“Nonlocality-of-a-single-photon” based Quantum Key Distribution and Random Number Generation schemes and their device-independent security analysis
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AbstractThe question of “non-locality of a single photon'', which started with a paper by Tan, Walls and Collett (TWC, 1991) stirred a thirty years long debate. This hampered attempts to use the TWC interferometric scheme in quantum cryptography. The scheme involves a single photon 50-50 beam-split into two modes propagating to two spatially separated observation stations at which weak homodyne measurements are made. The physics and non-classicality of such an arrangement has been understood only recently, and points out that an unquestionable Bell non-classicality, as was suggested by Hardy (1994), can be observed when the local measurement settings differ by the weak local oscillator being on or off, and additionally the homodyning for the on case is not balanced. Based on that, we present a single-photon based device-independent quantum key distribution scheme secure even against no-signaling eavesdropping. In our protocol the random bits of the cryptographic key are obtained by measurements on the single photon, that is for off settings at both Alice and Bob sides, while the security is positively tested if for eavesdropping testing runs one observes a violation of a specific Bell inequality involving the on and off weak homodyne measurements as alternative local settings. The security analysis presented here is based on a decomposition of the correlations into extreme points of a no-signaling polytope, which allows for identification of the optimal strategy for any eavesdropping constrained only by the no-signaling principle. For this strategy, the key rate is calculated, which is then connected with the violation of a specific Clauser-Horne inequality. We also adapt this analysis to propose a self-testing quantum random number generator based on the old idea that employs the randomness of reflection and transmission events of a quantum light impinged on a 50-50 beamsplitter.► BibTeX data@article{Schlichtholz2026nonlocalityof, doi = {10.22331/q-2026-04-28-2084}, url = {https://doi.org/10.22331/q-2026-04-28-2084}, title = {"{N}onlocality-of-a-single-photon" based {Q}uantum {K}ey {D}istribution and {R}andom {N}umber {G}eneration schemes and their device-independent security analysis}, author = {Schlichtholz, Konrad and Woloncewicz, Bianka and Das, Tamoghna and Markiewicz, Marcin and {\.{Z}}ukowski, Marek}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2084}, month = apr, year = {2026} }► References [1] S. M. Tan, D. F. Walls, and M. J. Collett. ``Nonlocality of a single photon''. Phys. Rev. Lett. 66, 252–255 (1991). https://doi.org/10.1103/PhysRevLett.66.252 [2] Emilio Santos. ``Comment on ``Nonlocality of a single photon''''. Phys. Rev. Lett. 68, 894–894 (1992). https://doi.org/10.1103/PhysRevLett.68.894 [3] Lucien Hardy. ``Nonlocality of a Single Photon Revisited''. Phys. Rev. Lett. 73, 2279–2283 (1994). https://doi.org/10.1103/PhysRevLett.73.2279 [4] Konrad Banaszek and Krzysztof Wódkiewicz. ``Testing Quantum Nonlocality in Phase Space''. Phys. Rev. Lett. 82, 2009–2013 (1999). https://doi.org/10.1103/PhysRevLett.82.2009 [5] Christopher C. Gerry. ``Nonlocality of a single photon in cavity QED''. Phys. Rev. A 53, 4583–4586 (1996). https://doi.org/10.1103/PhysRevA.53.4583 [6] Paolo Abiuso, Tamás Kriváchy, Emanuel-Cristian Boghiu, Marc-Olivier Renou, Alejandro Pozas-Kerstjens, and Antonio Acín. ``Single-photon nonlocality in quantum networks''. Phys. Rev. Research 4, L012041 (2022). https://doi.org/10.1103/PhysRevResearch.4.L012041 [7] P. Caspar, E. Verbanis, E. Oudot, N. Maring, F. Samara, M. Caloz, M. Perrenoud, P. Sekatski, A. Martin, N. Sangouard, H. Zbinden, and R. T. Thew. ``Heralded Distribution of Single-Photon Path Entanglement''. Phys. Rev. Lett. 