Self-testing of quantum systems: a review

Ivan Šupić1 and Joseph Bowles2

1Département de Physique Appliquée, Université de Genève, 1211 Genève, Switzerland
2ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


Self-testing is a method to infer the underlying physics of a quantum experiment in a black box scenario. As such it represents the strongest form of certification for quantum systems. In recent years a considerable self-testing literature has developed, leading to progress in related device-independent quantum information protocols and deepening our understanding of quantum correlations. In this work we give a thorough and self-contained introduction and review of self-testing and its application to other areas of quantum information.

► BibTeX data

► References

[1] A. Acín, A. Andrianov, L. Costa, E. Jané, J. I. Latorre, and R. Tarrach. Generalized Schmidt decomposition and classification of three-quantum-bit states. Phys. Rev. Lett., 85:1560–1563, Aug 2000. doi:10.1103/​PhysRevLett.85.1560.

[2] Ole Andersson, Piotr Badziąg, Ingemar Bengtsson, Irina Dumitru, and Adán Cabello. Self-testing properties of Gisin's elegant Bell inequality. Phys. Rev. A, 96:032119, Sep 2017. doi:10.1103/​PhysRevA.96.032119.

[3] Ole Andersson, Piotr Badziąg, Irina Dumitru, and Adán Cabello. Device-independent certification of two bits of randomness from one entangled bit and Gisin's elegant Bell inequality. Phys. Rev. A, 97:012314, Jan 2018. doi:10.1103/​PhysRevA.97.012314.

[4] R Augusiak, M Demianowicz, and A Acín. Local hidden variable models for entangled quantum states. Journal of Physics A: Mathematical and Theoretical, 47(42):424002, 2014. doi:10.1088/​1751-8113/​47/​42/​424002.

[5] Rotem Arnon-Friedman. Reductions to IID in Device-independent Quantum Information Processing. PhD thesis, 2018. arXiv:1812.10922.

[6] Rotem Arnon-Friedman and Jean-Daniel Bancal. Device-independent certification of one-shot distillable entanglement. New Journal of Physics, 21(3):033010, mar 2019. doi:10.1088/​1367-2630/​aafef6.

[7] Rotem Arnon-Friedman, Renato Renner, and Thomas Vidick. Simple and tight device-independent security proofs. SIAM Journal on Computing, 48(1):181–225, 2019. URL: https:/​/​​doi/​abs/​10.1137/​18M1174726, doi:10.1137/​18M1174726.

[8] Rotem Arnon-Friedman and Henry Yuen. Noise-Tolerant Testing of High Entanglement of Formation. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018), volume 107 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1–11:12, Dagstuhl, Germany, 2018. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. doi:10.4230/​LIPIcs.ICALP.2018.11.

[9] Emily Adlam and Adrian Kent. Device-independent relativistic quantum bit commitment. Phys. Rev. A, 92:022315, Aug 2015. doi:10.1103/​PhysRevA.92.022315.

[10] Antonio Acín and Lluis Masanes. Certified randomness in quantum physics. Nature, 540:213–219, 2016. doi:10.1038/​nature20119.

[11] Antonio Acín, Serge Massar, and Stefano Pironio. Randomness versus nonlocality and entanglement. Physical Review Letters, 108(10), Sep 2012. doi:10.1103/​physrevlett.108.100402.

[12] N Aharon, S Massar, S Pironio, and J Silman. Device-independent bit commitment based on the CHSH inequality. New Journal of Physics, 18(2):025014, feb 2016. doi:10.1088/​1367-2630/​18/​2/​025014.

[13] A. Ambainis, A. Nayak, A. Ta-Shama, and U. Vazirani. Dense quantum coding and a lower bound for 1-way quantum automata. Proceedings of 31st ACM Symposium on Theory of Computing, page 376, 1999. doi:10.1145/​301250.301347.

[14] Antonio Acín, Stefano Pironio, Tamás Vértesi, and Peter Wittek. Optimal randomness certification from one entangled bit. Phys. Rev. A, 93:040102, Apr 2016. doi:10.1103/​PhysRevA.93.040102.

[15] Mateus Araújo, Marco Túlio Quintino, Costantino Budroni, Marcelo Terra Cunha, and Adán Cabello. All noncontextuality inequalities for the $n$-cycle scenario. Phys. Rev. A, 88:022118, Aug 2013. doi:10.1103/​PhysRevA.88.022118.

[16] M. Ardehali. Bell inequalities with a magnitude of violation that grows exponentially with the number of particles. Phys. Rev. A, 46:5375–5378, Nov 1992. doi:10.1103/​PhysRevA.46.5375.

[17] Jonathan Barrett. Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality. Phys. Rev. A, 65:042302, Mar 2002. doi:10.1103/​PhysRevA.65.042302.

[18] F Baccari, R Augusiak, I Šupić, J Tura, and A Acín. Scalable bell inequalities for qubit graph states and robust self-testing. Physical Review Letters, 124(2):020402, 2020. doi:10.1103/​PhysRevLett.124.020402.

[19] Charles Bennett and Gilles Brassard. Quantum cryptography: Public key distribution and coin tossing. In Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, 1984, pages 175–179, 01 1984. doi:10.1016/​j.tcs.2014.05.025.

[20] C. H. Bennett, G. Brassard, S. Breidbart, and S. Wiesner. Eavesdrop-detecting quantum communications channel. IBM technical disclosure bulletin, 26(8):4363–4366, 01 1984.

[21] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical Review Letters, 70:1895–1899, Mar 1993. doi:10.1103/​PhysRevLett.70.1895.

[22] Piotr Badziąg, Ingemar Bengtsson, Adán Cabello, Helena Granström, and Jan-Åke Larsson. Pentagrams and paradoxes. Foundations of Physics, 41:414–423, 02 2011. doi:10.1007/​s10701-010-9433-3.

[23] Gilles Brassard, Anne Broadbent, and Alain Tapp. Quantum pseudo-telepathy. Foundations of Physics, 35(11):1877–1907, Nov 2005. doi:10.1007/​s10701-005-7353-4.

[24] Samuel L Braunstein and Carlton M Caves. Wringing out better Bell inequalities. Annals of Physics, 202(1):22 – 56, 1990. doi:10.1016/​0003-4916(90)90339-P.

[25] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. Bell nonlocality. Rev. Mod. Phys., 86:419–478, Apr 2014. doi:10.1103/​RevModPhys.86.419.

[26] Francesco Buscemi and Nilanjana Datta. Distilling entanglement from arbitrary resources. Journal of Mathematical Physics, 51(10):102201, 2010. doi:10.1063/​1.3483717.

[27] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entanglement and quantum error correction. Phys. Rev. A, 54:3824–3851, Nov 1996. doi:10.1103/​PhysRevA.54.3824.

[28] J. S. Bell. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1:195–200, Nov 1964. doi:10.1103/​PhysicsPhysiqueFizika.1.195.

[29] Joseph Bowles, Jérémie Francfort, Mathieu Fillettaz, Flavien Hirsch, and Nicolas Brunner. Genuinely multipartite entangled quantum states with fully local hidden variable models and hidden multipartite nonlocality. Phys. Rev. Lett., 116:130401, Mar 2016. doi:10.1103/​PhysRevLett.116.130401.

[30] Joseph Bowles, Flavien Hirsch, Marco Túlio Quintino, and Nicolas Brunner. Sufficient criterion for guaranteeing that a two-qubit state is unsteerable. Phys. Rev. A, 93:022121, Feb 2016. doi:10.1103/​PhysRevA.93.022121.

