{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,11,13]],"date-time":"2026-11-13T11:02:23Z","timestamp":1794567743710,"version":"4.9.1"},"reference-count":96,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,3,16]],"date-time":"2023-03-16T00:00:00Z","timestamp":1678924800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FWF","award":["M 2812-N"],"award-info":[{"award-number":["M 2812-N"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"\n In the minimal scenario of quantum correlations, two parties can choose from two observables with two possible outcomes each. Probabilities are specified by four marginals and four correlations. The resulting four-dimensional convex body of correlations, denoted\n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n , is fundamental for quantum information theory. We review and systematize what is known about\n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n , and add many details, visualizations, and complete proofs. In particular, we provide a detailed description of the boundary, which consists of three-dimensional faces isomorphic to elliptopes and sextic algebraic manifolds of exposed extreme points. These patches are separated by cubic surfaces of non-exposed extreme points. We provide a trigonometric parametrization of all extreme points, along with their exposing Tsirelson inequalities and quantum models. All non-classical extreme points (exposed or not) are self-testing, i.e., realized by an essentially unique quantum model.Two principles, which are specific to the minimal scenario, allow a quick and complete overview: The first is the pushout transformation, i.e., the application of the sine function to each coordinate. This transforms the classical correlation polytope exactly into the correlation body\n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n , also identifying the boundary structures. The second principle, self-duality, is an isomorphism between\n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n and its polar dual, i.e., the set of affine inequalities satisfied by all quantum correlations (\u201cTsirelson inequalities''). The same isomorphism links the polytope of classical correlations contained in\n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n to the polytope of no-signalling correlations, which contains\n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n .We also discuss the sets of correlations achieved with fixed Hilbert space dimension, fixed state or fixed observables, and establish a new non-linear inequality for\n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n involving the determinant of the correlation matrix.\n <\/jats:p>","DOI":"10.22331\/q-2023-03-16-947","type":"journal-article","created":{"date-parts":[[2023,3,16]],"date-time":"2023-03-16T09:43:45Z","timestamp":1678959825000},"page":"947","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":32,"title":["Quantum Correlations in the Minimal Scenario"],"prefix":"10.22331","volume":"7","author":[{"given":"Thinh P.","family":"Le","sequence":"first","affiliation":[{"name":"Institute for Quantum Optics and Quantum Information Vienna, Boltzmanngasse 3 1090 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chiara","family":"Meroni","sequence":"additional","affiliation":[{"name":"Institute for Computational and Experimental Research in Mathematics, 121 South Main Street Providence RI 02903, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bernd","family":"Sturmfels","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences Leipzig, Inselstrasse 22 04103 Leipzig, Germany"},{"name":"Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840 Berkeley CA 94720-3840, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Reinhard F.","family":"Werner","sequence":"additional","affiliation":[{"name":"Insitute f\u00fcr Theoretische Physik, Leibniz Universit\u00e4t Hannover, Appelstrasse 2 30167 Hannover, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Timo","family":"Ziegler","sequence":"additional","affiliation":[{"name":"Insitute f\u00fcr Theoretische Physik, Leibniz Universit\u00e4t Hannover, Appelstrasse 2 30167 Hannover, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2023,3,16]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Alain Aspect, Philippe Grangier, and G\u00e9rard Roger. ``Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A new violation of Bell's inequalities''. Phys. Rev. Lett. 49, 91\u201394 (1982).","DOI":"10.1103\/PhysRevLett.49.91"},{"key":"1","doi-asserted-by":"publisher","unstructured":"B. Hensen, R. Hanson, et al. ``Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres''. Nature 526, 682 EP \u2013 (2015). arXiv:1508.05949.","DOI":"10.1038\/nature15759"},{"key":"2","doi-asserted-by":"publisher","unstructured":"N. Sangouard, J.-D. Bancal, N. Gisin, W. Rosenfeld, P. Sekatski, M. Weber, and H. Weinfurter. ``Loophole-free Bell test with one atom and less than one photon on average''. Phys. Rev. A 84, 052122 (2011). arXiv:1108.1027.","DOI":"10.1103\/PhysRevA.84.052122"},{"key":"3","doi-asserted-by":"publisher","unstructured":"J. S. Bell. ``On the Einstein Podolsky Rosen paradox''. Physics 1, 195\u2013200 (1964).","DOI":"10.1103\/PhysicsPhysiqueFizika.1.195"},{"key":"4","doi-asserted-by":"publisher","unstructured":"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\u2013884 (1969).","DOI":"10.1103\/PhysRevLett.23.880"},{"key":"5","unstructured":"R. F. Werner et al. ``Open quantum problems''. url: https:\/\/oqp.iqoqi.oeaw.ac.at\/."},{"key":"6","doi-asserted-by":"publisher","unstructured":"Boris S. Tsirelson. ``Quantum analogues of the Bell inequalities. the case of two spatially separated domains''. J. Soviet Math. 36, 557\u2013570 (1987).","DOI":"10.1007\/BF01663472"},{"key":"7","doi-asserted-by":"publisher","unstructured":"R. F. Werner and M. M. Wolf. ``All multipartite Bell-correlation inequalities for two dichotomic observables per site''. Phys. Rev. A 64, 032112 (2001). arXiv:quant-ph\/0102024.","DOI":"10.1103\/PhysRevA.64.032112"},{"key":"8","doi-asserted-by":"publisher","unstructured":"William Slofstra. ``The set of quantum correlations is not closed''. Forum of Mathematics, Pi 7, e1 (2019). arXiv:1703.08618.","DOI":"10.1017\/fmp.2018.3"},{"key":"9","unstructured":"Volkher B. Scholz and R. F. Werner. ``Tsirelson's problem'' (2008). arXiv:0812.4305."},{"key":"10","unstructured":"Boris S Tsirelson. ``Some results and problems on quantum Bell-type inequalities''. Hadronic Journal Supplement 8, 329\u2013345 (1993). url: https:\/\/www.tau.ac.il\/ tsirel\/download\/hadron.html."},{"key":"11","doi-asserted-by":"publisher","unstructured":"Miguel Navascues, Stefano Pironio, and Antonio Ac\u00edn. ``A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations''. New J. Phys. 10, 073013 (2008).","DOI":"10.1088\/1367-2630\/10\/7\/073013"},{"key":"12","doi-asserted-by":"publisher","unstructured":"M. Junge, M. Navascues, C. Palazuelos, D. Perez-Garcia, V. B. Scholz, and R. F. Werner. ``Connes' embedding problem and Tsirelson's problem''. J. Math. Phys. 52, 012102 (2011). arXiv:1008.1142.","DOI":"10.1063\/1.3514538"},{"key":"13","doi-asserted-by":"publisher","unstructured":"Tobias Fritz. ``Tsirelson's problem and Kirchberg's conjecture''. Rev. Math. Phys. 24, 1250012 (2012). arXiv:1008.1168.","DOI":"10.1142\/S0129055X12500122"},{"key":"14","unstructured":"Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. ``MIP*=RE'' (2020). arXiv:2001.04383."