125, 110506 (2020). https://doi.org/10.1103/PhysRevLett.125.110506 [8] W. J. Munro. ``Optimal states for Bell-inequality violations using quadrature-phase homodyne measurements''. Phys. Rev. A 59, 4197–4201 (1999). https://doi.org/10.1103/PhysRevA.59.4197 [9] S. J. van Enk. ``Single-particle entanglement''. Phys. Rev. A 72, 064306 (2005). https://doi.org/10.1103/PhysRevA.72.064306 [10] T. Guerreiro, F. Monteiro, A. Martin, J. B. Brask, T. Vértesi, B. Korzh, M. Caloz, F. Bussières, V. B. Verma, A. E. Lita, R. P. Mirin, S. W. Nam, F. Marsilli, M. D. Shaw, N. Gisin, N. Brunner, H. Zbinden, and R. T. Thew. ``Demonstration of Einstein-Podolsky-Rosen Steering Using Single-Photon Path Entanglement and Displacement-Based Detection''. Phys. Rev. Lett. 117, 070404 (2016). https://doi.org/10.1103/PhysRevLett.117.070404 [11] Tamoghna Das, Marcin Karczewski, Antonio Mandarino, Marcin Markiewicz, Bianka Woloncewicz, and Marek Żukowski. ``Wave–particle complementarity: detecting violation of local realism with photon-number resolving weak-field homodyne measurements''. New Journal of Physics 24, 033017 (2022). https://doi.org/10.1088/1367-2630/ac54c8 [12] Tamoghna Das, Marcin Karczewski, Antonio Mandarino, Marcin Markiewicz, Bianka Woloncewicz, and Marek Żukowski. ``Can single photon excitation of two spatially separated modes lead to a violation of Bell inequality via weak-field homodyne measurements?''. New Journal of Physics 23, 073042 (2021). https://doi.org/10.1088/1367-2630/ac0ffe [13] Tamoghna Das, Marcin Karczewski, Antonio Mandarino, Marcin Markiewicz, and Marek Żukowski. ``Optimal Interferometry for Bell Nonclassicality Induced by a Vacuum–One-Photon Qubit''. Phys. Rev. Appl. 18, 034074 (2022). https://doi.org/10.1103/PhysRevApplied.18.034074 [14] Konrad Schlichtholz and Marcin Markiewicz. ``Generalization of Gisin’s theorem to quantum fields''. New Journal of Physics 26, 023048 (2024). https://doi.org/10.1088/1367-2630/ad2821 [15] 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 [16] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. ``Bell nonlocality''. Rev. Mod. Phys. 86, 419–478 (2014). https://doi.org/10.1103/RevModPhys.86.419 [17] D. Mayers and A. Yao. ``Self testing quantum apparatus''. Quantum Inf. Comp. 4, 273 (2004). https://doi.org/10.26421/QIC4.4-3 [18] A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani. ``Device-independent security of quantum cryptography against collective attacks''.
Physical Review Letters 98, 230501 (2007). https://doi.org/10.1103/PhysRevLett.98.230501 [19] L. Masanes, S. Pironio, and A. Acín. ``Secure device-independent quantum key distribution with causally independent measurement devices''. Nat. Commun. 2 (2011). https://doi.org/10.1038/ncomms1244 [20] R. Arnon-Friedman, R. Renner, and T. Vidick. ``Simple and tight device-independent security proofs''. SIAM Journal on Computing 48, 181–225 (2019). https://doi.org/10.1137/18M1174726 [21] J. Barrett, L. Hardy, and A. Kent. ``No signaling and quantum key distribution''. Phys. Rev. Lett 95, 010503 (2005). https://doi.org/10.1103/PhysRevLett.95.010503 [22] A. Acín, N. Gisin, and L. Masanes. ``From Bell's Theorem to Secure Quantum Key Distribution''. Phys. Rev. Lett 97, 120405 (2006). https://doi.org/10.1103/PhysRevLett.97.120405 [23] Lluis 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 [24] A. Acín, S. Massar, and S. Pironio. ``Efficient quantum key distribution secure against no-signaling eavesdroppers''. New J. Phys. 8, 126 (2006). https://doi.org/10.1088/1367-2630/8/8/126 [25] Esther Hänggi, Renato Renner, and Stefan Wolf. ``Efficient device-independent quantum key distribution''.
In Henri Gilbert, editor, Advances in Cryptology – EUROCRYPT 2010. Pages 216–234. Berlin, Heidelberg (2010).
Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-13190-5_11 [26] Charles H. Bennett and Gilles Brassard. ``Quantum cryptography: Public key distribution and coin tossing''.