[31] A V Belinskii and D N Klyshko. Interference of light and Bell's theorem. Physics-Uspekhi, 36(8):653, 1993. doi:10.1070/​pu1993v036n08abeh002299.

[32] Spencer Breiner, Amir Kalev, and Carl A. Miller. Parallel self-testing of the GHZ state with a proof by diagrams. In Peter Selinger and Giulio Chiribella, editors, Proceedings of the 15th International Conference on Quantum Physics and Logic, Halifax, Canada, 3-7th June 2018, volume 287 of Electronic Proceedings in Theoretical Computer Science, pages 43–66. Open Publishing Association, 2019. doi:10.4204/​EPTCS.287.3.

[33] C.-E. Bardyn, T. C. H. Liew, S. Massar, M. McKague, and V. Scarani. Device-independent state estimation based on Bell's inequalities. Phys. Rev. A, 80:062327, Dec 2009. doi:10.1103/​PhysRevA.80.062327.

[34] Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/​correcting with applications to numerical problems. J. Comput. Syst. Sci., 47(3):549–595, December 1993. doi:10.1016/​0022-0000(93)90044-W.

[35] H. Buhrman and S. Massar. Causality and Tsirelson's bounds. Phys. Rev. A, 72:052103, Nov 2005. doi:10.1103/​PhysRevA.72.052103.

[36] Cédric Bamps, Serge Massar, and Stefano Pironio. Device-independent randomness generation with sublinear shared quantum resources. Quantum, 2:86, August 2018. doi:10.22331/​q-2018-08-22-86.

[37] Samuel L. Braunstein, A. Mann, and M. Revzen. Maximal violation of Bell inequalities for mixed states. Phys. Rev. Lett., 68:3259–3261, Jun 1992. doi:10.1103/​PhysRevLett.68.3259.

[38] Jean-Daniel Bancal, Miguel Navascués, Valerio Scarani, Tamás Vértesi, and Tzyh Haur Yang. Physical characterization of quantum devices from nonlocal correlations. Phys. Rev. A, 91:022115, Feb 2015. doi:10.1103/​PhysRevA.91.022115.

[39] Cédric Bamps and Stefano Pironio. Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalities and their application to self-testing. Phys. Rev. A, 91:052111, May 2015. doi:10.1103/​PhysRevA.91.052111.

[40] Joseph Bowles, Marco Túlio Quintino, and Nicolas Brunner. Certifying the dimension of classical and quantum systems in a prepare-and-measure scenario with independent devices. Phys. Rev. Lett., 112:140407, Apr 2014. doi:10.1103/​PhysRevLett.112.140407.

[41] Cyril Branciard, Denis Rosset, Nicolas Gisin, and Stefano Pironio. Bilocal versus nonbilocal correlations in entanglement-swapping experiments. Phys. Rev. A, 85:032119, Mar 2012. doi:10.1103/​PhysRevA.85.032119.

[42] Cyril Branciard, Denis Rosset, Yeong-Cherng Liang, and Nicolas Gisin. Measurement-device-independent entanglement witnesses for all entangled quantum states. Phys. Rev. Lett., 110:060405, Feb 2013. doi:10.1103/​PhysRevLett.110.060405.

[43] Jean-Daniel Bancal, Kai Redeker, Pavel Sekatski, Wenjamin Rosenfeld, and Nicolas Sangouard. Device-independent certification of an elementary quantum network link, 2018. arXiv:1812.09117.

[44] Dagmar Bruß. Optimal eavesdropping in quantum cryptography with six states. Phys. Rev. Lett., 81:3018–3021, Oct 1998. doi:10.1103/​PhysRevLett.81.3018.

[45] Kishor Bharti, Maharshi Ray, Antonios Varvitsiotis, Naqueeb Ahmad Warsi, Adán Cabello, and Leong-Chuan Kwek. Robust self-testing of quantum systems via noncontextuality inequalities. Physical review letters, 122(25):250403, 2019. doi:10.1103/​PhysRevLett.122.250403.

[46] Joseph Bowles, Ivan Šupić, Daniel Cavalcanti, and Antonio Acín. Device-independent entanglement certification of all entangled states. Phys. Rev. Lett., 121:180503, Oct 2018. doi:10.1103/​PhysRevLett.121.180503.

[47] Joseph Bowles, Ivan Šupić, Daniel Cavalcanti, and Antonio Acín. Self-testing of Pauli observables for device-independent entanglement certification. Phys. Rev. A, 98:042336, Oct 2018. doi:10.1103/​PhysRevA.98.042336.

[48] Jean-Daniel Bancal, Nicolas Sangouard, and Pavel Sekatski. Noise-resistant device-independent certification of Bell state measurements. Phys. Rev. Lett., 121:250506, Dec 2018. doi:10.1103/​PhysRevLett.121.250506.

[49] Francesco Buscemi. All entangled quantum states are nonlocal. Phys. Rev. Lett., 108:200401, May 2012. doi:10.1103/​PhysRevLett.108.200401.

[50] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. doi:10.1017/​CBO9780511804441.

[51] 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. doi:10.1103/​PhysRevLett.69.2881.

[52] Daniel Cavalcanti, Mafalda L. Almeida, Valerio Scarani, and Antonio Acín. Quantum networks reveal quantum nonlocality. Nature News, Feb 2011. doi:10.1038/​ncomms1193.

[53] Wan Cong, Yu Cai, Jean-Daniel Bancal, and Valerio Scarani. Witnessing irreducible dimension. Phys. Rev. Lett., 119:080401, Aug 2017. doi:10.1103/​PhysRevLett.119.080401.

[54] Andrea Coladangelo, Alex Grilo, Stacey Jeffery, and Thomas Vidick. Verifier-on-a-leash: new schemes for verifiable delegated quantum computation, with quasilinear resources, 2017. arXiv:1708.07359.

[55] Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar, and Sandu Popescu. Bell inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett., 88:040404, Jan 2002. doi:10.1103/​PhysRevLett.88.040404.

[56] Andrea Coladangelo, Koon Tong Goh, and Valerio Scarani. All pure bipartite entangled states can be self-tested. Nature Communications, 8:15485, may 2017. doi:10.1038/​ncomms15485.

[57] Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra and its Applications, 10(3):285–290, 1975. doi:10.1016/​0024-3795(75)90075-0.

[58] E. G. Cavalcanti, Q. Y. He, M. D. Reid, and H. M. Wiseman. Unified criteria for multipartite quantum nonlocality. Phys. Rev. A, 84:032115, Sep 2011. doi:10.1103/​PhysRevA.84.032115.

[59] 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. doi:10.1103/​PhysRevLett.23.880.

[60] André Chailloux and Iordanis Kerenidis. Optimal bounds for quantum bit commitment. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 354–362. IEEE, 2011. doi:10.1109/​FOCS.2011.42.

[61] Roger Colbeck and Adrian Kent. Private randomness expansion with untrusted devices. Journal of Physics A: Mathematical and Theoretical, 44(9):095305, 2011. doi:10.1088/​1751-8113/​44/​9/​095305.

[62] Bob Coecke and Aleks Kissinger. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, 2017. doi:10.1017/​9781316219317.

[63] Tim Coopmans, Jędrzej Kaniewski, and Christian Schaffner. Robust self-testing of two-qubit states, 2019. arXiv:1902.00870. doi:10.1103/​PhysRevA.99.052123.

[64] Richard Cleve, Li Liu, and William Slofstra. Perfect commuting-operator strategies for linear system games. Journal of Mathematical Physics, 58(1):012202, 2017. doi:10.1063/​1.4973422.