},{"key":"15","doi-asserted-by":"publisher","unstructured":"G\u00fcnther M. Ziegler. ``Lectures on polytopes''. Springer. Berlin (1995).","DOI":"10.1007\/978-1-4613-8431-1"},{"key":"16","doi-asserted-by":"publisher","unstructured":"Mateusz Micha\u0142ek and Bernd Sturmfels. ``Invitation to nonlinear algebra''. Volume 211 of Graduate Studies in Mathematics. AMS. (2021).","DOI":"10.1007\/s00591-022-00324-z"},{"key":"17","doi-asserted-by":"publisher","unstructured":"Grigoriy Blekherman, Pablo Parrilo, and Rekha Thomas. ``Semidefinite optimization and convex algebraic geometry''. MOS-SIAM Series on Optimization 13. SIAM. Philadelphia (2012).","DOI":"10.1137\/1.9781611972290"},{"key":"18","doi-asserted-by":"publisher","unstructured":"Bernd Sturmfels and Caroline Uhler. ``Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry''. Ann. Inst. Statist. Math. 62, 603\u2013638 (2010). arXiv:0906.3529.","DOI":"10.1007\/s10463-010-0295-4"},{"key":"19","doi-asserted-by":"publisher","unstructured":"Claus Scheiderer. ``Spectrahedral shadows''. SIAM J. Appl. Algebra Geometry 2, 26\u201344 (2018). arXiv:1612.07048.","DOI":"10.1137\/17M1118981"},{"key":"20","doi-asserted-by":"publisher","unstructured":"B. S. Cirel'son. ``Quantum generalizations of Bell's inequality''. Lett. Math. Phys. 4, 93\u2013100 (1980).","DOI":"10.1007\/BF00417500"},{"key":"21","doi-asserted-by":"publisher","unstructured":"Jukka Kiukas and Reinhard F. Werner. ``Maximal violation of Bell inequalities by position measurements''. J. Math. Phys. 51, 072105 (2010). arXiv:0912.3740.","DOI":"10.1063\/1.3447736"},{"key":"22","doi-asserted-by":"publisher","unstructured":"Lawrence J. Landau. ``Empirical two-point correlation functions''. Found. Phys. 18, 449\u2013460 (1988).","DOI":"10.1007\/BF00732549"},{"key":"23","unstructured":"L Masanes. ``Necessary and sufficient condition for quantum-generated correlations'' (2003) arXiv:quant-ph\/0309137."},{"key":"24","doi-asserted-by":"publisher","unstructured":"Yukun Wang, Xingyao Wu, and Valerio Scarani. ``All the self-testings of the singlet for two binary measurements''. New J. Phys. 18, 025021 (2016). arXiv:1511.04886.","DOI":"10.1088\/1367-2630\/18\/2\/025021"},{"key":"25","doi-asserted-by":"publisher","unstructured":"Andrew C Doherty, Yeong-Cherng Liang, Ben Toner, and Stephanie Wehner. ``The quantum moment problem and bounds on entangled multi-prover games''. In 23rd Annual IEEE Conference on Computational Complexity. Pages 199\u2013210. IEEE (2008). arXiv:0803.4373.","DOI":"10.1109\/CCC.2008.26"},{"key":"26","doi-asserted-by":"publisher","unstructured":"Tobias Fritz. ``Polyhedral duality in Bell scenarios with two binary observables''. J. Math. Phys. 53, 072202 (2012). arXiv:1202.0141.","DOI":"10.1063\/1.4734586"},{"key":"27","doi-asserted-by":"publisher","unstructured":"Dominic Mayers and Andrew Yao. ``Self testing quantum apparatus''. Quantum Info. Comput. 