Theoretical Computer Science 560, 7–11 (2014). https://doi.org/10.1016/j.tcs.2014.05.025 [27] Daniel Huber, Marcus Reindl, Yongheng Huo, Huiying Huang, Johannes S. Wildmann, Oliver G. Schmidt, Armando Rastelli, and Rinaldo Trotta. ``Highly indistinguishable and strongly entangled photons from symmetric gaas quantum dots''. Nature Communications 8, 15506 (2017). https://doi.org/10.1038/ncomms15506 [28] M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert. ````Event-ready-detectors'' Bell experiment via entanglement swapping''. Phys. Rev. Lett. 71, 4287–4290 (1993). https://doi.org/10.1103/PhysRevLett.71.4287 [29] B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson. ``Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres''. Nature 526, 682–686 (2015). https://doi.org/10.1038/nature15759 [30] Wei Zhang, Tim 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 [31] Beatrice Da Lio, Davide Bacco, Daniele Cozzolino, Nicola Biagi, Tummas Napoleon Arge, Emil Larsen, Karsten Rottwitt, Yunhong Ding, Alessandro Zavatta, and Leif Katsuo Oxenløwe. ``Stable transmission of high-dimensional quantum states over a 2-km multicore fiber''. IEEE Journal of Selected Topics in Quantum Electronics 26, 1–8 (2019). https://doi.org/10.1109/JSTQE.2019.2960937 [32] Arun Kumar Pati and Marek Zukowski. ``Interference due to coherence swapping''. Pramana 56, 393–401 (2001). https://doi.org/10.1007/s12043-001-0133-6 [33] Marek Winczewski, Tamoghna Das, John H. Selby, Karol Horodecki, Paweł Horodecki, Łukasz Pankowski, Marco Piani, and Ravishankar Ramanathan. ``Complete extension: the non-signaling analog of quantum purification''. Quantum 7, 1159 (2023). https://doi.org/10.22331/q-2023-11-03-1159 [34] Constantin Carathéodory. ``Über den Variabilitätsbereich der Fourier’schen Konstanten von positiven harmonischen Funktionen''.
Rendiconti Del Circolo Matematico di Palermo (1884-1940) 32, 193–217 (1911). https://doi.org/10.1007/BF03014795 [35] R Tyrrell Rockafellar. ``Convex analysis''. Volume 11.
Princeton University Press. (1997). [36] Yury Polyanskiy and Yihong Wu. ``Strong Data-Processing Inequalities for Channels and Bayesian Networks''.
In Eric Carlen, Mokshay Madiman, and Elisabeth M. Werner, editors, Convexity and Concentration. Pages 211–249. New York, NY (2017).
Springer New York. https://doi.org/10.1007/978-1-4939-7005-6_7 [37] Imre Csiszár and János Körner. ``Broadcast channels with confidential messages''. IEEE Trans. Inf. Theory 24, 339–348 (1978). https://doi.org/10.1109/TIT.1978.1055892 [38] Vaisakh Mannalatha, Sandeep Mishra, and Anirban Pathak. ``A Comprehensive Review of Quantum Random Number Generators: Concepts, Classification and the Origin of Randomness''.
