An Operational Framework for Nonclassicality in Quantum Communication Networks

Summarize this article with:
AbstractQuantum resources, such as entanglement or quantum communication, offer significant communication advantages in information processing. We develop an operational framework for realizing these communication advantages in resource-constrained quantum networks. The framework computes linear bounds on the input/output probabilities of classical networks with limited communication and globally shared randomness. Since the violation of these classical bounds witnesses nonclassicality, a measurable communication advantage, the framework maximizes the violation of the classical bound using variational quantum optimization methods tailored to the communication network and quantum resources. This operational framework for nonclassicality can be scaled on quantum computers or deployed in the field to optimize noisy quantum networks for communication advantages. Applying this framework, we investigate the nonclassicality of communication networks that are assisted by quantum resources. We find that entanglement between communication-constrained parties is sufficient for nonclassicality to be found, whereas in networks with multiple senders, quantum communication with no entanglement-assistance is sufficient for nonclassicality to be found. As a result, entanglement is necessary for nonclassicality when a single sender broadcasts to multiple receivers.Featured image: A classical processor optimizes quantum network hardware for a particular information processing task or demonstration of nonclassicality.Popular summaryQuantum communication resources, such as entanglement, can improve the information processing capabilities of communication networks. These communication advantages are directly related to a measurable property known as nonclassicality. In this work, we develop an approach for realizing nonclassicality by using a classical processor to optimize the communication advantage of simulated or real-world network. Our methods are compatible with quantum processors, making them scalable to large quantum networks and extensible to quantum networking hardware. We apply our operational framework for nonclassicality in a broad survey over all basic multiparty communication scenarios. In all cases, we find that entanglement is sufficient for communication advantage. When no entanglement is present, advantages can be realized if the network contains multiple senders who are equipped with point-to-point quantum communication. Furthermore, we prove that entanglement is necessary for communication advantage when a network contains a single sender who broadcasts to multiple receivers. In the future, our framework can be used optimize protocols or automate tasks in real-world quantum networks, helping to solve practical challenges with noisy quantum hardware.► BibTeX data@article{Doolittle2026operational, doi = {10.22331/q-2026-04-08-2052}, url = {https://doi.org/10.22331/q-2026-04-08-2052}, title = {An {O}perational {F}ramework for {N}onclassicality in {Q}uantum {C}ommunication {N}etworks}, author = {Doolittle, Brian and Leditzky, Felix and Chitambar, Eric}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2052}, month = apr, year = {2026} }► References [1] H. J. Kimble. The quantum internet. Nature, 453 (7198): 1023–1030, June 2008. 10.1038/nature07127. https://doi.org/10.1038/nature07127 [2] Rodney Van Meter. Quantum networking and internetworking. IEEE Network, 26 (4): 59–64, 2012. 10.1109/MNET.2012.6246754. https://doi.org/10.1109/MNET.2012.6246754 [3] Stephanie Wehner, David Elkouss, and Ronald Hanson. Quantum internet: A vision for the road ahead. Science, 362 (6412), 2018. 10.1126/science.aam9288. https://doi.org/10.1126/science.aam9288 [4] Amoldeep Singh, Kapal Dev, Harun Siljak, Hem Dutt Joshi, and Maurizio Magarini. Quantum internet—applications, functionalities, enabling technologies, challenges, and research directions. IEEE Communications Surveys & Tutorials, 23 (4): 2218–2247, 2021. 10.1109/COMST.2021.3109944. https://doi.org/10.1109/COMST.2021.3109944 [5] Gilles Brassard. Quantum communication complexity. Foundations of Physics, 33 (11): 1593–1616, 2003. ISSN 0015-9018. 10.1023/a:1026009100467. https://doi.org/10.1023/a:1026009100467 [6] Harry Buhrman, Richard Cleve, Serge Massar, and Ronald de Wolf. Nonlocality and communication complexity. Rev. Mod. Phys., 82: 665–698, Mar 2010. 10.1103/RevModPhys.82.665. https://doi.org/10.1103/RevModPhys.82.665 [7] C.H. Bennett, P.W. Shor, J.A. Smolin, and A.V. Thapliyal. Entanglement-assisted capacity of a quantum channel and the reverse shannon theorem. IEEE Transactions on Information Theory, 48 (10): 2637–2655, October 2002. 10.1109/TIT.2002.802612. https://doi.org/10.1109/TIT.2002.802612 [8] Andreas Winter. Compression of sources of probability distributions and density operators. arXiv preprint, 2002. 10.48550/arXiv.quant-ph/0208131. https://doi.org/10.48550/arXiv.quant-ph/0208131 arXiv:quant-ph/0208131 [9] Charles H. Bennett, Igor Devetak, Aram W. Harrow, Peter W. Shor, and Andreas Winter. The quantum reverse shannon theorem and resource tradeoffs for simulating quantum channels. IEEE Transactions on Information Theory, 60 (5): 2926–2959, 2014. 10.1109/TIT.2014.2309968. https://doi.org/10.1109/TIT.2014.2309968 [10] Toby S. Cubitt, Debbie Leung, William Matthews, and Andreas Winter. Zero-error channel capacity and simulation assisted by non-local correlations. IEEE Transactions on Information Theory, 57 (8): 5509–5523, 2011. 10.1109/TIT.2011.2159047. https://doi.org/10.1109/TIT.2011.2159047 [11] Alexander Semenovich Holevo. Bounds for the quantity of information transmitted by a quantum communication channel.
Problemy Peredachi Informatsii, 9 (3): 3–11, 1973. URL https://www.mathnet.ru/eng/ppi903. https://www.mathnet.ru/eng/ppi903 [12] Péter E. Frenkel and Mihály Weiner. Classical information storage in an n-level quantum system. Communications in Mathematical Physics, 340 (2): 563–574, September 2015. 10.1007/s00220-015-2463-0. https://doi.org/10.1007/s00220-015-2463-0 [13] Joseph Bowles, Nicolas Brunner, and Marcin Pawłowski. Testing dimension and nonclassicality in communication networks. Phys. Rev. A, 92: 022351, Aug 2015. 10.1103/PhysRevA.92.022351. https://doi.org/10.1103/PhysRevA.92.022351 [14] John S Bell. On the Einstein Podolsky Rosen paradox.
