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1445},
  month = aug,
  year = {2024}
}

► References

[1] Charles H. Bennett and Gilles Brassard. ``Quantum cryptography: Public key distribution and coin tossing''. In Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing. Pages 175–179. IEEENew York (1984). arXiv:2003.06557.
https://doi.org/10.1016/j.tcs.2014.05.025
arXiv:2003.06557

[2] Lars Lydersen, Carlos Wiechers, Christoffer Wittmann, Dominique Elser, Johannes Skaar, and Vadim Makarov. ``Hacking commercial quantum cryptography systems by tailored bright illumination''. Nature Photonics 4, 686–689 (2010). arXiv:1008.4593.
https://doi.org/10.1038/nphoton.2010.214
arXiv:1008.4593

[3] Lluís Masanes, Antonio Acín, and Nicolas Gisin. ``General properties of nonsignaling theories''. Physical Review A 73, 012112 (2006). arXiv:quant-ph/0508016.
https://doi.org/10.1103/PhysRevA.73.012112
arXiv:quant-ph/0508016

[4] Nicolas Brunner, Stefano Pironio, Antonio Acin, Nicolas Gisin, André Allan Méthot, and Valerio Scarani. ``Testing the dimension of Hilbert spaces''. Physical review letters 100, 210503 (2008). arXiv:0802.0760.
https://doi.org/10.1103/PhysRevLett.100.210503
arXiv:0802.0760

[5] Harry Buhrman, Richard Cleve, Serge Massar, and Ronald De Wolf. ``Nonlocality and communication complexity''. Reviews of modern physics 82, 665 (2010). arXiv:0907.3584.
https://doi.org/10.1103/RevModPhys.82.665
arXiv:0907.3584

[6] Ivan Šupić and Joseph Bowles. ``Self-testing of quantum systems: a review''. Quantum 4, 337 (2020). arXiv:1904.10042.
https://doi.org/10.22331/q-2020-09-30-337
arXiv:1904.10042

[7] Dominic Mayers and Andrew Yao. ``Quantum cryptography with imperfect apparatus''. In Proceedings of the 39th Annual Symposium on Foundations of Computer Science (FOCS-98). Pages 503–509. Los Alamitos, CA, USA (1998). IEEE Computer Society. arXiv:quant-ph/9809039.
https://doi.org/10.1109/SFCS.1998.743501
arXiv:quant-ph/9809039

[8] Artur K. Ekert. ``Quantum cryptography based on Bell's theorem''. Physical Review Letters 67, 661–663 (1991).
https://doi.org/10.1103/PhysRevLett.67.661

[9] Frédéric Dupuis, Omar Fawzi, and Renato Renner. ``Entropy accumulation''. Communications in Mathematical Physics 379, 1–47 (2020). arXiv:1607.01796.
https://doi.org/10.1007/s00220-020-03839-5
arXiv:1607.01796

[10] Frédéric Dupuis and Omar Fawzi. ``Entropy accumulation with improved second-order term''. IEEE Transactions on information theory 65, 7596–7612 (2019). arXiv:1805.11652.
https://doi.org/10.1109/TIT.2019.2929564
arXiv:1805.11652

[11] Emanuel Knill, Yanbao Zhang, and Honghao Fu. ``Quantum probability estimation for randomness with quantum side information'' (2018). arXiv:1806.04553.
arXiv:1806.04553

[12] 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

[13] Rotem Arnon-Friedman, Renato Renner, and Thomas Vidick. ``Simple and tight device-independent security proofs''. SIAM Journal on Computing 48, 181–225 (2019). arXiv:1607.01797.
https://doi.org/10.1137/18M1174726
arXiv:1607.01797

[14] C. Jordan. ``Essai sur la géométrie à n dimensions''. Bulletin de la S. M. F. 3, 103–174 (1875).
https://doi.org/10.24033/bsmf.90

[15] 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). arXiv:0903.4460.
https://doi.org/10.1088/1367-2630/11/4/045021
arXiv:0903.4460

[16] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. ``Proposed experiment to test local hidden-variable theories''. Physical Review Letters 23, 880–884 (1969).
https://doi.org/10.1103/PhysRevLett.23.880