Quantum Information Processing 22, 439 (2023). https://doi.org/10.1007/s11128-023-04175-y [39] Robert Konig, Renato Renner, and Christian Schaffner. ``The Operational Meaning of Min- and Max-Entropy''. IEEE Transactions on Information Theory 55, 4337–4347 (2009). https://doi.org/10.1109/TIT.2009.2025545 [40] Renato Renner. ``Security of quantum key distribution''. International Journal of Quantum Information 06, 1–127 (2008). https://doi.org/10.1142/S0219749908003256 [41] Umesh Vazirani and Thomas Vidick. ``Certifiable quantum dice: or, true random number generation secure against quantum adversaries''. In Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing. Page 61–76. STOC '12New York, NY, USA (2012). Association for Computing Machinery. https://doi.org/10.1145/2213977.2213984 [42] Anindya De, Christopher Portmann, Thomas Vidick, and Renato Renner. ``Trevisan's Extractor in the Presence of Quantum Side Information''. SIAM Journal on Computing 41, 915–940 (2012). https://doi.org/10.1137/100813683 [43] Mario Berta, Omar Fawzi, and Stephanie Wehner. ``Quantum to classical randomness extractors''. IEEE Transactions on Information Theory 60, 1168–1192 (2014). https://doi.org/10.1109/TIT.2013.2291780 [44] G. S. Thekkadath, D. S. Phillips, J. F. F. Bulmer, W. R. Clements, A. Eckstein, B. A. Bell, J. Lugani, T. A. W. Wolterink, A. Lita, S. W. Nam, T. Gerrits, C. G. Wade, and I. A. Walmsley. ``Tuning between photon-number and quadrature measurements with weak-field homodyne detection''. Phys. Rev. A 101, 031801 (2020). https://doi.org/10.1103/PhysRevA.101.031801 [45] 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 (2011). https://doi.org/10.1103/RevModPhys.83.33 [46] Jonathan Barrett, Noah Linden, Serge Massar, Stefano Pironio, Sandu Popescu, and David Roberts. ``Nonlocal correlations as an information-theoretic resource''. Phys. Rev. A 71, 022101 (2005). https://doi.org/10.1103/PhysRevA.71.022101Cited by[1] Paweł Cieśliński, Jan-Åke Larsson, Marcin Markiewicz, Konrad Schlichtholz, and Marek Żukowski, "Unquestionable Bell Theorem for Interwoven Frustrated Downconversion Processes", Physical Review Letters 136 9, 090206 (2026). [2] Konrad Schlichtholz and Marcin Markiewicz, "Generalization of Gisin's theorem to quantum fields", New Journal of Physics 26 2, 023048 (2024). [3] Konrad Schlichtholz and Łukasz Rudnicki, "Open dynamics of entanglement in mesoscopic bosonic systems", New Journal of Physics 26 5, 053022 (2024). [4] Marek Żukowski, Paweł Cieśliński, Marcin Markiewicz, and Konrad Schlichtholz, "Bell-GHZ nonclassicality of many-observer interwoven frustrated down conversions", arXiv:2602.18381, (2026). [5] Daniel Kun, Teodor Strömberg, Borivoje Dakić, Philip Walther, and Lee A. Rozema, "Testing single-photon entanglement using self-referential measurements", Optica 13 4, 745 (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-04-28 07:00:35). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-04-28 07:00:33: Could not fetch cited-by data for 10.22331/q-2026-04-28-2084 from Crossref. This is normal if the DOI was registered recently.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. AbstractThe question of “non-locality of a single photon'', which started with a paper by Tan, Walls and Collett (TWC, 1991) stirred a thirty years long debate. This hampered attempts to use the TWC interferometric scheme in quantum cryptography. The scheme involves a single photon 50-50 beam-split into two modes propagating to two spatially separated observation stations at which weak homodyne measurements are made. The physics and non-classicality of such an arrangement has been understood only recently, and points out that an unquestionable Bell non-classicality, as was suggested by Hardy (1994), can be observed when the local measurement settings differ by the weak local oscillator being on or off, and additionally the homodyning for the on case is not balanced. Based on that, we present a single-photon based device-independent quantum key distribution scheme secure even against no-signaling eavesdropping. In our protocol the random bits of the cryptographic key are obtained by measurements on the single photon, that is for off settings at both Alice and Bob sides, while the security is positively tested if for eavesdropping testing runs one observes a violation of a specific Bell inequality involving the on and off weak homodyne measurements as alternative local settings. The security analysis presented here is based on a decomposition of the correlations into extreme points of a no-signaling polytope, which allows for identification of the optimal strategy for any eavesdropping constrained only by the no-signaling principle. For this strategy, the key rate is calculated, which is then connected with the violation of a specific Clauser-Horne inequality. We also adapt this analysis to propose a self-testing quantum random number generator based on the old idea that employs the randomness of reflection and transmission events of a quantum light impinged on a 50-50 beamsplitter.