Physics Physique Fizika, 1 (3): 195, 1964. 10.1103/PhysicsPhysiqueFizika.1.195. https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195 [15] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. Bell nonlocality. Rev. Mod. Phys., 86: 419–478, Apr 2014. 10.1103/RevModPhys.86.419. https://doi.org/10.1103/RevModPhys.86.419 [16] Serge Massar, Dave Bacon, Nicolas J. Cerf, and Richard Cleve. Classical simulation of quantum entanglement without local hidden variables. Phys. Rev. A, 63: 052305, Apr 2001. 10.1103/PhysRevA.63.052305. https://doi.org/10.1103/PhysRevA.63.052305 [17] D. Bacon and B. F. Toner. Bell inequalities with auxiliary communication. Phys. Rev. Lett., 90: 157904, Apr 2003. 10.1103/PhysRevLett.90.157904. https://doi.org/10.1103/PhysRevLett.90.157904 [18] B. F. Toner and D. Bacon. Communication cost of simulating bell correlations. Phys. Rev. Lett., 91: 187904, Oct 2003. 10.1103/PhysRevLett.91.187904. https://doi.org/10.1103/PhysRevLett.91.187904 [19] Oded Regev and Ben Toner. Simulating quantum correlations with finite communication. SIAM Journal on Computing, 39 (4): 1562–1580, 2010. 10.1137/080723909. https://doi.org/10.1137/080723909 [20] Katherine Maxwell and Eric Chitambar. Bell inequalities with communication assistance. Phys. Rev. A, 89: 042108, Apr 2014. 10.1103/PhysRevA.89.042108. https://doi.org/10.1103/PhysRevA.89.042108 [21] J B Brask and R Chaves. Bell scenarios with communication. Journal of Physics A: Mathematical and Theoretical, 50 (9): 094001, January 2017. ISSN 1751-8121. 10.1088/1751-8121/aa5840. https://doi.org/10.1088/1751-8121/aa5840 [22] Emmanuel Zambrini Cruzeiro and Nicolas Gisin. Bell inequalities with one bit of communication. Entropy, 21 (2): 171, February 2019. ISSN 1099-4300. 10.3390/e21020171. https://doi.org/10.3390/e21020171 [23] Mir Alimuddin, Ananya Chakraborty, Govind Lal Sidhardh, Ram Krishna Patra, Samrat Sen, Snehasish Roy Chowdhury, Sahil Gopalkrishna Naik, and Manik Banik. Advantage of hardy's nonlocal correlation in reverse zero-error channel coding. Phys. Rev. A, 108: 052430, Nov 2023. 10.1103/PhysRevA.108.052430. https://doi.org/10.1103/PhysRevA.108.052430 [24] Péter E. Frenkel and Mihály Weiner. On entanglement assistance to a noiseless classical channel. Quantum, 6: 662, March 2022. ISSN 2521-327X. 10.22331/q-2022-03-01-662. https://doi.org/10.22331/q-2022-03-01-662 [25] Michele Dall'Arno, Sarah Brandsen, Alessandro Tosini, Francesco Buscemi, and Vlatko Vedral. No-hypersignaling principle. Phys. Rev. Lett., 119: 020401, Jul 2017. 10.1103/PhysRevLett.119.020401. https://doi.org/10.1103/PhysRevLett.119.020401 [26] Teiko Heinosaari and Oskari Kerppo. Communication of partial ignorance with qubits. Journal of Physics A: Mathematical and Theoretical, 52 (39): 395301, September 2019. ISSN 1751-8121. 10.1088/1751-8121/ab3ae4. https://doi.org/10.1088/1751-8121/ab3ae4 [27] Teiko Heinosaari, Oskari Kerppo, and Leevi Leppäjärvi. Communication tasks in operational theories. Journal of Physics A: Mathematical and Theoretical, 53 (43): 435302, October 2020. ISSN 1751-8121. 10.1088/1751-8121/abb5dc. https://doi.org/10.1088/1751-8121/abb5dc [28] Davide Poderini, Samuraí Brito, Ranieri Nery, Fabio Sciarrino, and Rafael Chaves. Criteria for nonclassicality in the prepare-and-measure scenario. Phys. Rev. Res., 2: 043106, Oct 2020. 10.1103/PhysRevResearch.2.043106. https://doi.org/10.1103/PhysRevResearch.2.043106 [29] Brian Doolittle and Eric Chitambar. Certifying the classical simulation cost of a quantum channel. Phys. Rev. Res., 3: 043073, Oct 2021. 10.1103/PhysRevResearch.3.043073. https://doi.org/10.1103/PhysRevResearch.3.043073 [30] Armin Tavakoli, Jef Pauwels, Erik Woodhead, and Stefano Pironio. Correlations in entanglement-assisted prepare-and-measure scenarios. PRX Quantum, 2: 040357, Dec 2021. 10.1103/PRXQuantum.2.040357. https://doi.org/10.1103/PRXQuantum.2.040357 [31] Martin J. Renner, Armin Tavakoli, and Marco Túlio Quintino. Classical cost of transmitting a qubit. Phys. Rev. Lett., 130: 120801, Mar 2023. 10.1103/PhysRevLett.130.120801. https://doi.org/10.1103/PhysRevLett.130.120801 [32] Andrzej Grudka, Michał Horodecki, Ryszard Horodecki, and Antoni Wójcik. Nonsignaling quantum random access-code boxes. Phys. Rev. A, 92: 052312, Nov 2015. 10.1103/PhysRevA.92.052312. https://doi.org/10.1103/PhysRevA.92.052312 [33] Armin Tavakoli, Alley Hameedi, Breno Marques, and Mohamed Bourennane. Quantum random access codes using single $d$-level systems. Phys. Rev. Lett., 114: 170502, Apr 2015. 10.1103/PhysRevLett.114.170502. https://doi.org/10.1103/PhysRevLett.114.170502 [34] Teiko Heinosaari and Leevi Leppäjärvi. Random access test as an identifier of nonclassicality. Journal of Physics A: Mathematical and Theoretical, 55 (17): 174003, April 2022. ISSN 1751-8121. 10.1088/1751-8121/ac5b91. https://doi.org/10.1088/1751-8121/ac5b91 [35] Amélie Piveteau, Jef Pauwels, Emil Håkansson, Sadiq Muhammad, Mohamed Bourennane, and Armin Tavakoli. Entanglement-assisted quantum communication with simple measurements. Nature Communications, 13 (1), December 2022. ISSN 2041-1723. 10.1038/s41467-022-33922-5. https://doi.org/10.1038/s41467-022-33922-5 [36] Nitica Sakharwade, Michał Studziński, Michał Eckstein, and Paweł Horodecki. Two instances of random access code in the quantum regime. New Journal of Physics, 25 (5): 053038, May 2023. ISSN 1367-2630. 10.1088/1367-2630/acd716. https://doi.org/10.1088/1367-2630/acd716 [37] Pedro Lauand, Davide Poderini, Ranieri Nery, George Moreno, Lucas Pollyceno, Rafael Rabelo, and Rafael Chaves. Witnessing nonclassicality in a causal structure with three observable variables. PRX Quantum, 4: 020311, Apr 2023. 10.1103/PRXQuantum.4.020311. https://doi.org/10.1103/PRXQuantum.4.020311 [38] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles. Variational quantum algorithms.