[17] Federico Grasselli, Gláucia Murta, Hermann Kampermann, and Dagmar Bruß. ``Entropy bounds for multiparty device-independent cryptography''. PRX Quantum 2, 010308 (2021). arXiv:2004.14263.
https://doi.org/10.1103/PRXQuantum.2.010308
arXiv:2004.14263

[18] Federico Grasselli, Gláucia Murta, Hermann Kampermann, and Dagmar Bruß. ``Boosting device-independent cryptography with tripartite nonlocality''. Quantum 7, 980 (2023). arXiv:2209.12828.
https://doi.org/10.22331/q-2023-04-13-980
arXiv:2209.12828

[19] Erik Woodhead, Antonio Acín, and Stefano Pironio. ``Device-independent quantum key distribution with asymmetric CHSH inequalities''. Quantum 5, 443 (2021). arXiv:2007.16146.
https://doi.org/10.22331/q-2021-04-26-443
arXiv:2007.16146

[20] Jean-Daniel Bancal, Lana Sheridan, and Valerio Scarani. ``More randomness from the same data''. New Journal of Physics 16, 033011 (2014). arXiv:1309.3894.
https://doi.org/10.1088/1367-2630/16/3/033011
arXiv:1309.3894

[21] Olmo Nieto-Silleras, Stefano Pironio, and Jonathan Silman. ``Using complete measurement statistics for optimal device-independent randomness evaluation''. New Journal of Physics 16, 013035 (2014). arXiv:1309.3930.
https://doi.org/10.1088/1367-2630/16/1/013035
arXiv:1309.3930

[22] Robert König, Renato Renner, and Christian Schaffner. ``The operational meaning of min- and max-entropy''. IEEE Transactions on Information Theory 55, 4337–4347 (2009). arXiv:0807.1338.
https://doi.org/10.1109/TIT.2009.2025545
arXiv:0807.1338

[23] Miguel Navascués, Stefano Pironio, and Antonio Acín. ``Bounding the set of quantum correlations''. Physical Review Letters 98, 010401 (2007). arXiv:quant-ph/0607119.
https://doi.org/10.1103/PhysRevLett.98.010401
arXiv:quant-ph/0607119

[24] Stefano Pironio, Miguel Navascués, and Antonio Acín. ``Convergent relaxations of polynomial optimization problems with noncommuting variables''. SIAM Journal on Optimization 20, 2157–2180 (2010). arXiv:0903.4368.
https://doi.org/10.1137/090760155
arXiv:0903.4368

[25] 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, 158 (2021). arXiv:1908.11372.
https://doi.org/10.1038/s41534-021-00494-z
arXiv:1908.11372

[26] Peter Brown, Hamza Fawzi, and Omar Fawzi. ``Computing conditional entropies for quantum correlations''. Nature communications 12, 1–12 (2021). arXiv:2007.12575.
https://doi.org/10.1038/s41467-020-20018-1
arXiv:2007.12575

[27] Hideki Kosaki. ``Relative entropy of states: a variational expression''. Journal of Operator TheoryPages 335–348 (1986). url: https://www.jstor.org/stable/24714805.
https://www.jstor.org/stable/24714805

[28] Matthew J Donald. ``On the relative entropy''. Communications in Mathematical Physics 105, 13–34 (1986).
https://doi.org/10.1007/BF01212339

[29] Masanori Ohya and Dénes Petz. ``Quantum entropy and its use''. Springer Science & Business Media. (2004).

[30] Wen-Zhao Liu, Ming-Han Li, Sammy Ragy, Si-Ran Zhao, Bing Bai, Yang Liu, Peter J Brown, Jun Zhang, Roger Colbeck, Jingyun Fan, et al. ``Device-independent randomness expansion against quantum side information''. Nature PhysicsPages 1–4 (2021). arXiv:1912.11159.
https://doi.org/10.1038/s41567-020-01147-2
arXiv:1912.11159