► BibTeX data@article{Schlichtholz2026nonlocalityof, doi = {10.22331/q-2026-04-28-2084}, url = {https://doi.org/10.22331/q-2026-04-28-2084}, title = {"{N}onlocality-of-a-single-photon" based {Q}uantum {K}ey {D}istribution and {R}andom {N}umber {G}eneration schemes and their device-independent security analysis}, author = {Schlichtholz, Konrad and Woloncewicz, Bianka and Das, Tamoghna and Markiewicz, Marcin and {\.{Z}}ukowski, Marek}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2084}, month = apr, year = {2026} }► References [1] S. M. Tan, D. F. Walls, and M. J. Collett. ``Nonlocality of a single photon''. Phys. Rev. Lett. 66, 252–255 (1991). https://doi.org/10.1103/PhysRevLett.66.252 [2] Emilio Santos. ``Comment on ``Nonlocality of a single photon''''. Phys. Rev. Lett. 68, 894–894 (1992). https://doi.org/10.1103/PhysRevLett.68.894 [3] Lucien Hardy. ``Nonlocality of a Single Photon Revisited''. Phys. Rev. Lett. 73, 2279–2283 (1994). https://doi.org/10.1103/PhysRevLett.73.2279 [4] Konrad Banaszek and Krzysztof Wódkiewicz. ``Testing Quantum Nonlocality in Phase Space''. Phys. Rev. Lett. 82, 2009–2013 (1999). https://doi.org/10.1103/PhysRevLett.82.2009 [5] Christopher C. Gerry. ``Nonlocality of a single photon in cavity QED''. Phys. Rev. A 53, 4583–4586 (1996). https://doi.org/10.1103/PhysRevA.53.4583 [6] Paolo Abiuso, Tamás Kriváchy, Emanuel-Cristian Boghiu, Marc-Olivier Renou, Alejandro Pozas-Kerstjens, and Antonio Acín. ``Single-photon nonlocality in quantum networks''. Phys. Rev. Research 4, L012041 (2022). https://doi.org/10.1103/PhysRevResearch.4.L012041 [7] P. Caspar, E. Verbanis, E. Oudot, N. Maring, F. Samara, M. Caloz, M. Perrenoud, P. Sekatski, A. Martin, N. Sangouard, H. Zbinden, and R. T. Thew. ``Heralded Distribution of Single-Photon Path Entanglement''. Phys. Rev. Lett. 125, 110506 (2020). https://doi.org/10.1103/PhysRevLett.125.110506 [8] W. J. Munro. ``Optimal states for Bell-inequality violations using quadrature-phase homodyne measurements''. Phys. Rev. A 59, 4197–4201 (1999). https://doi.org/10.1103/PhysRevA.59.4197 [9] S. J. van Enk. ``Single-particle entanglement''. Phys. Rev. A 72, 064306 (2005). https://doi.org/10.1103/PhysRevA.72.064306 [10] T. Guerreiro, F. Monteiro, A. Martin, J. B. Brask, T. Vértesi, B. Korzh, M. Caloz, F. Bussières, V. B. Verma, A. E. Lita, R. P. Mirin, S. W. Nam, F. Marsilli, M. D. Shaw, N. Gisin, N. Brunner, H. Zbinden, and R. T. Thew. ``Demonstration of Einstein-Podolsky-Rosen Steering Using Single-Photon Path Entanglement and Displacement-Based Detection''. Phys. Rev. Lett. 117, 070404 (2016). https://doi.org/10.1103/PhysRevLett.117.070404 [11] Tamoghna Das, Marcin Karczewski, Antonio Mandarino, Marcin Markiewicz, Bianka Woloncewicz, and Marek Żukowski. ``Wave–particle complementarity: detecting violation of local realism with photon-number resolving weak-field homodyne measurements''. New Journal of Physics 24, 033017 (2022). https://doi.org/10.1088/1367-2630/ac54c8 [12] Tamoghna Das, Marcin Karczewski, Antonio Mandarino, Marcin Markiewicz, Bianka Woloncewicz, and Marek Żukowski. ``Can single photon excitation of two spatially separated modes lead to a violation of Bell inequality via weak-field homodyne measurements?''. New Journal of Physics 23, 073042 (2021). https://doi.org/10.1088/1367-2630/ac0ffe [13] Tamoghna Das, Marcin Karczewski, Antonio Mandarino, Marcin Markiewicz, and Marek Żukowski. ``Optimal Interferometry for Bell Nonclassicality Induced by a Vacuum–One-Photon Qubit''. Phys. Rev. Appl. 18, 034074 (2022). https://doi.org/10.1103/PhysRevApplied.18.034074 [14] Konrad Schlichtholz and Marcin Markiewicz. ``Generalization of Gisin’s theorem to quantum fields''. New Journal of Physics 26, 023048 (2024). https://doi.org/10.1088/1367-2630/ad2821 [15] 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 [16] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. ``Bell nonlocality''. Rev. Mod. Phys. 86, 419–478 (2014). https://doi.org/10.1103/RevModPhys.86.419 [17] D. Mayers and A. Yao. ``Self testing quantum apparatus''. Quantum Inf. Comp. 4, 273 (2004). https://doi.org/10.26421/QIC4.4-3 [18] A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani. ``Device-independent security of quantum cryptography against collective attacks''.