4, 273\u2013286 (2004). arXiv:quant-ph\/0307205.","DOI":"10.26421\/QIC4.4-3"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Stephen J. Summers and Reinhard F. Werner. ``Maximal violation of Bell's inequalities is generic in quantum field theory''. Commun. Math. Phys. 110, 247\u2013259 (1987).","DOI":"10.1007\/BF01207366"},{"key":"29","unstructured":"L Masanes. ``Extremal quantum correlations for n parties with two dichotomic observables per site'' (2005) arXiv:quant-ph\/0512100."},{"key":"30","doi-asserted-by":"publisher","unstructured":"Le Phuc Thinh, Antonios Varvitsiotis, and Yu Cai. ``Geometric structure of quantum correlators via semidefinite programming''. Phys. Rev. A 99, 052108 (2019). arXiv:1809.10886.","DOI":"10.1103\/PhysRevA.99.052108"},{"key":"31","doi-asserted-by":"publisher","unstructured":"Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. ``Bell nonlocality''. Rev. Mod. Phys. 86, 419\u2013478 (2014). arXiv:1303.2849.","DOI":"10.1103\/RevModPhys.86.419"},{"key":"32","doi-asserted-by":"publisher","unstructured":"Koon Tong Goh, J\u0119drzej Kaniewski, Elie Wolfe, Tam\u00e1s V\u00e9rtesi, Xingyao Wu, Yu Cai, Yeong-Cherng Liang, and Valerio Scarani. ``Geometry of the set of quantum correlations''. Phys. Rev. A 97, 022104 (2018). arXiv:1710.05892.","DOI":"10.1103\/PhysRevA.97.022104"},{"key":"33","doi-asserted-by":"publisher","unstructured":"Ivan \u0160upi\u0107 and Joseph Bowles. ``Self-testing of quantum systems: a review''. Quantum 4, 337 (2020). arXiv:1904.10042.","DOI":"10.22331\/q-2020-09-30-337"},{"key":"34","doi-asserted-by":"publisher","unstructured":"Rene Schwonnek, Koon Tong Goh, Ignatius W. Primaatmaja, Ernest Y. Z. Tan, Ramona Wolf, Valerio Scarani, and Charles C. W. Lim. ``Device-independent quantum key distribution with random key basis''. Nat. Commun. 12, 2880 (2020). arXiv:2005.02691.","DOI":"10.1038\/s41467-021-23147-3"},{"key":"35","doi-asserted-by":"publisher","unstructured":"Ernest Y. Z. Tan, Ren\u00e9 Schwonnek, Koon Tong Goh, Ignatius William Primaatmaja, and Charles C. W. Lim. ``Computing secure key rates for quantum key distribution with untrusted devices''. npj Quantum Inf. 7, 158 (2021). arXiv:1908.11372.","DOI":"10.1038\/s41534-021-00494-z"},{"key":"36","doi-asserted-by":"publisher","unstructured":"K. G. H. Vollbrecht and R. F. Werner. ``Entanglement measures under symmetry''. Phys. Rev. A 64, 062307 (2001). arXiv:quant-ph\/0010095.","DOI":"10.1103\/PhysRevA.64.062307"},{"key":"37","doi-asserted-by":"publisher","unstructured":"Peter Bierhorst. ``Geometric decompositions of Bell polytopes with practical applications''. J. Phys. A 49, 215301 (2016). arXiv:1511.04127.","DOI":"10.1088\/1751-8113\/49\/21\/215301"},{"key":"38","doi-asserted-by":"publisher","unstructured":"Monique Laurent. ``The real positive semidefinite completion problem for series-parallel graphs''. Linear Algebra and its Applications 252, 347\u2013366 (1997).","DOI":"10.1016\/0024-3795(95)00741-5"},{"key":"39","doi-asserted-by":"publisher","unstructured":"Vaughan F. R. Jones and J. H. Przytycki. ``Lissajous knots and billiard knots''. Banach Cent. Pub. 42, 145\u2013163 (1998).","DOI":"10.4064\/-42-1-145-163"},{"key":"40","doi-asserted-by":"publisher","unstructured":"Kaie Kubjas, Pablo A Parrilo, and Bernd Sturmfels. ``How to flatten a soccer ball''. In Aldo Conca, Joseph Gubeladze, and Tim R\u00f6mer, editors, Homological and Computational Methods in Commutative Algebra. Volume 20 of INdAM Ser., pages 141\u2013162. Springer (2017).","DOI":"10.1007\/978-3-319-61943-9_9"},{"key":"41","doi-asserted-by":"publisher","unstructured":"Kathleen S. Gibbons, Matthew J. Hoffman, and William K. Wootters. ``Discrete phase space based on finite fields''. Phys. Rev. A 70, 062101 (2004). arXiv:quant-ph\/0401155.","DOI":"10.1103\/physreva.70.062101"},{"key":"42","doi-asserted-by":"publisher","unstructured":"Reinhard F. Werner. ``Uncertainty relations for general phase spaces''. Frontiers of Physics 11, 1\u201310 (2016). arXiv:arxiv:1601.03843.","DOI":"10.1007\/s11467-016-0558-5"},{"key":"43","doi-asserted-by":"publisher","unstructured":"Amritanshu Prasad, Ilya Shapiro, and M.K. Vemuri. ``Locally compact abelian groups with symplectic self-duality''. Adv. Math. 225, 2429\u20132454 (2010). arXiv:0906.4397.","DOI":"10.1016\/j.aim.2010.04.023"},{"key":"44","doi-asserted-by":"publisher","unstructured":"Daniel Ciripoi, Nidhi Kaihnsa, Andreas L\u00f6hne, and Bernd Sturmfels. ``Computing convex hulls of trajectories''. Rev. Un. Mat. Argentina 60, 637\u2013662 (2019). arXiv:1810.03547.","DOI":"10.33044\/revuma.v60n2a22"},{"key":"45","doi-asserted-by":"publisher","unstructured":"Daniel Plaumann, Rainer Sinn, and Jannik Lennart Wesner. ``Families of faces and the normal cycle of a convex semi-algebraic set''. Beitr. Algebra Geom. (2022). arXiv:2104.13306.","DOI":"10.1007\/s13366-022-00657-9"},{"key":"46","unstructured":"Daniel R. Grayson and Michael E. Stillman. ``Macaulay2, a software system for research in algebraic geometry''. Available at http:\/\/www.math.uiuc.edu\/Macaulay2\/."},{"key":"47","doi-asserted-by":"publisher","unstructured":"John Ottem, Kristian Ranestad, Bernd Sturmfels, and Cynthia Vinzant. ``Quartic spectrahedra''. Mathematical Programming, Ser. B 151, 585\u2013612 (2015). arXiv:1311.3675.","DOI":"10.1007\/s10107-014-0844-3"},{"key":"48","doi-asserted-by":"publisher","unstructured":"Ad\u00e1n Cabello. ``How much larger quantum correlations are than classical ones''. Phys. Rev. A 72, 012113 (2005). arXiv:quant-ph\/0409192.","DOI":"10.1103\/PhysRevA.72.012113"},{"key":"49","doi-asserted-by":"publisher","unstructured":"C. E. Gonz\u00e1lez-Guill\u00e9n, C. H. Jim\u00e9nez, C. Palazuelos, and I. Villanueva. ``Sampling quantum nonlocal correlations with high probability''. Commun. Math. Phys. 344, 141\u2013154 (2016). arXiv:1412.4010.","DOI":"10.1007\/s00220-016-2625-8"},{"key":"50","doi-asserted-by":"publisher","unstructured":"C. R. Johnson and G. N\u00e6vdal. ``The probability that a (partial) matrix is positive semidefinite''. In I. Gohberg, R. Mennicken, and C. Tretter, editors, Recent Progress in Operator Theory. Pages 171\u2013182. Basel (1998). Birkh\u00e4user Basel.","DOI":"10.1007\/978-3-0348-8793-9_10"},{"key":"51","doi-asserted-by":"publisher","unstructured":"H. H Schaefer and M. P Wolff. ``Topological vector spaces''. Springer. (1999).","DOI":"10.1007\/978-1-4612-1468-7"},{"key":"52","doi-asserted-by":"publisher","unstructured":"Wojciech Tadej and Karol Z\u0300yczkowski. ``A concise guide to complex Hadamard matrices''. Open Systems & Information Dynamics 13, 133\u2013177 (2006). arXiv:quant-ph\/0512154.","DOI":"10.1007\/s11080-006-8220-2"},{"key":"53","doi-asserted-by":"publisher","unstructured":"H. Barnum, C.P. Gaebler, and A. Wilce. ``Ensemble steering, weak self-duality, and the structure of probabilistic theories''. Found. Phys 43, 1411\u20131427 (2013). arXiv:0912.5532.","DOI":"10.1007\/s10701-013-9752-2"},{"key":"54","doi-asserted-by":"publisher","unstructured":"Nikos Yannakakis. ``Stampacchia's property, self-duality and orthogonality relations''. Set-Valued and Variational Analysis 19, 555\u2013567 (2011). arXiv:1008.4958.","DOI":"10.1007\/s11228-011-0175-y"},{"key":"55","doi-asserted-by":"publisher","unstructured":"Jacek