[31] Lynden K Shalm, Yanbao Zhang, Joshua C Bienfang, Collin Schlager, Martin J Stevens, Michael D Mazurek, Carlos Abellán, Waldimar Amaya, Morgan W Mitchell, Mohammad A Alhejji, et al. ``Device-independent randomness expansion with entangled photons''. Nature PhysicsPages 1–5 (2021). arXiv:1912.11158.
https://doi.org/10.1038/s41567-020-01153-4
arXiv:1912.11158

[32] Oisín Faust and Hamza Fawzi. ``Rational approximations of operator monotone and operator convex functions'' (2023). arXiv:2305.12405.
arXiv:2305.12405

[33] Carl A. Miller and Yaoyun Shi. ``Universal security for randomness expansion from the spot-checking protocol''. Siam Journal of Computing 46, 1304–1335 (2017). arXiv:1411.6608.
https://doi.org/10.1137/15M1044333
arXiv:1411.6608

[34] 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). arXiv:quant-ph/0306078.
https://doi.org/10.1098/rspa.2004.1372
arXiv:quant-ph/0306078

[35] Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. ``MIP*= RE''. Communications of the ACM 64, 131–138 (2021). arXiv:2001.04383.
https://doi.org/10.1145/3485628
arXiv:2001.04383

[36] Rutvij Bhavsar, Sammy Ragy, and Roger Colbeck. ``Improved device-independent randomness expansion rates using two sided randomness''. New Journal of Physics 25, 093035 (2023). arXiv:2103.07504.
https://doi.org/10.1088/1367-2630/acf393
arXiv:2103.07504

[37] Peter J Brown, Sammy Ragy, and Roger Colbeck. ``A framework for quantum-secure device-independent randomness expansion''. IEEE Transactions on Information Theory 66, 2964–2987 (2019). arXiv:1810.13346.
https://doi.org/10.1109/TIT.2019.2960252
arXiv:1810.13346

[38] Peter Wittek. ``Algorithm 950: Ncpol2sdpa—sparse semidefinite programming relaxations for polynomial optimization problems of noncommuting variables''. ACM Transactions on Mathematical Software (TOMS) 41, 1–12 (2015). arXiv:1308.6029.
https://doi.org/10.1145/2699464
arXiv:1308.6029

[39] MOSEK ApS. ``The MOSEK optimization suite 10.1.11''. (2023). url: https://docs.mosek.com/latest/intro/index.html.
https://docs.mosek.com/latest/intro/index.html

[40] Peter Brown. ``Maintained fork of ncpol2sdpa''. https://github.com/peterjbrown519/ncpol2sdpa.
https://github.com/peterjbrown519/ncpol2sdpa

[41] Peter Brown. ``Example scripts for computing rates of device-independent protocols''. Available at https://github.com/peterjbrown519/DI-rates (2021).
https://github.com/peterjbrown519/DI-rates

[42] Gláucia Murta, Federico Grasselli, Hermann Kampermann, and Dagmar Bruß. ``Quantum conference key agreement: A review''. Advanced Quantum Technologies 3, 2000025 (2020). arXiv:2003.10186.
https://doi.org/10.1002/qute.202000025
arXiv:2003.10186

[43] Xiongfeng Ma and Norbert Lütkenhaus. ``Improved data post-processing in quantum key distribution and application to loss thresholds in device independent qkd''. Quantum Information & Computation 12, 203–214 (2012). arXiv:1109.1203.
arXiv:1109.1203

[44] M Ho, P Sekatski, EY-Z Tan, R Renner, J-D Bancal, and N Sangouard. ``Noisy preprocessing facilitates a photonic realization of device-independent quantum key distribution''. Physical Review Letters 124, 230502 (2020). arXiv:2005.13015.
https://doi.org/10.1103/PhysRevLett.124.230502
arXiv:2005.13015

[45] Wiesław Pusz and Stanisław L Woronowicz. ``Functional calculus for sesquilinear forms and the purification map''. Reports on Mathematical Physics 8, 159–170 (1975).
https://doi.org/10.1016/0034-4877(75)90061-0

[46] Wiesław Pusz and Stanisław L Woronowicz. ``Form convex functions and the WYDL and other inequalities''. Letters in Mathematical Physics 2, 505–512 (1978).
https://doi.org/10.1007/BF00398504