Nature Reviews Physics, 3 (9): 625–644, August 2021. ISSN 2522-5820. 10.1038/s42254-021-00348-9. https://doi.org/10.1038/s42254-021-00348-9 [39] Brian Doolittle, Thomas R. Bromley, Nathan Killoran, and Eric Chitambar. Variational quantum optimization of nonlocality in noisy quantum networks. IEEE Transactions on Quantum Engineering, pages 1–28, 2023. 10.1109/TQE.2023.3243849. https://doi.org/10.1109/TQE.2023.3243849 [40] Brian Doolittle and Tom Bromley. qnetvo: the quantum network variational optimizer. March 2022. 10.5281/zenodo.6345834. URL https://github.com/ChitambarLab/qNetVO. https://doi.org/10.5281/zenodo.6345834 https://github.com/ChitambarLab/qNetVO [41] Brian Doolittle. Nonclassicality in Noisy Quantum Networks. Phd thesis, University of Illinois Urbana-Champaign, September 2023. URL https://hdl.handle.net/2142/121945. https://hdl.handle.net/2142/121945 [42] Daniel T. Chen, Brian Doolittle, Jeffrey Larson, Zain H. Saleem, and Eric Chitambar. Inferring quantum network topology using local measurements. PRX Quantum, 4: 040347, Dec 2023. 10.1103/PRXQuantum.4.040347. https://doi.org/10.1103/PRXQuantum.4.040347 [43] Ziqi Ma, Pranav Gokhale, Tian-Xing Zheng, Sisi Zhou, Xiaofei Yu, Liang Jiang, Peter Maurer, and Frederic T. Chong. Adaptive circuit learning for quantum metrology. In 2021 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 419–430, 2021. 10.1109/QCE52317.2021.00063. https://doi.org/10.1109/QCE52317.2021.00063 [44] Teiko Heinosaari, Oskari Kerppo, Leevi Leppäjärvi, and Martin Plávala. Simple information-processing tasks with unbounded quantum advantage. Phys. Rev. A, 109: 032627, Mar 2024. 10.1103/PhysRevA.109.032627. https://doi.org/10.1103/PhysRevA.109.032627 [45] Armin Tavakoli, Alejandro Pozas-Kerstjens, Ming-Xing Luo, and Marc-Olivier Renou. Bell nonlocality in networks. Reports on Progress in Physics, 85 (5): 056001, Mar 2022. ISSN 1361-6633. 10.1088/1361-6633/ac41bb. https://doi.org/10.1088/1361-6633/ac41bb [46] S. G. A. Brito, B. Amaral, and R. Chaves. Quantifying bell nonlocality with the trace distance. Phys. Rev. A, 97: 022111, Feb 2018. 10.1103/PhysRevA.97.022111. https://doi.org/10.1103/PhysRevA.97.022111 [47] G.M. Ziegler. Lectures on Polytopes. Graduate Texts in Mathematics.
Springer New York, 2012. 10.1007/978-1-4613-8431-1. https://doi.org/10.1007/978-1-4613-8431-1 [48] Denis Rosset, Jean-Daniel Bancal, and Nicolas Gisin. Classifying 50 years of bell inequalities. Journal of Physics A: Mathematical and Theoretical, 47 (42): 424022, October 2014. 10.1088/1751-8113/47/42/424022. https://doi.org/10.1088/1751-8113/47/42/424022 [49] Thomas Christof and Andreas Löbel. Porta, 1997. URL http://porta.zib.de/. http://porta.zib.de/ [50] Brian Doolittle. https://github.com/ChitambarLab/nonclassicality-in-quantum-communication-networks-supplemental-code (v0.2.0). Feb 2024. 10.5281/zenodo.10780789. https://doi.org/10.5281/zenodo.10780789 https://github.com/ChitambarLab/nonclassicality-in-quantum-communication-networks-supplemental-code [51] Rajarshi Pal and Sibasish Ghosh. Non-locality breaking qubit channels: the case for chsh inequality. Journal of Physics A: Mathematical and Theoretical, 48 (15): 155302, mar 2015. 10.1088/1751-8113/48/15/155302. https://doi.org/10.1088/1751-8113/48/15/155302 [52] Yujie Zhang, Rodrigo Araiza Bravo, Virginia O Lorenz, and Eric Chitambar. Channel activation of chsh nonlocality. New Journal of Physics, 22 (4): 043003, apr 2020. 10.1088/1367-2630/ab7bef. https://doi.org/10.1088/1367-2630/ab7bef [53] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, Shahnawaz Ahmed, Vishnu Ajith, M Sohaib Alam, Guillermo Alonso-Linaje, B AkashNarayanan, Ali Asadi, et al. Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint, 2018. 10.48550/arXiv.1811.04968. https://doi.org/10.48550/arXiv.1811.04968 [54] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information.