[65] Richard Cleve and Rajat Mittal. Characterization of binary constraint system games. Automata, Languages, and Programming Lecture Notes in Computer Science, pages 320–331, 2014. doi:10.1007/​978-3-662-43948-7_27.

[66] M. Coudron and A. Natarajan. The parallel-repeated magic square game is rigid, 2016. arXiv:1609.06306.

[67] Roger Colbeck. Quantum And Relativistic Protocols For Secure Multi-Party Computation. PhD thesis, University of Cambridge, 2006. arXiv:0911.3814.

[68] Andrea Coladangelo. Parallel self-testing of (tilted) epr pairs via copies of (tilted) chsh and the magic square game. Quantum Info. Comput., 17(9-10):831–865, August 2017. URL: http:/​/​​citation.cfm?id=3179561.3179567.

[69] Andrea Coladangelo. Generalization of the Clauser-Horne-Shimony-Holt inequality self-testing maximally entangled states of any local dimension. Phys. Rev. A, 98:052115, Nov 2018. doi:10.1103/​PhysRevA.98.052115.

[70] Andrea Coladangelo. A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations. Quantum, 4:282, June 2020. URL: https:/​/​​10.22331/​q-2020-06-18-282, doi:10.22331/​q-2020-06-18-282.

[71] Rui Chao, Ben W. Reichardt, Chris Sutherland, and Thomas Vidick. Overlapping Qubits. In Christos H. Papadimitriou, editor, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017), volume 67 of Leibniz International Proceedings in Informatics (LIPIcs), pages 48:1–48:21, Dagstuhl, Germany, 2017. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. doi:10.4230/​LIPIcs.ITCS.2017.48.

[72] Rui Chao, Ben W. Reichardt, Chris Sutherland, and Thomas Vidick. Test for a large amount of entanglement, using few measurements. Quantum, 2:92, September 2018. doi:10.22331/​q-2018-09-03-92.

[73] Daniel Cavalcanti and Paul Skrzypczyk. Quantum steering: a review with focus on semidefinite programming. Reports on Progress in Physics, 80(2):024001, 2017. doi:10.1088/​1361-6633/​80/​2/​024001.

[74] Andrea Coladangelo and Jalex Stark. Robust self-testing for linear constraint system games, 2017. arXiv:1709.09267.

[75] Andrea Coladangelo and Jalex Stark. Separation of finite and infinite-dimensional quantum correlations, with infinite question or answer sets, 2017. arXiv:1708.06522.

[76] Andrea Coladangelo and Jalex Stark. Unconditional separation of finite and infinite-dimensional quantum correlations, 2018. arXiv:1804.05116.

[77] Matthew Coudron and Henry Yuen. Infinite randomness expansion with a constant number of devices. In Proceedings of the Forty-sixth Annual ACM Symposium on Theory of Computing, STOC '14, pages 427–436, New York, NY, USA, 2014. ACM. doi:10.1145/​2591796.2591873.

[78] Frederic Dupuis, Omar Fawzi, and Renato Renner. Entropy accumulation, 2016. arXiv:1607.01796.

[79] Chirag Dhara, Giuseppe Prettico, and Antonio Acín. Maximal quantum randomness in Bell tests. Phys. Rev. A, 88:052116, Nov 2013. doi:10.1103/​PhysRevA.88.052116.

[80] Ken Dykema, Vern I. Paulsen, and Jitendra Prakash. Non-closure of the set of quantum correlations via graphs. Communications in Mathematical Physics, 365(3):1125–1142, Feb 2019. doi:10.1007/​s00220-019-03301-1.

[81] Artur K. Ekert. Quantum cryptography based on Bell's theorem. Phys. Rev. Lett., 67:661–663, Aug 1991. doi:10.1103/​PhysRevLett.67.661.

[82] Artur Ekert and Renato Renner. The ultimate physical limits of privacy. Nature News, Mar 2014. doi:10.1038/​nature13132.

[83] Matteo Fadel. Self-testing Dicke states, 2017. arXiv:1707.01215.

[84] Máté Farkas and Jędrzej Kaniewski. Self-testing mutually unbiased bases in the prepare-and-measure scenario. Phys. Rev. A, 99:032316, Mar 2019. doi:10.1103/​PhysRevA.99.032316.

[85] Suchetana Goswami, Bihalan Bhattacharya, Debarshi Das, Souradeep Sasmal, C. Jebaratnam, and A. S. Majumdar. One-sided device-independent self-testing of any pure two-qubit entangled state. Phys. Rev. A, 98:022311, Aug 2018. doi:10.1103/​PhysRevA.98.022311.

[86] Mariami Gachechiladze, Costantino Budroni, and Otfried Gühne. Extreme violation of local realism in quantum hypergraph states. Phys. Rev. Lett., 116:070401, Feb 2016. doi:10.1103/​PhysRevLett.116.070401.

[87] Koon Tong Goh, Jean-Daniel Bancal, and Valerio Scarani. Measurement-device-independent quantification of entanglement for given Hilbert space dimension. New Journal of Physics, 18(4):045022, apr 2016. doi:10.1088/​1367-2630/​18/​4/​045022.

[88] Daniel Gottesman and Isaac L Chuang. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature, 402:390, 1999. doi:10.1038/​46503.

[89] W T Gowers and O Hatami. Inverse and stability theorems for approximate representations of finite groups. Sbornik: Mathematics, 208(12):1784–1817, 2017. doi:10.1070/​sm8872.

[90] Nicolas Gisin. Bell inequalities: Many questions, a few answers. In Wayne C. Myrvold and Joy Christian, editors, Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle, pages 125–138. Springer, 2009. doi:10.1007/​978-1-4020-9107-0_9.

[91] Alexandru Gheorghiu, Theodoros Kapourniotis, and Elham Kashefi. Verification of quantum computation: An overview of existing approaches. Theory of Computing Systems, pages 1–94, 2018. doi:10.1007/​s00224-018-9872-3.

[92] Alexandru Gheorghiu, Elham Kashefi, and Petros Wallden. Robustness and device independence of verifiable blind quantum computing. New Journal of Physics, 17(8):083040, 2015. doi:10.1088/​1367-2630/​17/​8/​083040.

[93] Koon Tong Goh, Jędrzej Kaniewski, Elie Wolfe, Tamás Vértesi, Xingyao Wu, Yu Cai, Yeong-Cherng Liang, and Valerio Scarani. Geometry of the set of quantum correlations. Phys. Rev. A, 97:022104, Feb 2018. doi:10.1103/​PhysRevA.97.022104.

[94] S. Gómez, A. Mattar, I. Machuca, E. S. Gómez, D. Cavalcanti, O. Jiménez Farías, A. Acín, and G. Lima. Experimental investigation of partially entangled states for device-independent randomness generation and self-testing protocols. Phys. Rev. A, 99:032108, Mar 2019. doi:10.1103/​PhysRevA.99.032108.

[95] Koon Tong Goh, Chithrabhanu Perumangatt, Zhi Xian Lee, Alexander Ling, and Valerio Scarani. Experimental comparison of tomography and self-testing in certifying entanglement. Physical Review A, 100(2):022305, 2019. doi:10.1103/​PhysRevA.100.022305.

[96] Otfried Gühne, Géza Tóth, Philipp Hyllus, and Hans J. Briegel. Bell inequalities for graph states. Phys. Rev. Lett., 95:120405, Sep 2005. doi:10.1103/​PhysRevLett.95.120405.