[47] Konrad Schmüdgen. ``Unbounded self-adjoint operators on Hilbert space''. Volume 265. Springer Science & Business Media. (2012).
https://doi.org/10.1007/978-94-007-4753-1

[48] Hisaharu Umegaki. ``Conditional expectation in an operator algebra, iv (entropy and information)''. In Kodai Mathematical Seminar Reports. Volume 14, pages 59–85. Department of Mathematics, Tokyo Institute of Technology (1962).
https://doi.org/10.2996/kmj/1138844604

[49] Edward G Effros. ``A matrix convexity approach to some celebrated quantum inequalities''. Proceedings of the National Academy of Sciences 106, 1006–1008 (2009). arXiv:0802.1234.
https://doi.org/10.1073/pnas.0807965106
arXiv:0802.1234

[50] Naresh Sharma. ``Equality conditions for quantum quasi-entropies under monotonicity and joint-convexity''. In 2014 Twentieth National Conference on Communications (NCC). Pages 1–6. (2014).
https://doi.org/10.1109/NCC.2014.6811277

[51] Dénes Petz. ``Quasi-entropies for finite quantum systems''. Reports on Mathematical Physics 23, 57–65 (1986).
https://doi.org/10.1016/0034-4877(86)90067-4

[52] Hamza Fawzi, James Saunderson, and Pablo A. Parrilo. ``Semidefinite approximations of the matrix logarithm''. Foundations of Computational Mathematics 19, 259––296 (2019). arXiv:1705.00812.
https://doi.org/10.1007/s10208-018-9385-0
arXiv:1705.00812

[53] Philip J Davis and Philip Rabinowitz. ``Methods of numerical integration''. Academic Press. (1984). 2nd edition.
https://doi.org/10.1016/C2013-0-10566-1

[54] Gene H Golub. ``Some modified matrix eigenvalue problems''. SIAM review 15, 318–334 (1973).
https://doi.org/10.1137/1015032

[55] Hamza Fawzi and Omar Fawzi. ``Semidefinite programming lower bounds on the squashed entanglement'' (2022). arXiv:2203.03394.
arXiv:2203.03394

[56] Flemming Topsøe. ``Some bounds for the logarithmic function''. Inequality theory and applications 4 (2007).

[57] Walter Gautschi. ``Gauss–Radau formulae for Jacobi and Laguerre weight functions''. Mathematics and Computers in Simulation 54, 403–412 (2000).
https://doi.org/10.1016/S0378-4754(00)00179-8

[58] Wei Zhang, Tim van Leent, Kai Redeker, Robert Garthoff, René Schwonnek, Florian Fertig, Sebastian Eppelt, Wenjamin Rosenfeld, Valerio Scarani, Charles C-W Lim, et al. ``A device-independent quantum key distribution system for distant users''. Nature 607, 687–691 (2022). arXiv:2110.00575.
https://doi.org/10.1038/s41586-022-04891-y
arXiv:2110.00575

[59] DP Nadlinger, P Drmota, BC Nichol, G Araneda, D Main, R Srinivas, DM Lucas, CJ Ballance, K Ivanov, EY-Z Tan, et al. ``Experimental quantum key distribution certified by Bell's theorem''. Nature 607, 682–686 (2022). arXiv:2109.14600.
https://doi.org/10.1038/s41586-022-04941-5
arXiv:2109.14600

[60] Wen-Zhao Liu, Yu-Zhe Zhang, Yi-Zheng Zhen, Ming-Han Li, Yang Liu, Jingyun Fan, Feihu Xu, Qiang Zhang, and Jian-Wei Pan. ``Toward a photonic demonstration of device-independent quantum key distribution''. Physical Review Letters 129, 050502 (2022). arXiv:2110.01480.
https://doi.org/10.1103/PhysRevLett.129.050502
arXiv:2110.01480

[61] Denis Rosset. ``Symdpoly: symmetry-adapted moment relaxations for noncommutative polynomial optimization'' (2018). arXiv:1808.09598.
arXiv:1808.09598

[62] Gert K Pedersen. ``On the operator equation HT+ TH= 2K''. Indiana University Mathematics Journal 25, 1029–1033 (1976). url: https://www.jstor.org/stable/24891356.
https://www.jstor.org/stable/24891356