Cambridge University Press, 2009. 10.1017/cbo9780511976667. https://doi.org/10.1017/cbo9780511976667 [55] David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Learning representations by back-propagating errors. Nature, 323 (6088): 533–536, 1986. https://doi.org/10.1038/323533a0. https://doi.org/10.1038/323533a0 [56] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Phys. Rev. A, 99: 032331, Mar 2019. 10.1103/PhysRevA.99.032331. https://doi.org/10.1103/PhysRevA.99.032331 [57] Andrea Mari, Thomas R. Bromley, and Nathan Killoran. Estimating the gradient and higher-order derivatives on quantum hardware. Phys. Rev. A, 103: 012405, Jan 2021. 10.1103/PhysRevA.103.012405. https://doi.org/10.1103/PhysRevA.103.012405 [58] David Wierichs, Josh Izaac, Cody Wang, and Cedric Yen-Yu Lin. General parameter-shift rules for quantum gradients. Quantum, 6: 677, March 2022. 10.22331/q-2022-03-30-677. https://doi.org/10.22331/q-2022-03-30-677 [59] Oleksandr Kyriienko and Vincent E. Elfving. Generalized quantum circuit differentiation rules. Phys. Rev. A, 104: 052417, Nov 2021. 10.1103/PhysRevA.104.052417. https://doi.org/10.1103/PhysRevA.104.052417 [60] Sebastian Ruder. An overview of gradient descent optimization algorithms. arXiv preprint, 2016. 10.48550/arXiv.1609.04747. https://doi.org/10.48550/arXiv.1609.04747 [61] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint, 2014. 10.48550/arXiv.1412.6980. https://doi.org/10.48550/arXiv.1412.6980 [62] Lennart Bittel and Martin Kliesch. Training variational quantum algorithms is np-hard. Phys. Rev. Lett., 127: 120502, Sep 2021. 10.1103/PhysRevLett.127.120502. https://doi.org/10.1103/PhysRevLett.127.120502 [63] Charles H. Bennett and Stephen J. Wiesner. Communication via one- and two-particle operators on einstein-podolsky-rosen states. Phys. Rev. Lett., 69: 2881–2884, Nov 1992. 10.1103/PhysRevLett.69.2881. https://doi.org/10.1103/PhysRevLett.69.2881 [64] Eric Chitambar, Ian George, Brian Doolittle, and Marius Junge. The communication value of a quantum channel. IEEE Transactions on Information Theory, 69 (3): 1660–1679, 2023. 10.1109/TIT.2022.3218540. https://doi.org/10.1109/TIT.2022.3218540 [65] Andris Ambainis, Debbie Leung, Laura Mancinska, and Maris Ozols. Quantum random access codes with shared randomness. arXiv preprint, 2008. 10.48550/arXiv.0810.2937. https://doi.org/10.48550/arXiv.0810.2937 [66] Rodrigo Gallego, Nicolas Brunner, Christopher Hadley, and Antonio Acín. Device-independent tests of classical and quantum dimensions. Phys. Rev. Lett., 105: 230501, Nov 2010. 10.1103/PhysRevLett.105.230501. https://doi.org/10.1103/PhysRevLett.105.230501 [67] Marcin Pawłowski and Andreas Winter. ``hyperbits'': The information quasiparticles. Phys. Rev. A, 85: 022331, Feb 2012. 10.1103/PhysRevA.85.022331. https://doi.org/10.1103/PhysRevA.85.022331 [68] Jef Pauwels, Stefano Pironio, Emmanuel Zambrini Cruzeiro, and Armin Tavakoli. Adaptive advantage in entanglement-assisted communications. Phys. Rev. Lett., 129: 120504, Sep 2022. 10.1103/PhysRevLett.129.120504. https://doi.org/10.1103/PhysRevLett.129.120504 [69] Marcin Pawłowski and Nicolas Brunner. Semi-device-independent security of one-way quantum key distribution. Phys. Rev. A, 84: 010302, Jul 2011. 10.1103/PhysRevA.84.010302. https://doi.org/10.1103/PhysRevA.84.010302 [70] Hong-Wei Li, Zhen-Qiang Yin, Yu-Chun Wu, Xu-Bo Zou, Shuang Wang, Wei Chen, Guang-Can Guo, and Zheng-Fu Han. Semi-device-independent random-number expansion without entanglement. Phys. Rev. A, 84: 034301, Sep 2011. 10.1103/PhysRevA.84.034301. https://doi.org/10.1103/PhysRevA.84.034301 [71] George Moreno, Ranieri Nery, Carlos de Gois, Rafael Rabelo, and Rafael Chaves. Semi-device-independent certification of entanglement in superdense coding. Phys. Rev. A, 103: 022426, Feb 2021. 10.1103/PhysRevA.103.022426. https://doi.org/10.1103/PhysRevA.103.022426 [72] Armin Tavakoli, Jędrzej Kaniewski, Tamás Vértesi, Denis Rosset, and Nicolas Brunner. Self-testing quantum states and measurements in the prepare-and-measure scenario. Phys. Rev. A, 98: 062307, Dec 2018. 10.1103/PhysRevA.98.062307. https://doi.org/10.1103/PhysRevA.98.062307 [73] Sandu Popescu and Daniel Rohrlich. Quantum nonlocality as an axiom. Foundations of Physics, 24 (3): 379–385, March 1994. ISSN 1572-9516. 10.1007/bf02058098. https://doi.org/10.1007/bf02058098 [74] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23: 880–884, Oct 1969. 10.1103/PhysRevLett.23.880. https://doi.org/10.1103/PhysRevLett.23.880 [75] Brian Doolittle. https://github.com/ChitambarLab/BellScenario.jl (v0.1.3). 2020. 10.5281/zenodo.10277572. https://doi.org/10.5281/zenodo.10277572 https://github.com/ChitambarLab/BellScenario.jl [76] Brian Doolittle and Benoı̂t Legat. https://github.com/JuliaPolyhedra/XPORTA.jl (v0.1.3), 2020. https://github.com/JuliaPolyhedra/XPORTA.jl [77] Benoı̂t Legat, Robin Deits, Gustavo Goretkin, Twan Koolen, Joey Huchette, Daisuke Oyama, and Marcelo Forets. https://github.com/JuliaPolyhedra/Polyhedra.jl (v0.6.16). 2021. 10.5281/zenodo.1214290. https://doi.org/10.5281/zenodo.1214290 https://github.com/JuliaPolyhedra/Polyhedra.jl [78] Qi Huangfu and JA Julian Hall. Parallelizing the dual revised simplex method.