[97] Marissa Giustina, Marijn A. M. Versteegh, Sören Wengerowsky, Johannes Handsteiner, Armin Hochrainer, Kevin Phelan, Fabian Steinlechner, Johannes Kofler, Jan-Åke Larsson, Carlos Abellán, Waldimar Amaya, Valerio Pruneri, Morgan W. Mitchell, Jörn Beyer, Thomas Gerrits, Adriana E. Lita, Lynden K. Shalm, Sae Woo Nam, Thomas Scheidl, Rupert Ursin, Bernhard Wittmann, and Anton Zeilinger. Significant-loophole-free test of bell's theorem with entangled photons. Phys. Rev. Lett., 115:250401, Dec 2015. doi:10.1103/​PhysRevLett.115.250401.

[98] Alexandru Gheorghiu, Petros Wallden, and Elham Kashefi. Rigidity of quantum steering and one-sided device-independent verifiable quantum computation. New Journal of Physics, 19(2):023043, 2017. doi:10.1088/​1367-2630/​aa5cff.

[99] Lucien Hardy. Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories. Phys. Rev. Lett., 68:2981–2984, May 1992. doi:10.1103/​PhysRevLett.68.2981.

[100] Lucien Hardy. Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett., 71:1665–1668, Sep 1993. doi:10.1103/​PhysRevLett.71.1665.

[101] 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, and et al. Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 526(7575):682–686, 2015. doi:10.1038/​nature15759.

[102] Masahito Hayashi and Michal Hajdušek. Self-guaranteed measurement-based quantum computation. Phys. Rev. A, 97:052308, May 2018. doi:10.1103/​PhysRevA.97.052308.

[103] B. Hensen, N. Kalb, M. S. Blok, A. E. Dréau, A. Reiserer, R. F. L. Vermeulen, R. N. Schouten, M. Markham, D. J. Twitchen, K. Goodenough, and et al. Loophole-free Bell test using electron spins in diamond: second experiment and additional analysis. Scientific Reports, 6(1), 2016. doi:10.1038/​srep30289.

[104] Teiko Heinosaari, Jukka Kiukas, and Daniel Reitzner. Noise robustness of the incompatibility of quantum measurements. Phys. Rev. A, 92:022115, Aug 2015. doi:10.1103/​PhysRevA.92.022115.

[105] Flavien Hirsch, Marco Túlio Quintino, Joseph Bowles, Tamas Vértesi, and Nicolas Brunner. Entanglement without hidden nonlocality. New Journal of Physics, 18(11):113019, 2016. doi:10.1088/​1367-2630/​18/​11/​113019.

[106] Flavien Hirsch, Marco Túlio Quintino, Joseph Bowles, and Nicolas Brunner. Genuine hidden quantum nonlocality. Phys. Rev. Lett., 111:160402, Oct 2013. doi:10.1103/​PhysRevLett.111.160402.

[107] Sania Jevtic, Michael J. W. Hall, Malcolm R. Anderson, Marcin Zwierz, and Howard M. Wiseman. Einstein–Podolsky–Rosen steering and the steering ellipsoid. J. Opt. Soc. Am. B, 32(4):A40–A49, Apr 2015. doi:10.1364/​JOSAB.32.000A40.

[108] Rahul Jain, Carl A. Miller, and Yaoyun Shi. Parallel Device-Independent Quantum Key Distribution, 2017. arXiv:1703.05426.

[109] Rahul Jain, Carl A Miller, and Yaoyun Shi. Parallel device-independent quantum key distribution. IEEE Transactions on Information Theory, 2020. doi:10.1109/​TIT.2020.2986740.

[110] Egbert R. Van Kampen. On some lemmas in the theory of groups. American Journal of Mathematics, 55(1):268–273, 1933. doi:10.2307/​2371129.

[111] Jędrzej Kaniewski. Analytic and nearly optimal self-testing bounds for the Clauser-Horne-Shimony-Holt and Mermin inequalities. Phys. Rev. Lett., 117:070402, Aug 2016. doi:10.1103/​PhysRevLett.117.070402.

[112] Jędrzej Kaniewski. Self-testing of binary observables based on commutation. Phys. Rev. A, 95:062323, Jun 2017. doi:10.1103/​PhysRevA.95.062323.

[113] Alexander A. Klyachko, M. Ali Can, Sinem Binicioğlu, and Alexander S. Shumovsky. Simple test for hidden variables in spin-1 systems. Phys. Rev. Lett., 101:020403, Jul 2008. doi:10.1103/​PhysRevLett.101.020403.

[114] Amir Kalev and Carl A Miller. Rigidity of the magic pentagram game. Quantum Science and Technology, 3(1):015002, 2018. doi:10.1088/​2058-9565/​aa931d.

[115] Adrian Kent, William J. Munro, and Timothy P. Spiller. Quantum tagging: Authenticating location via quantum information and relativistic signaling constraints. Phys. Rev. A, 84:012326, Jul 2011. doi:10.1103/​PhysRevA.84.012326.

[116] B. Kraus. Local unitary equivalence of multipartite pure states. Phys. Rev. Lett., 104:020504, Jan 2010. doi:10.1103/​PhysRevLett.104.020504.

[117] S. Kochen and E. Specker. The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics, 17(1):59–87, 1967. doi:10.1512/​iumj.1968.17.17004.

[118] Jędrzej Kaniewski, Ivan Šupić, Jordi Tura, Flavio Baccari, Alexia Salavrakos, and Remigiusz Augusiak. Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems. Quantum, 3:198, October 2019. URL: https:/​/​​10.22331/​q-2019-10-24-198, doi:10.22331/​q-2019-10-24-198.

[119] Jędrzej Kaniewski and Stephanie Wehner. Device-independent two-party cryptography secure against sequential attacks. New Journal of Physics, 18(5):055004, may 2016. doi:10.1088/​1367-2630/​18/​5/​055004.

[120] R. Konig, S. Wehner, and J. Wullschleger. Unconditional security from noisy quantum storage. IEEE Transactions on Information Theory, 58(3):1962–1984, March 2012. doi:10.1109/​TIT.2011.2177772.

[121] Hoi-Kwong Lo and Hoi Fung Chau. Is quantum bit commitment really possible? Physical Review Letters, 78(17):3410, 1997. doi:10.1103/​physrevlett.78.3410.

[122] Xinhui Li, Yu Cai, Yunguang Han, Qiaoyan Wen, and Valerio Scarani. Self-testing using only marginal information. Phys. Rev. A, 98:052331, Nov 2018. doi:10.1103/​PhysRevA.98.052331.

[123] Thomas Lawson, Noah Linden, and Sandu Popescu. Biased nonlocal quantum games, 2010. arXiv:1011.6245v1.

[124] Jian Li, Tong-Jun Liu, Si Wang, C. Jebarathinam, and Qin Wang. Experimental violation of mermin steering inequality by three-photon entangled states with nontrivial ghz-fidelity. Opt. Express, 27(9):13559–13567, Apr 2019. doi:10.1364/​OE.27.013559.

[125] L. Lovasz. On the Shannon capacity of a graph. IEEE Transactions on Information Theory, 25(1):1–7, January 1979. doi:10.1109/​TIT.1979.1055985.

[126] Pei-Sheng Lin, Denis Rosset, Yanbao Zhang, Jean-Daniel Bancal, and Yeong-Cherng Liang. Device-independent point estimation from finite data and its application to device-independent property estimation. Physical Review A, 97(3):032309, 2018. doi:10.1103/​PhysRevA.97.032309.