[63] Timo Holz, Hermann Kampermann, and Dagmar Bruß. ``Genuine multipartite Bell inequality for device-independent conference key agreement''. Physical Review Research 2, 023251 (2020). arXiv:1910.11360.
https://doi.org/10.1103/PhysRevResearch.2.023251
arXiv:1910.11360

[64] Renato Renner. ``Security of quantum key distribution''. PhD thesis. Swiss Federal Institute of Technology, Zurich. (2005). arXiv:quant-ph/0512258.
arXiv:quant-ph/0512258

[65] Stephen Boyd and Lieven Vandenberghe. ``Convex optimization''. Cambridge university press. (2004).
https://doi.org/10.1017/CBO9780511804441

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[1] Hyejung H. Jee, Florian J. Curchod, and Mafalda L. Almeida, "More entropy from shorter experiments using polytope approximations to the quantum set", Quantum 10, 2019 (2026).

[2] Makoto Ishihara, Anthony Brendan, Wojciech Roga, Ulrik L. Andersen, and Masahiro Takeoka, "Long-distance device-independent quantum key distribution with standard optics tools", Optica Quantum 3 6, 535 (2025).

[3] Wenjie Wang, Mengyan Li, Fenzhuo Guo, Yukun Wang, and Fei Gao, "Self-testing positive operator-valued measures and certifying randomness", Quantum Information Processing 24 6, 157 (2025).

[4] Gereon Koßmann and Mark M. Wilde, "Semidefinite Optimization of the Quantum Relative Entropy of Channels", IEEE Transactions on Information Theory 72 4, 2378 (2026).

[5] Tristan Le Roy-Deloison, Edwin Peter Lobo, Jef Pauwels, and Stefano Pironio, "Device-Independent Quantum Key Distribution Based on Routed Bell Tests", PRX Quantum 6 2, 020311 (2025).

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[8] Hamid Arian Zad, Michal Jaščur, Asad Ali, Saif Al-Kuwari, and Saeed Haddadi, "Jahn-Teller effect for controlling quantum correlations in hexanuclear Fe$$^{3+}$$ magnets", Scientific Reports 15 1, 30381 (2025).

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[36] 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 7920, 682 (2022).

[37] 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 1, 10 (2023).

[38] Wen-Zhao Liu, Yu-Zhe Zhang, Yi-Zheng Zhen, Ming-Han Li, Yang Liu, Jingyun Fan, Feihu Xu, Qiang Zhang, and Jian-Wei Pan, "Toward a Photonic Demonstration of Device-Independent Quantum Key Distribution", Physical Review Letters 129 5, 050502 (2022).

[39] Renato Renner and Ramona Wolf, "Quantum Advantage in Cryptography", AIAA Journal 61 5, 1895 (2023).

[40] Tony Metger and Renato Renner, "Security of quantum key distribution from generalised entropy accumulation", Nature Communications 14, 5272 (2023).

[41] Lewis Wooltorton, Peter Brown, and Roger Colbeck, "Tight Analytic Bound on the Trade-Off between Device-Independent Randomness and Nonlocality", Physical Review Letters 129 15, 150403 (2022).

[42] 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 1, 158 (2021).

[43] Feihu Xu, Yu-Zhe Zhang, Qiang Zhang, and Jian-Wei Pan, "Device-Independent Quantum Key Distribution with Random Postselection", Physical Review Letters 128 11, 110506 (2022).

[44] 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).

[45] Piotr Mironowicz, "Semi-definite programming and quantum information", Journal of Physics A Mathematical General 57 16, 163002 (2024).

[46] Eva M. González-Ruiz, Javier Rivera-Dean, Marina F. B. Cenni, Anders S. Sørensen, Antonio Acín, and Enky Oudot, "Device-independent quantum key distribution with realistic single-photon source implementations", Optics Express 32 8, 13181 (2024).

[47] Marie Ioannou, Pavel Sekatski, Alastair A. Abbott, Denis Rosset, Jean-Daniel Bancal, and Nicolas Brunner, "Receiver-device-independent quantum key distribution protocols", New Journal of Physics 24 6, 063006 (2022).