Mathematical Programming Computation, 10 (1): 119–142, 2018. 10.1007/s12532-017-0130-5. https://doi.org/10.1007/s12532-017-0130-5 [79] Iain Dunning, Joey Huchette, and Miles Lubin. Jump: A modeling language for mathematical optimization. SIAM Review, 59 (2): 295–320, 2017. 10.1137/15M1020575. https://doi.org/10.1137/15M1020575Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-08 08:17:16: Could not fetch cited-by data for 10.22331/q-2026-04-08-2052 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-08 08:17:16: 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 resources, such as entanglement or quantum communication, offer significant communication advantages in information processing. We develop an operational framework for realizing these communication advantages in resource-constrained quantum networks. The framework computes linear bounds on the input/output probabilities of classical networks with limited communication and globally shared randomness. Since the violation of these classical bounds witnesses nonclassicality, a measurable communication advantage, the framework maximizes the violation of the classical bound using variational quantum optimization methods tailored to the communication network and quantum resources. This operational framework for nonclassicality can be scaled on quantum computers or deployed in the field to optimize noisy quantum networks for communication advantages. Applying this framework, we investigate the nonclassicality of communication networks that are assisted by quantum resources. We find that entanglement between communication-constrained parties is sufficient for nonclassicality to be found, whereas in networks with multiple senders, quantum communication with no entanglement-assistance is sufficient for nonclassicality to be found. As a result, entanglement is necessary for nonclassicality when a single sender broadcasts to multiple receivers.Featured image: A classical processor optimizes quantum network hardware for a particular information processing task or demonstration of nonclassicality.Popular summaryQuantum communication resources, such as entanglement, can improve the information processing capabilities of communication networks. These communication advantages are directly related to a measurable property known as nonclassicality. In this work, we develop an approach for realizing nonclassicality by using a classical processor to optimize the communication advantage of simulated or real-world network. Our methods are compatible with quantum processors, making them scalable to large quantum networks and extensible to quantum networking hardware. We apply our operational framework for nonclassicality in a broad survey over all basic multiparty communication scenarios. In all cases, we find that entanglement is sufficient for communication advantage. When no entanglement is present, advantages can be realized if the network contains multiple senders who are equipped with point-to-point quantum communication. Furthermore, we prove that entanglement is necessary for communication advantage when a network contains a single sender who broadcasts to multiple receivers. In the future, our framework can be used optimize protocols or automate tasks in real-world quantum networks, helping to solve practical challenges with noisy quantum hardware.► BibTeX data@article{Doolittle2026operational, doi = {10.22331/q-2026-04-08-2052}, url = {https://doi.org/10.22331/q-2026-04-08-2052}, title = {An {O}perational {F}ramework for {N}onclassicality in {Q}uantum {C}ommunication {N}etworks}, author = {Doolittle, Brian and Leditzky, Felix and Chitambar, Eric}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2052}, month = apr, year = {2026} }► References [1] H. J. Kimble. The quantum internet. Nature, 453 (7198): 1023–1030, June 2008. 10.1038/nature07127. https://doi.org/10.1038/nature07127 [2] Rodney Van Meter. Quantum networking and internetworking. IEEE Network, 26 (4): 59–64, 2012. 10.1109/MNET.2012.6246754. https://doi.org/10.1109/MNET.2012.6246754 [3] Stephanie Wehner, David Elkouss, and Ronald Hanson. Quantum internet: A vision for the road ahead. Science, 362 (6412), 2018. 10.1126/science.aam9288. https://doi.org/10.1126/science.aam9288 [4] Amoldeep Singh, Kapal Dev, Harun Siljak, Hem Dutt Joshi, and Maurizio Magarini. Quantum internet—applications, functionalities, enabling technologies, challenges, and research directions. IEEE Communications Surveys & Tutorials, 23 (4): 2218–2247, 2021. 10.1109/COMST.2021.3109944. https://doi.org/10.1109/COMST.2021.3109944 [5] Gilles Brassard. Quantum communication complexity. Foundations of Physics, 33 (11): 1593–1616, 2003. ISSN 0015-9018. 10.1023/a:1026009100467. https://doi.org/10.1023/a:1026009100467 [6] Harry Buhrman, Richard Cleve, Serge Massar, and Ronald de Wolf. Nonlocality and communication complexity. Rev. Mod. Phys., 82: 665–698, Mar 2010. 10.1103/RevModPhys.82.665. https://doi.org/10.1103/RevModPhys.82.665 [7] C.H. Bennett, P.W. Shor, J.A. Smolin, and A.V. Thapliyal. Entanglement-assisted capacity of a quantum channel and the reverse shannon theorem. IEEE Transactions on Information Theory, 48 (10): 2637–2655, October 2002. 10.1109/TIT.2002.802612. https://doi.org/10.1109/TIT.2002.802612 [8] Andreas Winter. Compression of sources of probability distributions and density operators. arXiv preprint, 2002. 10.48550/arXiv.quant-ph/0208131. https://doi.org/10.48550/arXiv.quant-ph/0208131 arXiv:quant-ph/0208131 [9] Charles H. Bennett, Igor Devetak, Aram W. Harrow, Peter W. Shor, and Andreas Winter. The quantum reverse shannon theorem and resource tradeoffs for simulating quantum channels. IEEE Transactions on Information Theory, 60 (5): 2926–2959, 2014. 10.1109/TIT.2014.2309968. https://doi.org/10.1109/TIT.2014.2309968 [10] Toby S. Cubitt, Debbie Leung, William Matthews, and Andreas Winter. Zero-error channel capacity and simulation assisted by non-local correlations. IEEE Transactions on Information Theory, 57 (8): 5509–5523, 2011. 10.1109/TIT.2011.2159047. https://doi.org/10.1109/TIT.2011.2159047 [11] Alexander Semenovich Holevo. Bounds for the quantity of information transmitted by a quantum communication channel.
Problemy Peredachi Informatsii, 9 (3): 3–11, 1973. URL https://www.mathnet.ru/eng/ppi903. https://www.mathnet.ru/eng/ppi903 [12] Péter E. Frenkel and Mihály Weiner. Classical information storage in an n-level quantum system. Communications in Mathematical Physics, 340 (2): 563–574, September 2015. 10.1007/s00220-015-2463-0. https://doi.org/10.1007/s00220-015-2463-0 [13] Joseph Bowles, Nicolas Brunner, and Marcin Pawłowski. Testing dimension and nonclassicality in communication networks. Phys. Rev. A, 92: 022351, Aug 2015. 10.1103/PhysRevA.92.022351. https://doi.org/10.1103/PhysRevA.92.022351 [14] John S Bell. On the Einstein Podolsky Rosen paradox.