[127] Yeong-Cherng Liang, Robert W. Spekkens, and Howard M. Wiseman. Specker's parable of the overprotective seer: A road to contextuality, nonlocality and complementarity. Physics Reports, 506(1):1 – 39, 2011. doi:10.1016/​j.physrep.2011.05.001.

[128] Xinhui Li, Yukun Wang, Yunguang Han, Sujuan Qin, Fei Gao, and Qiaoyan Wen. Analytic robustness bound for self-testing of the singlet with two binary measurements. J. Opt. Soc. Am. B, 36(2):457–463, Feb 2019. doi:10.1364/​JOSAB.36.000457.

[129] U. Mahadev. Classical verification of quantum computations. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 259–267, Oct 2018. doi:10.1109/​FOCS.2018.00033.

[130] Robert A. Malaney. Location-dependent communications using quantum entanglement. Phys. Rev. A, 81:042319, Apr 2010. doi:10.1103/​PhysRevA.81.042319.

[131] Laura Mančinska. Maximally Entangled State in Pseudo-Telepathy Games, pages 200–207. Springer International Publishing, Cham, 2014. doi:10.1007/​978-3-319-13350-8_15.

[132] Dominic Mayers. Unconditionally secure quantum bit commitment is impossible. Physical review letters, 78(17):3414, 1997. doi:10.1103/​PhysRevLett.78.3414.

[133] Nikolai Miklin, Borkała Borkała, and Marcin Pawłowski. Self-testing of unsharp measurements, 2019. arXiv:1903.12533. doi:10.10.1103/​PhysRevResearch.2.033014.

[134] Matthew McKague. Quantum Information Processing with Adversarial Devices. PhD thesis, University of Waterloo, 2010. URL: http:/​/​​10012/​5259.

[135] Mathew McKague. Self-testing graph states. In D. Bacon, M. Martin-Delgado, and M. Roetteler, editors, Theory of Quantum Computation, Communication, and Cryptography ,, volume 6745 of Lecture Notes in Computer Science, pages 104–120. Springer, Berlin, Heidelberg, 2014. doi:https:/​/​​10.1007/​978-3-642-54429-3_7.

[136] Matthew McKague. Interactive proofs for $\mathsf{BQP}$ via self-tested graph states. Theory of Computing, 12(3):1–42, 2016. doi:10.4086/​toc.2016.v012a003.

[137] Matthew McKague. Self-testing in parallel. New Journal of Physics, 18:045013, 2016. doi:10.1088/​1367-2630/​18/​4/​045013.

[138] Matthew McKague. Self-testing in parallel with CHSH. Quantum, 1:1, April 2017. doi:10.22331/​q-2017-04-25-1.

[139] Ashley Montanaro and Ronald de Wolf. A survey of quantum property testing. Theory of Computing Graduate Surveys, 7, 2016. doi:10.4086/​

[140] N. David Mermin. Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett., 65:1838–1840, Oct 1990. doi:10.1103/​PhysRevLett.65.1838.

[141] N. David Mermin. Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett., 65:3373–3376, Dec 1990. doi:10.1103/​PhysRevLett.65.3373.

[142] M. McKague and M. Mosca. Generalized self-testing and the security of the 6-state protocol. In W. van Dam, V. M. Kendon, and S. Severini, editors, Theory of Quantum Computation, Communication, and Cryptography, volume 6519 of Lecture Notes in Computer Science, pages 113–130. Springer, Berlin, Heidelberg, 2011. doi:10.1007/​978-3-642-18073-6_10.

[143] Frédéric Magniez, Dominic Mayers, Michele Mosca, and Harold Ollivier. Self-testing of quantum circuits. In Michele Bugliesi, Bart Preneel, Vladimiro Sassone, and Ingo Wegener, editors, Automata, Languages and Programming, pages 72–83, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg. doi:10.1007/​11786986_8.

[144] Mehdi Mhalla and Simon Perdrix. Graph States, Pivot Minor, and Universality of ($X,Z$)-measurements. International Journal of Unconventional Computing, 9(1-2):153–171, 2013. Special Issue: New Worlds of Computation. URL: https:/​/​​hal-00934104.

[145] Piotr Mironowicz and Marcin Pawłowski. Experimentally feasible semi-device-independent certification of four-outcome positive-operator-valued measurements. Physical Review A, 100(3):030301, 2019. doi:10.1103/​PhysRevA.100.030301.

[146] Magdalena Musat and Mikael Rørdam. Non-closure of quantum correlation matrices and factorizable channels that require infinite dimensional ancilla (with an appendix by narutaka ozawa). Communications in Mathematical Physics, pages 1–16, 2019. doi:10.1007/​s00220-019-03449-w.

[147] C. A. Miller and Y. Shi. Optimal robust self-testing by binary nonlocal XOR games. Leibniz Int. Proc. Informat., 22(254), 2013. doi:10.4230/​LIPIcs.TQC.2013.254.

[148] Carl A. Miller and Yaoyun Shi. Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices. J. ACM, 63(4):33:1–33:63, October 2016. doi:10.1145/​2885493.

[149] Dominic Mayers and Christian Tourenne. Violation of Locality and Self-Checking Source: A Brief Account, pages 269–276. Springer US, Boston, MA, 2002. doi:10.1007/​0-306-47114-0_43.

[150] D. Mayers and A. Yao. Quantum cryptography with imperfect apparatus. Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280), 1998. doi:10.1109/​sfcs.1998.743501.

[151] D. Mayers and A. Yao. Self testing quantum apparatus. Quantum Info. Comput., 4:273, 2004. arXiv:quant-ph/​0307205.

[152] M. McKague, T. H. Yang, and V. Scarani. Robust self-testing of the singlet. Journal of Mathematical Physics, 45(45):455304, 2012. doi:10.1088/​1751-8113/​45/​45/​455304.

[153] A. Nayak. Optimal lower bounds for quantum automata and random access codes. Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), page 369, 1999. doi:10.1109/​SFFCS.1999.814608.

[154] Michael A. Nielsen and Isaac L. Chuang. Quantum computation and quantum information. Cambridge University Press, 2018. doi:10.1017/​CBO9780511976667.

[155] Miguel Navascués, Stefano Pironio, and Antonio Acín. Bounding the set of quantum correlations. Phys. Rev. Lett., 98:010401, Jan 2007. doi:10.1103/​PhysRevLett.98.010401.

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

[157] Anand Natarajan and Thomas Vidick. A quantum linearity test for robustly verifying entanglement. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 1003–1015, New York, NY, USA, 2017. ACM. doi:10.1145/​3055399.3055468.

[158] A. Natarajan and T. Vidick. Low-degree testing for quantum states, and a quantum entangled games PCP for QMA. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 731–742, Oct 2018. doi:10.1109/​FOCS.2018.00075.

[159] Dimiter Ostrev and Thomas Vidick. The structure of nearly-optimal quantum strategies for the CHSH (n) XOR games. Quantum Information & Computation, 16(13-14), pp.(13-14):1191–1211, 2016.

[160] Jonathan Oppenheim and Stephanie Wehner. The uncertainty principle determines the nonlocality of quantum mechanics. Science, 330(6007):1072–1074, 2010. doi:10.1126/​science.1192065.

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

[162] Carlos Palazuelos. Superactivation of quantum nonlocality. Phys. Rev. Lett., 109:190401, Nov 2012. doi:10.1103/​PhysRevLett.109.190401.

[163] Stefano Pironio, Antonio Acín, Serge Massar, A Boyer de La Giroday, Dzimitry N Matsukevich, Peter Maunz, Steven Olmschenk, David Hayes, Le Luo, T Andrew Manning, et al. Random numbers certified by Bell's theorem. Nature, 464(7291):1021–1024, 2010. doi:10.1038/​nature09008.