Bochnak, Michel Coste, and Marie-Fran\u00e7oise Roy. ``Real algebraic geometry''. Volume 36 of A Series of Modern Surveys in Mathematics. Springer Berlin, Heidelberg. (2013).","DOI":"10.1007\/978-3-662-03718-8"},{"key":"56","doi-asserted-by":"publisher","unstructured":"Joseph H. G. Fu. ``Algebraic integral geometry''. Pages 47\u2013112. Springer Basel. Basel (2014). arXiv:1103.6256.","DOI":"10.1007\/978-3-0348-0874-3_2"},{"key":"57","doi-asserted-by":"publisher","unstructured":"Herbert Federer. ``Curvature measures''. Trans. Amer. Math. Soc. 93, 418\u2013491 (1959).","DOI":"10.2307\/1993504"},{"key":"58","unstructured":"Peter Wintgen. ``Normal cycle and integral curvature for polyhedra in Riemannian manifolds''. In Gy. Soos and J. Szenthe, editors, Differential Geometry. Volume 21. North-Holland, Amsterdam (1982)."},{"key":"59","doi-asserted-by":"publisher","unstructured":"Martina Z\u00e4hle. ``Integral and current representation of Federer's curvature measures''. Arch. Math. 46, 557\u2013567 (1986).","DOI":"10.1007\/BF01195026"},{"key":"60","doi-asserted-by":"publisher","unstructured":"David Cohen-Steiner and Jean-Marie Morvan. ``Restricted Delaunay triangulations and normal cycle''. In SCG '03: Proceedings of the nineteenth annual symposium on Computational geometry. Pages 312\u2013321. (2003).","DOI":"10.1145\/777792.777839"},{"key":"61","doi-asserted-by":"publisher","unstructured":"Pierre Roussillon and Joan Alexis Glaun\u00e8s. ``Surface matching using normal cycles''. In Frank Nielsen and Fr\u00e9d\u00e9ric Barbaresco, editors, Geometric Science of Information. Pages 73\u201380. Cham (2017). Springer International Publishing.","DOI":"10.1007\/978-3-319-68445-1_9"},{"key":"62","doi-asserted-by":"publisher","unstructured":"Kehua Su, Na Lei, Wei Chen, Li Cui, Hang Si, Shikui Chen, and Xianfeng Gu. ``Curvature adaptive surface remeshing by sampling normal cycle''. Computer-Aided Design 111, 1\u201312 (2019).","DOI":"10.1016\/j.cad.2019.01.004"},{"key":"63","doi-asserted-by":"publisher","unstructured":"David A. Cox, John Little, and Donal O'Shea. ``Ideals, varieties, and algorithms''. Undergraduate Texts in Mathematics. Springer Cham. (2015). Fourth edition.","DOI":"10.1007\/978-3-319-16721-3"},{"key":"64","doi-asserted-by":"publisher","unstructured":"Guido A. Raggio. ``A remark on Bell's inequality and decomposable normal states''. Lett. Math. Phys. 15, 27\u201329 (1988).","DOI":"10.1007\/BF00416568"},{"key":"65","doi-asserted-by":"publisher","unstructured":"Marc-Olivier Renou, David Trillo, Mirjam Weilenmann, Thinh P. Le, Armin Tavakoli, Nicolas Gisin, Antonio Ac\u00edn, and Miguel Navascu\u00e9s. ``Quantum theory based on real numbers can be experimentally falsified''. Nature 600, 625\u2013629 (2021). arXiv:2101.10873.","DOI":"10.1038\/s41586-021-04160-4"},{"key":"66","doi-asserted-by":"publisher","unstructured":"Andrea Coladangelo, Koon Tong Goh, and Valerio Scarani. ``All pure bipartite entangled states can be self-tested''. Nature Commun. 8, 15485 (2017). arXiv:1611.08062.","DOI":"10.1038\/ncomms15485"},{"key":"67","doi-asserted-by":"publisher","unstructured":"Charles H. Bennett and Gilles Brassard. ``Quantum cryptography: Public key distribution and coin tossing''. Theoret. Comp. Sci. 560, 7\u201311 (2014). arXiv:2003.06557.","DOI":"10.1016\/j.tcs.2014.05.025"},{"key":"68","doi-asserted-by":"publisher","unstructured":"T. Franz, F. Furrer, and R. F. Werner. ``Extremal quantum correlations and cryptographic security''. Phys. Rev. Lett. 106, 250502 (2011). arXiv:1010.1131.","DOI":"10.1103\/PhysRevLett.106.250502"},{"key":"69","doi-asserted-by":"publisher","unstructured":"J\u0119drzej Kaniewski. ``Weak form of self-testing''. Phys. Rev. Research 2, 033420 (2020). arXiv:1910.00706.","DOI":"10.1103\/PhysRevResearch.2.033420"},{"key":"70","doi-asserted-by":"publisher","unstructured":"C. H. Bennett, G. Brassard, C. Crepeau, and U. M. Maurer. ``Generalized privacy amplification''. IEEE Transactions on Information Theory 41, 1915\u20131923 (1995).","DOI":"10.1109\/18.476316"},{"key":"71","doi-asserted-by":"publisher","unstructured":"Pavel Sekatski, Jean-Daniel Bancal, Xavier Valcarce, Ernest Y.-Z. Tan, Renato Renner, and Nicolas Sangouard. ``Device-independent quantum key distribution from generalized CHSH inequalities''. Quantum 5, 444 (2021). arXiv:2009.01784.","DOI":"10.22331\/q-2021-04-26-444"},{"key":"72","doi-asserted-by":"publisher","unstructured":"Ernest Y.-Z. Tan, Pavel Sekatski, Jean-Daniel Bancal, Ren\u00e9 Schwonnek, Renato Renner, Nicolas Sangouard, and Charles C.-W. Lim. ``Improved DIQKD protocols with finite-size analysis''. Quantum 6, 880 (2022). arXiv:2012.08714.","DOI":"10.22331\/q-2022-12-22-880"},{"key":"73","doi-asserted-by":"publisher","unstructured":"Marissa Giustina et al. ``Significant-loophole-free test of Bell's theorem with entangled photons''. Phys. Rev. Lett. 115, 250401 (2015). arXiv:1511.03190.","DOI":"10.1103\/PhysRevLett.115.250401"},{"key":"74","doi-asserted-by":"publisher","unstructured":"Lynden K. Shalm et al. ``Strong loophole-free test of local realism''. Phys. Rev. Lett. 115, 250402 (2015). arXiv:1511.03189.","DOI":"10.1103\/PhysRevLett.115.250402"},{"key":"75","doi-asserted-by":"publisher","unstructured":"D. P Nadlinger, J.-D. Bancal, and et al. ``Experimental quantum key distribution certified by Bell's theorem''. Nature 607, 682\u2013686 (2022). arXiv:2109.14600.","DOI":"10.1038\/s41586-022-04941-5"},{"key":"76","doi-asserted-by":"publisher","unstructured":"Wei Zhang, Harald Weinfurter, et al. ``A device-independent quantum key distribution system for distant users''. Nature 607, 687\u2013691 (2022). arXiv:2110.00575.","DOI":"10.1038\/s41586-022-04891-y"},{"key":"77","doi-asserted-by":"publisher","unstructured":"Feihu Xu, Yu-Zhe Zhang, Qiang Zhang, and Jian-Wei Pan. ``Device-independent quantum key distribution with random postselection''. Phys. Rev. Lett. 128, 110506 (2022). arXiv:2110.02701.","DOI":"10.1103\/PhysRevLett.128.110506"},{"key":"78","unstructured":"Wikipedia authors. ``Quantum key distribution''. url: https:\/\/en.wikipedia.org\/wiki\/Quantum_key_distribution. (accessed: 25-October-2021)."},{"key":"79","doi-asserted-by":"publisher","unstructured":"Armin Tavakoli, M\u00e1t\u00e9 Farkas, Denis Rosset, Jean-Daniel Bancal, and Jedrzej Kaniewski. ``Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments''. Science Advances 7, eabc3847 (2021). arXiv:1912.03225.","DOI":"10.1126\/sciadv.abc3847"},{"key":"80","unstructured":"Stephen J. Summers and Reinhard F. Werner. ``Maximal violation of Bell's inequalities for algebras of observables in tangent spacetime regions''. Ann. Inst. H. Poincar\u00e9. 49, 215\u2013243 (1988)."},{"key":"81","doi-asserted-by":"publisher","unstructured":"N. David Mermin. ``Is the moon there when nobody looks? Reality and the quantum theory''. Physics Today 38, 38\u201347 (1985).","DOI":"10.1063\/1.880968"},{"key":"82","unstructured":"Michael Janas, Michael E. Cuffaro, and Michel Janssen. ``Putting probabilities first. How Hilbert space generates and constrains them'' (2019) arXiv:1910.10688."