Physics Physique Fizika, 1 (3): 195, 1964. 10.1103/PhysicsPhysiqueFizika.1.195. https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195 [15] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. Bell nonlocality. Rev. Mod. Phys., 86: 419–478, Apr 2014. 10.1103/RevModPhys.86.419. https://doi.org/10.1103/RevModPhys.86.419 [16] Serge Massar, Dave Bacon, Nicolas J. Cerf, and Richard Cleve. Classical simulation of quantum entanglement without local hidden variables. Phys. Rev. A, 63: 052305, Apr 2001. 10.1103/PhysRevA.63.052305. https://doi.org/10.1103/PhysRevA.63.052305 [17] D. Bacon and B. F. Toner. Bell inequalities with auxiliary communication. Phys. Rev. Lett., 90: 157904, Apr 2003. 10.1103/PhysRevLett.90.157904. https://doi.org/10.1103/PhysRevLett.90.157904 [18] B. F. Toner and D. Bacon. Communication cost of simulating bell correlations. Phys. Rev. Lett., 91: 187904, Oct 2003. 10.1103/PhysRevLett.91.187904. https://doi.org/10.1103/PhysRevLett.91.187904 [19] Oded Regev and Ben Toner. Simulating quantum correlations with finite communication. SIAM Journal on Computing, 39 (4): 1562–1580, 2010. 10.1137/080723909. https://doi.org/10.1137/080723909 [20] Katherine Maxwell and Eric Chitambar. Bell inequalities with communication assistance. Phys. Rev. A, 89: 042108, Apr 2014. 10.1103/PhysRevA.89.042108. https://doi.org/10.1103/PhysRevA.89.042108 [21] J B Brask and R Chaves. Bell scenarios with communication. Journal of Physics A: Mathematical and Theoretical, 50 (9): 094001, January 2017. ISSN 1751-8121. 10.1088/1751-8121/aa5840. https://doi.org/10.1088/1751-8121/aa5840 [22] Emmanuel Zambrini Cruzeiro and Nicolas Gisin. Bell inequalities with one bit of communication. Entropy, 21 (2): 171, February 2019. ISSN 1099-4300. 10.3390/e21020171. https://doi.org/10.3390/e21020171 [23] Mir Alimuddin, Ananya Chakraborty, Govind Lal Sidhardh, Ram Krishna Patra, Samrat Sen, Snehasish Roy Chowdhury, Sahil Gopalkrishna Naik, and Manik Banik. Advantage of hardy's nonlocal correlation in reverse zero-error channel coding. Phys. Rev. A, 108: 052430, Nov 2023. 10.1103/PhysRevA.108.052430. https://doi.org/10.1103/PhysRevA.108.052430 [24] Péter E. Frenkel and Mihály Weiner. On entanglement assistance to a noiseless classical channel. Quantum, 6: 662, March 2022. ISSN 2521-327X. 10.22331/q-2022-03-01-662. https://doi.org/10.22331/q-2022-03-01-662 [25] Michele Dall'Arno, Sarah Brandsen, Alessandro Tosini, Francesco Buscemi, and Vlatko Vedral. No-hypersignaling principle. Phys. Rev. Lett., 119: 020401, Jul 2017. 10.1103/PhysRevLett.119.020401. https://doi.org/10.1103/PhysRevLett.119.020401 [26] Teiko Heinosaari and Oskari Kerppo. Communication of partial ignorance with qubits. Journal of Physics A: Mathematical and Theoretical, 52 (39): 395301, September 2019. ISSN 1751-8121. 10.1088/1751-8121/ab3ae4. https://doi.org/10.1088/1751-8121/ab3ae4 [27] Teiko Heinosaari, Oskari Kerppo, and Leevi Leppäjärvi. Communication tasks in operational theories. Journal of Physics A: Mathematical and Theoretical, 53 (43): 435302, October 2020. ISSN 1751-8121. 10.1088/1751-8121/abb5dc. https://doi.org/10.1088/1751-8121/abb5dc [28] Davide Poderini, Samuraí Brito, Ranieri Nery, Fabio Sciarrino, and Rafael Chaves. Criteria for nonclassicality in the prepare-and-measure scenario. Phys. Rev. Res., 2: 043106, Oct 2020. 10.1103/PhysRevResearch.2.043106. https://doi.org/10.1103/PhysRevResearch.2.043106 [29] Brian Doolittle and Eric Chitambar. Certifying the classical simulation cost of a quantum channel. Phys. Rev. Res., 3: 043073, Oct 2021. 10.1103/PhysRevResearch.3.043073. https://doi.org/10.1103/PhysRevResearch.3.043073 [30] Armin Tavakoli, Jef Pauwels, Erik Woodhead, and Stefano Pironio. Correlations in entanglement-assisted prepare-and-measure scenarios. PRX Quantum, 2: 040357, Dec 2021. 10.1103/PRXQuantum.2.040357. https://doi.org/10.1103/PRXQuantum.2.040357 [31] Martin J. Renner, Armin Tavakoli, and Marco Túlio Quintino. Classical cost of transmitting a qubit. Phys. Rev. Lett., 130: 120801, Mar 2023. 10.1103/PhysRevLett.130.120801. https://doi.org/10.1103/PhysRevLett.130.120801 [32] Andrzej Grudka, Michał Horodecki, Ryszard Horodecki, and Antoni Wójcik. Nonsignaling quantum random access-code boxes. Phys. Rev. A, 92: 052312, Nov 2015. 10.1103/PhysRevA.92.052312. https://doi.org/10.1103/PhysRevA.92.052312 [33] Armin Tavakoli, Alley Hameedi, Breno Marques, and Mohamed Bourennane. Quantum random access codes using single $d$-level systems. Phys. Rev. Lett., 114: 170502, Apr 2015. 10.1103/PhysRevLett.114.170502. https://doi.org/10.1103/PhysRevLett.114.170502 [34] Teiko Heinosaari and Leevi Leppäjärvi. Random access test as an identifier of nonclassicality. Journal of Physics A: Mathematical and Theoretical, 55 (17): 174003, April 2022. ISSN 1751-8121. 10.1088/1751-8121/ac5b91. https://doi.org/10.1088/1751-8121/ac5b91 [35] Amélie Piveteau, Jef Pauwels, Emil Håkansson, Sadiq Muhammad, Mohamed Bourennane, and Armin Tavakoli. Entanglement-assisted quantum communication with simple measurements. Nature Communications, 13 (1), December 2022. ISSN 2041-1723. 10.1038/s41467-022-33922-5. https://doi.org/10.1038/s41467-022-33922-5 [36] Nitica Sakharwade, Michał Studziński, Michał Eckstein, and Paweł Horodecki. Two instances of random access code in the quantum regime. New Journal of Physics, 25 (5): 053038, May 2023. ISSN 1367-2630. 10.1088/1367-2630/acd716. https://doi.org/10.1088/1367-2630/acd716 [37] Pedro Lauand, Davide Poderini, Ranieri Nery, George Moreno, Lucas Pollyceno, Rafael Rabelo, and Rafael Chaves. Witnessing nonclassicality in a causal structure with three observable variables. PRX Quantum, 4: 020311, Apr 2023. 10.1103/PRXQuantum.4.020311. https://doi.org/10.1103/PRXQuantum.4.020311 [38] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles. Variational quantum algorithms.