Physical Review Letters 98, 230501 (2007). https://doi.org/10.1103/PhysRevLett.98.230501 [19] L. Masanes, S. Pironio, and A. Acín. ``Secure device-independent quantum key distribution with causally independent measurement devices''. Nat. Commun. 2 (2011). https://doi.org/10.1038/ncomms1244 [20] R. Arnon-Friedman, R. Renner, and T. Vidick. ``Simple and tight device-independent security proofs''. SIAM Journal on Computing 48, 181–225 (2019). https://doi.org/10.1137/18M1174726 [21] J. Barrett, L. Hardy, and A. Kent. ``No signaling and quantum key distribution''. Phys. Rev. Lett 95, 010503 (2005). https://doi.org/10.1103/PhysRevLett.95.010503 [22] A. Acín, N. Gisin, and L. Masanes. ``From Bell's Theorem to Secure Quantum Key Distribution''. Phys. Rev. Lett 97, 120405 (2006). https://doi.org/10.1103/PhysRevLett.97.120405 [23] Lluis 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 [24] A. Acín, S. Massar, and S. Pironio. ``Efficient quantum key distribution secure against no-signaling eavesdroppers''. New J. Phys. 8, 126 (2006). https://doi.org/10.1088/1367-2630/8/8/126 [25] Esther Hänggi, Renato Renner, and Stefan Wolf. ``Efficient device-independent quantum key distribution''.
In Henri Gilbert, editor, Advances in Cryptology – EUROCRYPT 2010. Pages 216–234. Berlin, Heidelberg (2010).
Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-13190-5_11 [26] Charles H. Bennett and Gilles Brassard. ``Quantum cryptography: Public key distribution and coin tossing''.
Theoretical Computer Science 560, 7–11 (2014). https://doi.org/10.1016/j.tcs.2014.05.025 [27] Daniel Huber, Marcus Reindl, Yongheng Huo, Huiying Huang, Johannes S. Wildmann, Oliver G. Schmidt, Armando Rastelli, and Rinaldo Trotta. ``Highly indistinguishable and strongly entangled photons from symmetric gaas quantum dots''. Nature Communications 8, 15506 (2017). https://doi.org/10.1038/ncomms15506 [28] M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert. ````Event-ready-detectors'' Bell experiment via entanglement swapping''. Phys. Rev. Lett. 71, 4287–4290 (1993). https://doi.org/10.1103/PhysRevLett.71.4287 [29] B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson. ``Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres''. Nature 526, 682–686 (2015). https://doi.org/10.1038/nature15759 [30] Wei Zhang, Tim 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 [31] Beatrice Da Lio, Davide Bacco, Daniele Cozzolino, Nicola Biagi, Tummas Napoleon Arge, Emil Larsen, Karsten Rottwitt, Yunhong Ding, Alessandro Zavatta, and Leif Katsuo Oxenløwe. ``Stable transmission of high-dimensional quantum states over a 2-km multicore fiber''. IEEE Journal of Selected Topics in Quantum Electronics 26, 1–8 (2019). https://doi.org/10.1109/JSTQE.2019.2960937 [32] Arun Kumar Pati and Marek Zukowski. ``Interference due to coherence swapping''. Pramana 56, 393–401 (2001). https://doi.org/10.1007/s12043-001-0133-6 [33] Marek Winczewski, Tamoghna Das, John H. Selby, Karol Horodecki, Paweł Horodecki, Łukasz Pankowski, Marco Piani, and Ravishankar Ramanathan. ``Complete extension: the non-signaling analog of quantum purification''. Quantum 7, 1159 (2023). https://doi.org/10.22331/q-2023-11-03-1159 [34] Constantin Carathéodory. ``Über den Variabilitätsbereich der Fourier’schen Konstanten von positiven harmonischen Funktionen''.
Rendiconti Del Circolo Matematico di Palermo (1884-1940) 32, 193–217 (1911). https://doi.org/10.1007/BF03014795 [35] R Tyrrell Rockafellar. ``Convex analysis''. Volume 11.