[164] Philip M. Pearle. Hidden-variable example based upon data rejection. Phys. Rev. D, 2:1418–1425, Oct 1970. doi:10.1103/​PhysRevD.2.1418.

[165] Asher Peres. Incompatible results of quantum measurements. Physics Letters A, 151(3-4):107–108, 1990. doi:10.1016/​0375-9601(90)90172-k.

[166] S. Pironio, M. Navascués, and A. Acín. Convergent relaxations of polynomial optimization problems with noncommuting variables. SIAM Journal on Optimization, 20(5):2157–2180, 2010. doi:10.1137/​090760155.

[167] Sandu Popescu. Bell's inequalities and density matrices: Revealing ``hidden'' nonlocality. Phys. Rev. Lett., 74:2619–2622, Apr 1995. doi:10.1103/​PhysRevLett.74.2619.

[168] Sandu Popescu and Daniel Rohrlich. Which states violate Bell's inequality maximally? Physics Letters A, 169(6):411 – 414, 1992. doi:https:/​/​​10.1016/​0375-9601(92)90819-8.

[169] John Preskill. Quantum computation. California Institute of Technology, 1998. URL: http:/​/​​people/​preskill/​ph229.

[170] V. I. Paulsen and I. G. Todorov. Quantum chromatic numbers via operator systems. The Quarterly Journal of Mathematics, 66(2):677–692, Mar 2015. doi:10.1093/​qmath/​hav004.

[171] Károly F. Pál, Tamás Vértesi, and Miguel Navascués. Device-independent tomography of multipartite quantum states. Phys. Rev. A, 90:042340, Oct 2014. doi:10.1103/​PhysRevA.90.042340.

[172] Robert Raussendorf and Hans J. Briegel. A one-way quantum computer. Phys. Rev. Lett., 86:5188–5191, May 2001. doi:10.1103/​PhysRevLett.86.5188.

[173] Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel. Measurement-based quantum computation on cluster states. Phys. Rev. A, 68:022312, Aug 2003. doi:10.1103/​PhysRevA.68.022312.

[174] M Rossi, M Huber, D Bruß, and C Macchiavello. Quantum hypergraph states. New Journal of Physics, 15(11):113022, nov 2013. doi:10.1088/​1367-2630/​15/​11/​113022.

[175] Rafael Rabelo, Melvyn Ho, Daniel Cavalcanti, Nicolas Brunner, and Valerio Scarani. Device-independent certification of entangled measurements. Phys. Rev. Lett., 107:050502, Jul 2011. doi:10.1103/​PhysRevLett.107.050502.

[176] Marc Olivier Renou, Jędrzej Kaniewski, and Nicolas Brunner. Self-testing entangled measurements in quantum networks. Phys. Rev. Lett., 121:250507, Dec 2018. doi:10.1103/​PhysRevLett.121.250507.

[177] Ravishankar Ramanathan, Dardo , Sadiq Muhammad, Piotr Mironowicz, Marcus Grünfeld, Mohamed Bourennane, and Paweł Horodecki. Steering is an essential feature of non-locality in quantum theory. Nature Communications, 9, 2018. doi:10.1038/​s41467-018-06255-5.

[178] Ran Raz and Shmuel Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Proceedings of the Twenty-ninth Annual ACM Symposium on Theory of Computing, STOC '97, pages 475–484, New York, NY, USA, 1997. ACM. doi:10.1145/​258533.258641.

[179] Jérémy Ribeiro, Le Phuc Thinh, Jędrzej Kaniewski, Jonas Helsen, and Stephanie Wehner. Device independence for two-party cryptography and position verification with memoryless devices. Phys. Rev. A, 97:062307, Jun 2018. doi:10.1103/​PhysRevA.97.062307.

[180] Ben W. Reichardt, Falk Unger, and Umesh Vazirani. Classical command of quantum systems. Nature, 496:456, 2013. doi:10.1038/​nature12035.

[181] Rafael Rabelo, Law Yun Zhi, and Valerio Scarani. Device-independent bounds for Hardy's experiment. Phys. Rev. Lett., 109:180401, Oct 2012. doi:10.1103/​PhysRevLett.109.180401.

[182] I Šupić, R Augusiak, A Salavrakos, and A Acín. Self-testing protocols based on the chained Bell inequalities. New Journal of Physics, 18(3):035013, apr 2016. doi:10.1088/​1367-2630/​18/​3/​035013.

[183] Alexia Salavrakos, Remigiusz Augusiak, Jordi Tura, Peter Wittek, Antonio Acín, and Stefano Pironio. Bell inequalities tailored to maximally entangled states. Phys. Rev. Lett., 119:040402, Jul 2017. doi:10.1103/​PhysRevLett.119.040402.

[184] Pavel Sekatski, Jean-Daniel Bancal, Sebastian Wagner, and Nicolas Sangouard. Certifying the building blocks of quantum computers from Bell's theorem. Phys. Rev. Lett., 121:180505, Nov 2018. doi:10.1103/​PhysRevLett.121.180505.

[185] J. Silman, A. Chailloux, N. Aharon, I. Kerenidis, S. Pironio, and S. Massar. Fully distrustful quantum bit commitment and coin flipping. Phys. Rev. Lett., 106:220501, Jun 2011. doi:10.1103/​PhysRevLett.106.220501.

[186] Valerio Scarani. The device-independent outlook on quantum physics (Lecture notes on the power of Bell's theorem). Acta Physica Slovaca, 62, 2012. URL: http:/​/​​aps/​pub.php?y=2012&pub=aps-12-04, doi:10.2478/​v10155-012-0003-4.

[187] I Šupić, A Coladangelo, R Augusiak, and A Acín. Self-testing multipartite entangled states through projections onto two systems. New Journal of Physics, 20(8):083041, aug 2018. doi:10.1088/​1367-2630/​aad89b.

[188] Aditi Sen De, Ujjwal Sen, Časlav Brukner, Vladimír Bužek, and Marek Żukowski. Entanglement swapping of noisy states: A kind of superadditivity in nonclassicality. Phys. Rev. A, 72:042310, Oct 2005. doi:10.1103/​PhysRevA.72.042310.

[189] Ivan Šupić and Matty J Hoban. Self-testing through EPR-steering. New Journal of Physics, 18(7):075006, jul 2016. doi:10.1088/​1367-2630/​18/​7/​075006.

[190] William Slofstra. Lower bounds on the entanglement needed to play XOR non-local games. Journal of Mathematical Physics, 52(10):102202, 2011. doi:10.1063/​1.3652924.

[191] William Slofstra. The set of quantum correlations is not closed. Forum of Mathematics, Pi, 7, 2019. doi:10.1017/​fmp.2018.3.

[192] William Slofstra. Tsirelson’s problem and an embedding theorem for groups arising from non-local games. Journal of the American Mathematical Society, 33(1):1–56, 2020. doi:https:/​/​​10.1090/​jams/​929.

[193] Massimiliano Smania, Piotr Mironowicz, Mohamed Nawareg, Marcin Pawłowski, Adán Cabello, and Mohamed Bourennane. Experimental certification of an informationally complete quantum measurement in a device-independent protocol. Optica, 7(2):123–128, 2020. doi:10.1364/​OPTICA.377959.