Nature Reviews Physics, 3 (9): 625–644, August 2021. ISSN 2522-5820. 10.1038/s42254-021-00348-9. https://doi.org/10.1038/s42254-021-00348-9 [39] Brian Doolittle, Thomas R. Bromley, Nathan Killoran, and Eric Chitambar. Variational quantum optimization of nonlocality in noisy quantum networks. IEEE Transactions on Quantum Engineering, pages 1–28, 2023. 10.1109/TQE.2023.3243849. https://doi.org/10.1109/TQE.2023.3243849 [40] Brian Doolittle and Tom Bromley. qnetvo: the quantum network variational optimizer. March 2022. 10.5281/zenodo.6345834. URL https://github.com/ChitambarLab/qNetVO. https://doi.org/10.5281/zenodo.6345834 https://github.com/ChitambarLab/qNetVO [41] Brian Doolittle. Nonclassicality in Noisy Quantum Networks. Phd thesis, University of Illinois Urbana-Champaign, September 2023. URL https://hdl.handle.net/2142/121945. https://hdl.handle.net/2142/121945 [42] Daniel T. Chen, Brian Doolittle, Jeffrey Larson, Zain H. Saleem, and Eric Chitambar. Inferring quantum network topology using local measurements. PRX Quantum, 4: 040347, Dec 2023. 10.1103/PRXQuantum.4.040347. https://doi.org/10.1103/PRXQuantum.4.040347 [43] Ziqi Ma, Pranav Gokhale, Tian-Xing Zheng, Sisi Zhou, Xiaofei Yu, Liang Jiang, Peter Maurer, and Frederic T. Chong. Adaptive circuit learning for quantum metrology. In 2021 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 419–430, 2021. 10.1109/QCE52317.2021.00063. https://doi.org/10.1109/QCE52317.2021.00063 [44] Teiko Heinosaari, Oskari Kerppo, Leevi Leppäjärvi, and Martin Plávala. Simple information-processing tasks with unbounded quantum advantage. Phys. Rev. A, 109: 032627, Mar 2024. 10.1103/PhysRevA.109.032627. https://doi.org/10.1103/PhysRevA.109.032627 [45] Armin Tavakoli, Alejandro Pozas-Kerstjens, Ming-Xing Luo, and Marc-Olivier Renou. Bell nonlocality in networks. Reports on Progress in Physics, 85 (5): 056001, Mar 2022. ISSN 1361-6633. 10.1088/1361-6633/ac41bb. https://doi.org/10.1088/1361-6633/ac41bb [46] S. G. A. Brito, B. Amaral, and R. Chaves. Quantifying bell nonlocality with the trace distance. Phys. Rev. A, 97: 022111, Feb 2018. 10.1103/PhysRevA.97.022111. https://doi.org/10.1103/PhysRevA.97.022111 [47] G.M. Ziegler. Lectures on Polytopes. Graduate Texts in Mathematics.
Springer New York, 2012. 10.1007/978-1-4613-8431-1. https://doi.org/10.1007/978-1-4613-8431-1 [48] Denis Rosset, Jean-Daniel Bancal, and Nicolas Gisin. Classifying 50 years of bell inequalities. Journal of Physics A: Mathematical and Theoretical, 47 (42): 424022, October 2014. 10.1088/1751-8113/47/42/424022. https://doi.org/10.1088/1751-8113/47/42/424022 [49] Thomas Christof and Andreas Löbel. Porta, 1997. URL http://porta.zib.de/. http://porta.zib.de/ [50] Brian Doolittle. https://github.com/ChitambarLab/nonclassicality-in-quantum-communication-networks-supplemental-code (v0.2.0). Feb 2024. 10.5281/zenodo.10780789. https://doi.org/10.5281/zenodo.10780789 https://github.com/ChitambarLab/nonclassicality-in-quantum-communication-networks-supplemental-code [51] Rajarshi Pal and Sibasish Ghosh. Non-locality breaking qubit channels: the case for chsh inequality. Journal of Physics A: Mathematical and Theoretical, 48 (15): 155302, mar 2015. 10.1088/1751-8113/48/15/155302. https://doi.org/10.1088/1751-8113/48/15/155302 [52] Yujie Zhang, Rodrigo Araiza Bravo, Virginia O Lorenz, and Eric Chitambar. Channel activation of chsh nonlocality. New Journal of Physics, 22 (4): 043003, apr 2020. 10.1088/1367-2630/ab7bef. https://doi.org/10.1088/1367-2630/ab7bef [53] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, Shahnawaz Ahmed, Vishnu Ajith, M Sohaib Alam, Guillermo Alonso-Linaje, B AkashNarayanan, Ali Asadi, et al. Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint, 2018. 10.48550/arXiv.1811.04968. https://doi.org/10.48550/arXiv.1811.04968 [54] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information.