[48] Gereon Koßmann and René Schwonnek, "Reliable Entropy Estimation from Observed Statistics for Device-Independent Quantum Cryptography", arXiv:2411.04858, (2024).

[49] Rutvij Bhavsar, Sammy Ragy, and Roger Colbeck, "Improved device-independent randomness expansion rates using two sided randomness", New Journal of Physics 25 9, 093035 (2023).

[50] Mateus Araújo, Marcus Huber, Miguel Navascués, Matej Pivoluska, and Armin Tavakoli, "Quantum key distribution rates from semidefinite programming", Quantum 7, 1019 (2023).

[51] Alban Jean-Marie Seguinard, Amélie Piveteau, Piotr Mironowicz, and Mohamed Bourennane, "Experimental certification of more than one bit of quantum randomness in the two inputs and two outputs scenario", New Journal of Physics 25 11, 113022 (2023).

[52] Kerry He, James Saunderson, and Hamza Fawzi, "Interior Point Methods for Structured Quantum Relative Entropy Optimization Problems", arXiv:2407.00241, (2024).

[53] Lewis Wooltorton, Peter Brown, and Roger Colbeck, "Device-Independent Quantum Key Distribution with Arbitrarily Small Nonlocality", Physical Review Letters 132 21, 210802 (2024).

[54] Victor Barizien, Pavel Sekatski, and Jean-Daniel Bancal, "Custom Bell inequalities from formal sums of squares", Quantum 8, 1333 (2024).

[55] Gereon Koßmann and Mark M. Wilde, "Semidefinite optimization of the quantum relative entropy of channels", arXiv:2410.16362, (2024).

[56] Yuchen Lu and Kun Fang, "Estimating quantum relative entropies on quantum computers", arXiv:2501.07292, (2025).

[57] Chao Wang, Ignatius William Primaatmaja, Hong Jie Ng, Jing Yan Haw, Raymond Ho, Jianran Zhang, Gong Zhang, and Charles Lim, "Provably-secure quantum randomness expansion with uncharacterised homodyne detection", Nature Communications 14, 316 (2023).

[58] Andreas Fyrillas, Boris Bourdoncle, Alexandre Maïnos, Pierre-Emmanuel Emeriau, Kayleigh Start, Nico Margaria, Martina Morassi, Aristide Lemaître, Isabelle Sagnes, Petr Stepanov, Thi Huong Au, Sébastien Boissier, Niccolo Somaschi, Nicolas Maring, Nadia Belabas, and Shane Mansfield, "Certified Randomness in Tight Space", PRX Quantum 5 2, 020348 (2024).

[59] Piotr Mironowicz, Marcus Grünfeld, and Mohamed Bourennane, "Generalized measurements on qubits in quantum randomness certification and expansion", Physical Review Applied 22 4, 044041 (2024).

[60] Michele Masini, Stefano Pironio, and Erik Woodhead, "Simple and practical DIQKD security analysis via BB84-type uncertainty relations and Pauli correlation constraints", Quantum 6, 843 (2022).

[61] Karol Łukanowski, Maria Balanzó-Juandó, Máté Farkas, Antonio Acín, and Jan Kołodyński, "Upper bounds on key rates in device-independent quantum key distribution based on convex-combination attacks", Quantum 7, 1199 (2023).

[62] Giuseppe Viola, Nikolai Miklin, Mariami Gachechiladze, and Marcin Pawłowski, "Entanglement witnessing with untrusted detectors", Journal of Physics A Mathematical General 56 42, 425301 (2023).

[63] Rutvij Bhavsar, Hamid Tebyanian, and Roger Colbeck, "Higher rates for semi-device-independent randomness expansion by recycling input randomness", arXiv:2604.06457, (2026).

[64] Chun-Neng Chu, Wei-Fu Tseng, and Yen-Huan Li, "A Linearly Convergent Algorithm for Computing the Petz-Augustin Mean", arXiv:2502.06399, (2025).

[65] Andrew J. P. Garner and Mateus Araújo, "Introducing Moment: A toolkit for semi-definite programming with moment matrices", arXiv:2406.15559, (2024).

[66] 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", Physical Review Letters 131 8, 080801 (2023).