[194] Lynden K. Shalm, Evan Meyer-Scott, Bradley G. Christensen, Peter Bierhorst, Michael A. Wayne, Martin J. Stevens, Thomas Gerrits, Scott Glancy, Deny R. Hamel, Michael S. Allman, Kevin J. Coakley, Shellee D. Dyer, Carson Hodge, Adriana E. Lita, Varun B. Verma, Camilla Lambrocco, Edward Tortorici, Alan L. Migdall, Yanbao Zhang, Daniel R. Kumor, William H. Farr, Francesco Marsili, Matthew D. Shaw, Jeffrey A. Stern, Carlos Abellán, Waldimar Amaya, Valerio Pruneri, Thomas Jennewein, Morgan W. Mitchell, Paul G. Kwiat, Joshua C. Bienfang, Richard P. Mirin, Emanuel Knill, and Sae Woo Nam. Strong loophole-free test of local realism. Phys. Rev. Lett., 115:250402, Dec 2015. doi:10.1103/​PhysRevLett.115.250402.

[195] W. Forrest Stinespring. Positive functions on C*-algebras. Proceedings of the American Mathematical Society, 6(2):211–211, Jan 1955. doi:10.1090/​s0002-9939-1955-0069403-4.

[196] Jamie Sikora, Antonios Varvitsiotis, and Zhaohui Wei. Minimum dimension of a Hilbert space needed to generate a quantum correlation. Phys. Rev. Lett., 117:060401, Aug 2016. doi:10.1103/​PhysRevLett.117.060401.

[197] S. J. Summers and R. F. Werner. Maximal violation of Bell's inequalities is generic in quantum field theory. Communications in Mathematical Physics, 110(2):247–259, 1987. doi:10.1007/​BF01207366.

[198] 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. doi:10.1103/​PhysRevLett.114.170502.

[199] 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. doi:10.1103/​PhysRevA.98.062307.

[200] Tassius Temistocles, Rafael Rabelo, and Marcelo Terra Cunha. Measurement compatibility in bell nonlocality tests. Physical Review A, 99(4):042120, 2019. doi:10.1103/​PhysRevA.99.042120.

[201] Armin Tavakoli, Denis Rosset, and Marc-Olivier Renou. Enabling computation of correlation bounds for finite-dimensional quantum systems via symmetrization. Phys. Rev. Lett., 122:070501, Feb 2019. doi:10.1103/​PhysRevLett.122.070501.

[202] B. S. Tsirelson. Quantum analogues of the Bell inequalities. The case of two spatially separated domains. Journal of Soviet Mathematics, 36(4):557–570, Feb 1987. doi:10.1007/​BF01663472.

[203] Boris Tsirelson. Some results and problems on quantum Bell-type inequalities. Hadronis Journal Supplement, 8:329–45, 1993.

[204] J Tura, A B Sainz, T Vértesi, A Acín, M Lewenstein, and R Augusiak. Translationally invariant multipartite Bell inequalities involving only two-body correlators. Journal of Physics A: Mathematical and Theoretical, 47(42):424024, oct 2014. doi:10.1088/​1751-8113/​47/​42/​424024.

[205] Armin Tavakoli, Massimiliano Smania, Tamás Vértesi, Nicolas Brunner, and Mohamed Bourennane. Self-testing nonprojective quantum measurements in prepare-and-measure experiments. Science Advances, 6(16):eaaw6664, 2020. doi:10.1126/​sciadv.aaw6664.

[206] T. R. Tan, Y. Wan, S. Erickson, P. Bierhorst, D. Kienzler, S. Glancy, E. Knill, D. Leibfried, and D. J. Wineland. Chained Bell inequality experiment with high-efficiency measurements. Phys. Rev. Lett., 118:130403, Mar 2017. doi:10.1103/​PhysRevLett.118.130403.

[207] Roope Uola, Ana C. S. Costa, H. Chau Nguyen, and Otfried Gühne. Quantum Steering. arXiv, 2019. arXiv:1903.06663. doi:10.1103/​RevModPhys.92.015001.

[208] Wim van Dam and Patrick Hayden. Universal entanglement transformations without communication. Phys. Rev. A, 67:060302, Jun 2003. URL: https:/​/​​doi/​10.1103/​PhysRevA.67.060302, doi:10.1103/​PhysRevA.67.060302.

[209] Wim van Dam, Frédéric Magniez, Michele Mosca, and Miklos Santha. Self-testing of universal and fault-tolerant sets of quantum gates. SIAM Journal on Computing, 37(2):611–629, 2007. doi:10.1137/​s0097539702404377.

[210] Thomas Vidick. Parallel DIQKD from parallel repetition, 2017. arXiv:1703.08508.

[211] Xingyao Wu, Jean-Daniel Bancal, Matthew McKague, and Valerio Scarani. Device-independent parallel self-testing of two singlets. Phys. Rev. A, 93:062121, Jun 2016. doi:10.1103/​PhysRevA.93.062121.

[212] Sebastian Wagner, Jean-Daniel Bancal, Nicolas Sangouard, and Pavel Sekatski. Device-independent characterization of generalized measurements, 2018. URL: https:/​/​​abs/​1812.02628, arXiv:1812.02628. doi:10.22331/​q-2020-03-19-243.

[213] Xingyao Wu, Yu Cai, Tzyh Haur Yang, Huy Nguyen Le, Jean-Daniel Bancal, and Valerio Scarani. Robust self-testing of the three-qubit W–state. Phys. Rev. A, 90:042339, Oct 2014. doi:10.1103/​PhysRevA.90.042339.

[214] Reinhard F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A, 40:4277–4281, Oct 1989. doi:10.1103/​PhysRevA.40.4277.

[215] H. M. Wiseman, S. J. Jones, and A. C. Doherty. Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox. Phys. Rev. Lett., 98:140402, Apr 2007. doi:10.1103/​PhysRevLett.98.140402.

[216] Erik Woodhead, Jęndrzej Kaniewski, Boris Bourdoncle, Alexia Salavrakos, Joseph Bowles, Remigiusz Augusiak, and Antonio Acín. Maximal randomness from partially entangled states, 2019. arXiv:1901.06912.

[217] Erik Woodhead, Charles Ci Wen Lim, and Stefano Pironio. Semi-device-independent QKD based on BB84 and a CHSH-type estimation. In Kazuo Iwama, Yasuhito Kawano, and Mio Murao, editors, Theory of Quantum Computation, Communication, and Cryptography, pages 107–115, Berlin, Heidelberg, 2013. Springer Berlin Heidelberg. doi:10.1007/​978-3-642-35656-8_9.

[218] Jianwei Wang, Stefano Paesani, Yunhong Ding, Raffaele Santagati, Paul Skrzypczyk, Alexia Salavrakos, Jordi Tura, Remigiusz Augusiak, Laura Mančinska, Davide Bacco, Damien Bonneau, Joshua W. Silverstone, Qihuang Gong, Antonio Acín, Karsten Rottwitt, Leif K. Oxenløwe, Jeremy L. O’Brien, Anthony Laing, and Mark G. Thompson. Multidimensional quantum entanglement with large-scale integrated optics. Science, 2018. doi:10.1126/​science.aar7053.

[219] Xingyao Wu. Self-testing: walking on the boundary of the quantum set. PhD thesis, National University of Singapore, 2017. URL: http:/​/​​handle/​10635/​134729.

[220] Yukun Wang, Xingyao Wu, and Valerio Scarani. All the self-testings of the singlet for two binary measurements. New Journal of Physics, 18(2):025021, feb 2016. doi:10.1088/​1367-2630/​18/​2/​025021.