Cambridge University Press, 2009. 10.1017/cbo9780511976667. https://doi.org/10.1017/cbo9780511976667 [55] David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Learning representations by back-propagating errors. Nature, 323 (6088): 533–536, 1986. https://doi.org/10.1038/323533a0. https://doi.org/10.1038/323533a0 [56] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Phys. Rev. A, 99: 032331, Mar 2019. 10.1103/PhysRevA.99.032331. https://doi.org/10.1103/PhysRevA.99.032331 [57] Andrea Mari, Thomas R. Bromley, and Nathan Killoran. Estimating the gradient and higher-order derivatives on quantum hardware. Phys. Rev. A, 103: 012405, Jan 2021. 10.1103/PhysRevA.103.012405. https://doi.org/10.1103/PhysRevA.103.012405 [58] David Wierichs, Josh Izaac, Cody Wang, and Cedric Yen-Yu Lin. General parameter-shift rules for quantum gradients. Quantum, 6: 677, March 2022. 10.22331/q-2022-03-30-677. https://doi.org/10.22331/q-2022-03-30-677 [59] Oleksandr Kyriienko and Vincent E. Elfving. Generalized quantum circuit differentiation rules. Phys. Rev. A, 104: 052417, Nov 2021. 10.1103/PhysRevA.104.052417. https://doi.org/10.1103/PhysRevA.104.052417 [60] Sebastian Ruder. An overview of gradient descent optimization algorithms. arXiv preprint, 2016. 10.48550/arXiv.1609.04747. https://doi.org/10.48550/arXiv.1609.04747 [61] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint, 2014. 10.48550/arXiv.1412.6980. https://doi.org/10.48550/arXiv.1412.6980 [62] Lennart Bittel and Martin Kliesch. Training variational quantum algorithms is np-hard. Phys. Rev. Lett., 127: 120502, Sep 2021. 10.1103/PhysRevLett.127.120502. https://doi.org/10.1103/PhysRevLett.127.120502 [63] Charles H. Bennett and Stephen J. Wiesner. Communication via one- and two-particle operators on einstein-podolsky-rosen states. Phys. Rev. Lett., 69: 2881–2884, Nov 1992. 10.1103/PhysRevLett.69.2881. https://doi.org/10.1103/PhysRevLett.69.2881 [64] Eric Chitambar, Ian George, Brian Doolittle, and Marius Junge. The communication value of a quantum channel. IEEE Transactions on Information Theory, 69 (3): 1660–1679, 2023. 10.1109/TIT.2022.3218540. https://doi.org/10.1109/TIT.2022.3218540 [65] Andris Ambainis, Debbie Leung, Laura Mancinska, and Maris Ozols. Quantum random access codes with shared randomness. arXiv preprint, 2008. 10.48550/arXiv.0810.2937. https://doi.org/10.48550/arXiv.0810.2937 [66] Rodrigo Gallego, Nicolas Brunner, Christopher Hadley, and Antonio Acín. Device-independent tests of classical and quantum dimensions. Phys. Rev. Lett., 105: 230501, Nov 2010. 10.1103/PhysRevLett.105.230501. https://doi.org/10.1103/PhysRevLett.105.230501 [67] Marcin Pawłowski and Andreas Winter. ``hyperbits'': The information quasiparticles. Phys. Rev. A, 85: 022331, Feb 2012. 10.1103/PhysRevA.85.022331. https://doi.org/10.1103/PhysRevA.85.022331 [68] Jef Pauwels, Stefano Pironio, Emmanuel Zambrini Cruzeiro, and Armin Tavakoli. Adaptive advantage in entanglement-assisted communications. Phys. Rev. Lett., 129: 120504, Sep 2022. 10.1103/PhysRevLett.129.120504. https://doi.org/10.1103/PhysRevLett.129.120504 [69] Marcin Pawłowski and Nicolas Brunner. Semi-device-independent security of one-way quantum key distribution. Phys. Rev. A, 84: 010302, Jul 2011. 10.1103/PhysRevA.84.010302. https://doi.org/10.1103/PhysRevA.84.010302 [70] Hong-Wei Li, Zhen-Qiang Yin, Yu-Chun Wu, Xu-Bo Zou, Shuang Wang, Wei Chen, Guang-Can Guo, and Zheng-Fu Han. Semi-device-independent random-number expansion without entanglement. Phys. Rev. A, 84: 034301, Sep 2011. 10.1103/PhysRevA.84.034301. https://doi.org/10.1103/PhysRevA.84.034301 [71] George Moreno, Ranieri Nery, Carlos de Gois, Rafael Rabelo, and Rafael Chaves. Semi-device-independent certification of entanglement in superdense coding. Phys. Rev. A, 103: 022426, Feb 2021. 10.1103/PhysRevA.103.022426. https://doi.org/10.1103/PhysRevA.103.022426 [72] Armin Tavakoli, Jędrzej Kaniewski, Tamás Vértesi, Denis Rosset, and Nicolas Brunner. Self-testing quantum states and measurements in the prepare-and-measure scenario. Phys. Rev. A, 98: 062307, Dec 2018. 10.1103/PhysRevA.98.062307. https://doi.org/10.1103/PhysRevA.98.062307 [73] Sandu Popescu and Daniel Rohrlich. Quantum nonlocality as an axiom. Foundations of Physics, 24 (3): 379–385, March 1994. ISSN 1572-9516. 10.1007/bf02058098. https://doi.org/10.1007/bf02058098 [74] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23: 880–884, Oct 1969. 10.1103/PhysRevLett.23.880. https://doi.org/10.1103/PhysRevLett.23.880 [75] Brian Doolittle. https://github.com/ChitambarLab/BellScenario.jl (v0.1.3). 2020. 10.5281/zenodo.10277572. https://doi.org/10.5281/zenodo.10277572 https://github.com/ChitambarLab/BellScenario.jl [76] Brian Doolittle and Benoı̂t Legat. https://github.com/JuliaPolyhedra/XPORTA.jl (v0.1.3), 2020. https://github.com/JuliaPolyhedra/XPORTA.jl [77] Benoı̂t Legat, Robin Deits, Gustavo Goretkin, Twan Koolen, Joey Huchette, Daisuke Oyama, and Marcelo Forets. https://github.com/JuliaPolyhedra/Polyhedra.jl (v0.6.16). 2021. 10.5281/zenodo.1214290. https://doi.org/10.5281/zenodo.1214290 https://github.com/JuliaPolyhedra/Polyhedra.jl [78] Qi Huangfu and JA Julian Hall. Parallelizing the dual revised simplex method.
Mathematical Programming Computation, 10 (1): 119–142, 2018. 10.1007/s12532-017-0130-5. https://doi.org/10.1007/s12532-017-0130-5 [79] Iain Dunning, Joey Huchette, and Miles Lubin. Jump: A modeling language for mathematical optimization. SIAM Review, 59 (2): 295–320, 2017. 10.1137/15M1020575. https://doi.org/10.1137/15M1020575Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-08 08:17:16: Could not fetch cited-by data for 10.22331/q-2026-04-08-2052 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-08 08:17:16: 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.