[67] Ignatius William Primaatmaja, Wen Yu Kon, and Charles Lim, "Discrete-modulated continuous-variable quantum key distribution secure against general attacks", arXiv:2409.02630, (2024).

[68] Hong-Yi Su, "Monte Carlo approach to the evaluation of the security of device-independent quantum key distribution", New Journal of Physics 25 12, 123036 (2023).

[69] Michele Masini and Shubhayan Sarkar, "One-sided DI-QKD secure against coherent attacks over long distances", arXiv:2403.11850, (2024).

[70] X. Valcarce, P. Sekatski, E. Gouzien, A. Melnikov, and N. Sangouard, "Automated design of quantum-optical experiments for device-independent quantum key distribution", Physical Review A 107 6, 062607 (2023).

[71] Máté Farkas, "Unbounded Device-Independent Quantum Key Rates from Arbitrarily Small Nonlocality", Physical Review Letters 132 21, 210803 (2024).

[72] Javier Rivera-Dean, Anna Steffinlongo, Neil Parker-Sánchez, Antonio Acín, and Enky Oudot, "Device-Independent Quantum Key Distribution beyond qubits", arXiv:2402.00161, (2024).

[73] Mateus Araújo, Andrew J. P. Garner, and Miguel Navascues, "Non-commutative optimization problems with differential constraints", arXiv:2408.02572, (2024).

[74] Ignatius William Primaatmaja, Hong Jie Ng, and Koon Tong Goh, "Device-Independent Private Quantum Randomness Beacon", arXiv:2507.09963, (2025).

[75] José Jesus and Emmanuel Zambrini Cruzeiro, "Tight Bell inequalities from polytope slices", Physical Review A 108 5, 052220 (2023).

[76] Hamza Fawzi and Omar Fawzi, "Semidefinite programming lower bounds on the squashed entanglement", arXiv:2203.03394, (2022).

[77] Ernest Y.-Z. Tan, "Prospects for device-independent quantum key distribution", arXiv:2111.11769, (2021).

[78] Ravishankar Ramanathan, Yuan Liu, Yutian Wu, and Stefano Pironio, "No Bound Randomness in Quantum Nonlocality", arXiv:2509.08623, (2025).

[79] Yu-Zhe Zhang, Yi-Zheng Zhen, and Feihu Xu, "Upper bound on device-independent quantum key distribution with two way classical postprocessing under individual attack", New Journal of Physics 24 11, 113045 (2022).

[80] Oisín Faust and Hamza Fawzi, "Rational approximations of operator monotone and operator convex functions", arXiv:2305.12405, (2023).

[81] Martin Sandfuchs and Ramona Wolf, "Coherent attacks are stronger than collective attacks on DIQKD with random postselection", arXiv:2306.07364, (2023).

[82] Ernest Y.-Z. Tan, "Robustness of implemented device-independent protocols against constrained leakage", arXiv:2302.13928, (2023).

[83] Sarnava Datta, Hermann Kampermann, and Dagmar Bruß, "Device-independent secret key rates via a postselected Bell inequality", Physical Review A 105 3, 032451 (2022).

[84] Corentin Lanore, Xavier Valcarce, Jean Etesse, Anthony Martin, and Jean-Daniel Bancal, "Towards Device-Independent Quantum Key Distribution with Photonic Devices", arXiv:2601.14373, (2026).

[85] Chun-Yu Chen, Kai-Siang Chen, Kai-Min Chung, Min-Hsiu Hsieh, Yeong-Cherng Liang, and Gelo Noel M. Tabia, "Incorporating Zero-Probability Constraints to Device-Independent Randomness Expansion", arXiv:2401.08452, (2024).

[86] Syed M Arslan, Muhammad T Rahim, Asad Ali, Hashir Kuniyil, and Saif Al Kuwari, "Noise Resilient 1SDIQKD for Practical Quantum Networks", arXiv:2602.00916, (2026).

[87] Rutvij Bhavsar, Junguk Moon, and Joonwoo Bae, "Entropy Bounds via Hypothesis Testing and Its Applications to Two-Way Key Distillation in Quantum Cryptography", arXiv:2602.05870, (2026).

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