},{"key":"83","doi-asserted-by":"publisher","unstructured":"Nicolas Brunner, Stefano Pironio, Antonio Ac\u00edn, Nicolas Gisin, Andr\u00e9 Allan M\u00e9thot, and Valerio Scarani. ``Testing the dimension of Hilbert spaces''. Phys. Rev. Lett. 100, 210503 (2008). arXiv:0802.0760.","DOI":"10.1103\/PhysRevLett.100.210503"},{"key":"84","doi-asserted-by":"publisher","unstructured":"Yu Cai, Jean-Daniel Bancal, Jacquiline Romero, and Valerio Scarani. ``A new device-independent dimension witness and its experimental implementation''. J. Phys. A 49, 305301 (2016). arXiv:1606.01602.","DOI":"10.1088\/1751-8113\/49\/30\/305301"},{"key":"85","doi-asserted-by":"publisher","unstructured":"Wan Cong, Yu Cai, Jean-Daniel Bancal, and Valerio Scarani. ``Witnessing irreducible dimension''. Phys. Rev. Lett. 119, 080401 (2017). arXiv:1611.01258.","DOI":"10.1103\/PhysRevLett.119.080401"},{"key":"86","doi-asserted-by":"publisher","unstructured":"R. Horodecki, P. Horodecki, and M. Horodecki. ``Violating Bell inequality by mixed spin-1\/2 states: necessary and sufficient condition''. Phys. Lett. A 200, 340\u2013344 (1995).","DOI":"10.1016\/0375-9601(95)00214-N"},{"key":"87","doi-asserted-by":"publisher","unstructured":"N. Gisin. ``Bell's inequality holds for all non-product states''. Physics Letters A 154, 201\u2013202 (1991).","DOI":"10.1016\/0375-9601(91)90805-I"},{"key":"88","doi-asserted-by":"publisher","unstructured":"R. Grone, C.R. Johnson, E.M. S\u00e1, and H. Wolkowicz. ``Positive definite completions of partial Hermitian matrices''. Lin. Alg. Appl. 58, 109\u2013124 (1984).","DOI":"10.1016\/0024-3795(84)90207-6"},{"key":"89","doi-asserted-by":"publisher","unstructured":"Alexander Barvinok. ``A course in convexity''. Graduate Studies in Mathematics 54. AMS. Providence (2002).","DOI":"10.1090\/gsm\/054"},{"key":"90","unstructured":"J. Dixmier. ``C*-algebras''. North-Holland mathematical library. North-Holland. (1982)."},{"key":"91","unstructured":"M. Reed and B. Simon. ``Methods of modern mathematical physics IV: Analysis of operators''. Elsevier Science. (1978)."},{"key":"92","doi-asserted-by":"publisher","unstructured":"Iain Raeburn and Allan M. Sinclair. ``The C*-algebra generated by two projections.''. Math. Scand. 65, 278\u2013290 (1989).","DOI":"10.7146\/math.scand.a-12283"},{"key":"93","doi-asserted-by":"publisher","unstructured":"Roy Araiza, Travis Russell, and Mark Tomforde. ``A universal representation for quantum commuting correlations''. Ann. Henri Poinc. 23, 4489\u20134520 (2022). arXiv:2102.05827.","DOI":"10.1007\/s00023-022-01197-7"},{"key":"94","doi-asserted-by":"publisher","unstructured":"I. Pitowsky. ``Quantum probability \u2013 quantum logic''. Volume 321 of Lect.Notes Phys. Springer. (1989).","DOI":"10.1007\/BFb0021186"},{"key":"95","doi-asserted-by":"publisher","unstructured":"Dan Geiger, Christopher Meek, Bernd Sturmfels, et al. ``On the toric algebra of graphical models''. Ann. Statist. 34, 1463\u20131492 (2006). arXiv:math\/0608054.","DOI":"10.1214\/009053606000000263"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2023-03-16-947\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,3,16]],"date-time":"2023-03-16T09:43:56Z","timestamp":1678959836000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2023-03-16-947\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,16]]},"references-count":96,"URL":"https:\/\/doi.org\/10.22331\/q-2023-03-16-947","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,16]]},"article-number":"947"}}