Princeton University Press. (1997). [36] Yury Polyanskiy and Yihong Wu. ``Strong Data-Processing Inequalities for Channels and Bayesian Networks''.
In Eric Carlen, Mokshay Madiman, and Elisabeth M. Werner, editors, Convexity and Concentration. Pages 211–249. New York, NY (2017).
Springer New York. https://doi.org/10.1007/978-1-4939-7005-6_7 [37] Imre Csiszár and János Körner. ``Broadcast channels with confidential messages''. IEEE Trans. Inf. Theory 24, 339–348 (1978). https://doi.org/10.1109/TIT.1978.1055892 [38] Vaisakh Mannalatha, Sandeep Mishra, and Anirban Pathak. ``A Comprehensive Review of Quantum Random Number Generators: Concepts, Classification and the Origin of Randomness''.
Quantum Information Processing 22, 439 (2023). https://doi.org/10.1007/s11128-023-04175-y [39] Robert Konig, Renato Renner, and Christian Schaffner. ``The Operational Meaning of Min- and Max-Entropy''. IEEE Transactions on Information Theory 55, 4337–4347 (2009). https://doi.org/10.1109/TIT.2009.2025545 [40] Renato Renner. ``Security of quantum key distribution''. International Journal of Quantum Information 06, 1–127 (2008). https://doi.org/10.1142/S0219749908003256 [41] Umesh Vazirani and Thomas Vidick. ``Certifiable quantum dice: or, true random number generation secure against quantum adversaries''. In Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing. Page 61–76. STOC '12New York, NY, USA (2012). Association for Computing Machinery. https://doi.org/10.1145/2213977.2213984 [42] Anindya De, Christopher Portmann, Thomas Vidick, and Renato Renner. ``Trevisan's Extractor in the Presence of Quantum Side Information''. SIAM Journal on Computing 41, 915–940 (2012). https://doi.org/10.1137/100813683 [43] Mario Berta, Omar Fawzi, and Stephanie Wehner. ``Quantum to classical randomness extractors''. IEEE Transactions on Information Theory 60, 1168–1192 (2014). https://doi.org/10.1109/TIT.2013.2291780 [44] G. S. Thekkadath, D. S. Phillips, J. F. F. Bulmer, W. R. Clements, A. Eckstein, B. A. Bell, J. Lugani, T. A. W. Wolterink, A. Lita, S. W. Nam, T. Gerrits, C. G. Wade, and I. A. Walmsley. ``Tuning between photon-number and quadrature measurements with weak-field homodyne detection''. Phys. Rev. A 101, 031801 (2020). https://doi.org/10.1103/PhysRevA.101.031801 [45] 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 (2011). https://doi.org/10.1103/RevModPhys.83.33 [46] Jonathan Barrett, Noah Linden, Serge Massar, Stefano Pironio, Sandu Popescu, and David Roberts. ``Nonlocal correlations as an information-theoretic resource''. Phys. Rev. A 71, 022101 (2005). https://doi.org/10.1103/PhysRevA.71.022101Cited by[1] Paweł Cieśliński, Jan-Åke Larsson, Marcin Markiewicz, Konrad Schlichtholz, and Marek Żukowski, "Unquestionable Bell Theorem for Interwoven Frustrated Downconversion Processes", Physical Review Letters 136 9, 090206 (2026). [2] Konrad Schlichtholz and Marcin Markiewicz, "Generalization of Gisin's theorem to quantum fields", New Journal of Physics 26 2, 023048 (2024). [3] Konrad Schlichtholz and Łukasz Rudnicki, "Open dynamics of entanglement in mesoscopic bosonic systems", New Journal of Physics 26 5, 053022 (2024). [4] Marek Żukowski, Paweł Cieśliński, Marcin Markiewicz, and Konrad Schlichtholz, "Bell-GHZ nonclassicality of many-observer interwoven frustrated down conversions", arXiv:2602.18381, (2026). [5] Daniel Kun, Teodor Strömberg, Borivoje Dakić, Philip Walther, and Lee A. Rozema, "Testing single-photon entanglement using self-referential measurements", Optica 13 4, 745 (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-04-28 07:00:35). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-04-28 07:00:33: Could not fetch cited-by data for 10.22331/q-2026-04-28-2084 from Crossref. This is normal if the DOI was registered recently.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.