[221] Tzyh Haur Yang and Miguel Navascués. Robust self-testing of unknown quantum systems into any entangled two-qubit states. Phys. Rev. A, 87:050102, May 2013. doi:10.1103/​PhysRevA.87.050102.

[222] Tzyh Haur Yang, Tamás Vértesi, Jean-Daniel Bancal, Valerio Scarani, and Miguel Navascués. Robust and versatile black-box certification of quantum devices. Phys. Rev. Lett., 113:040401, Jul 2014. doi:10.1103/​PhysRevLett.113.040401.

[223] Wen-Hao Zhang, Geng Chen, Xing-Xiang Peng, Xiang-Jun Ye, Peng Yin, Ya Xiao, Zhi-Bo Hou, Ze-Di Cheng, Yu-Chun Wu, Jin-Shi Xu, Chuan-Feng Li, and Guang-Can Guo. Experimentally robust self-testing for bipartite and tripartite entangled states. Phys. Rev. Lett., 121:240402, Dec 2018. doi:10.1103/​PhysRevLett.121.240402.

[224] Wen-Hao Zhang, Geng Chen, Xing-Xiang Peng, Xiang-Jun Ye, Peng Yin, Xiao-Ye Xu, Jin-Shi Xu, Chuan-Feng Li, and Guang-Can Guo. Experimental realization of robust self-testing of Bell state measurements. Phys. Rev. Lett., 122:090402, Mar 2019. doi:10.1103/​PhysRevLett.122.090402.

[225] Wen-Hao Zhang, Geng Chen, Peng Yin, Xing-Xiang Peng, Xiao-Min Hu, Zhi-Bo Hou, Zhi-Yuan Zhou, Shang Yu, Xiang-Jun Ye, Zong-Quan Zhou, and et al. Experimental demonstration of robust self-testing for bipartite entangled states. npj Quantum Information, 5(1), Nov 2019. doi:10.1038/​s41534-018-0120-0.

[226] Yi-Zheng Zhen, Koon Tong Goh, Yu-Lin Zheng, Wen-Fei Cao, Xingyao Wu, Kai Chen, and Valerio Scarani. Nonlocal games and optimal steering at the boundary of the quantum set. Phys. Rev. A, 94:022116, Aug 2016. doi:10.1103/​PhysRevA.94.022116.

Cited by

[1] Shin-Liang Chen, Huan-Yu Ku, Wenbin Zhou, Jordi Tura, and Yueh-Nan Chen, "Robust self-testing of steerable quantum assemblages and its applications on device-independent quantum certification", Quantum 5, 552 (2021).

[2] Iris Agresti, Beatrice Polacchi, Davide Poderini, Emanuele Polino, Alessia Suprano, Ivan Šupić, Joseph Bowles, Gonzalo Carvacho, Daniel Cavalcanti, and Fabio Sciarrino, "Experimental Robust Self-Testing of the State Generated by a Quantum Network", PRX Quantum 2 2, 020346 (2021).

[3] Joseph Bowles, Flavio Baccari, and Alexia Salavrakos, "Bounding sets of sequential quantum correlations and device-independent randomness certification", Quantum 4, 344 (2020).

[4] Owidiusz Makuta and Remigiusz Augusiak, "Self-testing maximally-dimensional genuinely entangled subspaces within the stabilizer formalism", New Journal of Physics 23 4, 043042 (2021).

[5] Ivan Šupić, Daniel Cavalcanti, and Joseph Bowles, "Device-independent certification of tensor products of quantum states using single-copy self-testing protocols", Quantum 5, 418 (2021).

[6] Takahiko Satoh, Shota Nagayama, Shigeya Suzuki, Takaaki Matsuo, Michal Hajdusek, and Rodney Van Meter, "Attacking the Quantum Internet", IEEE Transactions on Quantum Engineering 2, 1 (2021).

[7] Máté Farkas, Nayda Guerrero, Jaime Cariñe, Gustavo Cañas, and Gustavo Lima, "Self-Testing Mutually Unbiased Bases in Higher Dimensions with Space-Division Multiplexing Optical Fiber Technology", Physical Review Applied 15 1, 014028 (2021).

[8] A. K. Pan, "Oblivious communication game, self-testing of projective and nonprojective measurements, and certification of randomness", Physical Review A 104 2, 022212 (2021).

[9] Tony Metger and Thomas Vidick, "Self-testing of a single quantum device under computational assumptions", arXiv:2001.09161, Quantum 5, 544 (2021).

[10] Austin K. Daniel and Akimasa Miyake, "Quantum Computational Advantage with String Order Parameters of One-Dimensional Symmetry-Protected Topological Order", Physical Review Letters 126 9, 090505 (2021).

[11] Fabian Bernards and Otfried Gühne, "Finding optimal Bell inequalities using the cone-projection technique", Physical Review A 104 1, 012206 (2021).

[12] Nikolai Miklin and Michał Oszmaniec, "A universal scheme for robust self-testing in the prepare-and-measure scenario", Quantum 5, 424 (2021).

[13] Armin Tavakoli, Máté Farkas, Denis Rosset, Jean-Daniel Bancal, and Jedrzej Kaniewski, "Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments", arXiv:1912.03225, Science Advances 7 7, eabc3847 (2021).

[14] Fabian Bernards and Otfried Gühne, "Generalizing Optimal Bell Inequalities", Physical Review Letters 125 20, 200401 (2020).

[15] Albert Aloy, Matteo Fadel, and Jordi Tura, "The quantum marginal problem for symmetric states: applications to variational optimization, nonlocality and self-testing", New Journal of Physics 23 3, 033026 (2021).

[16] Ricardo Faleiro and Manuel Goulão, "Device-independent quantum authorization based on the Clauser-Horne-Shimony-Holt game", Physical Review A 103 2, 022430 (2021).

[17] Ashutosh Rai, Matej Pivoluska, Martin Plesch, Souradeep Sasmal, Manik Banik, and Sibasish Ghosh, "Device-independent bounds from Cabello's nonlocality argument", Physical Review A 103 6, 062219 (2021).

[18] Hamid Tebyanian, Mujtaba Zahidy, Marco Avesani, Andrea Stanco, Paolo Villoresi, and Giuseppe Vallone, "Semi-device independent randomness generation based on quantum state’s indistinguishability", Quantum Science and Technology 6 4, 045026 (2021).

[19] 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).

[20] Ramita Sarkar, Supriyo Dutta, Subhashish Banerjee, and Prasanta K Panigrahi, "Phase squeezing of quantum hypergraph states", Journal of Physics B: Atomic, Molecular and Optical Physics 54 13, 135501 (2021).

[21] Jose Carrasco, Andreas Elben, Christian Kokail, Barbara Kraus, and Peter Zoller, "Theoretical and Experimental Perspectives of Quantum Verification", PRX Quantum 2 1, 010102 (2021).

[22] Sean A. Adamson and Petros Wallden, "Quantum magic rectangles: Characterization and application to certified randomness expansion", Physical Review Research 2 4, 043317 (2020).

[23] Adam Bene Watts, Nicole Yunger Halpern, and Aram Harrow, "Nonlinear Bell inequality for macroscopic measurements", arXiv:1911.09122, Physical Review A 103 1, L010202 (2021).

[24] Matteo Lostaglio, "Certifying Quantum Signatures in Thermodynamics and Metrology via Contextuality of Quantum Linear Response", Physical Review Letters 125 23, 230603 (2020).

[25] George Moreno, Ranieri Nery, Carlos de Gois, Rafael Rabelo, and Rafael Chaves, "Semi-device-independent certification of entanglement in superdense coding", Physical Review A 103 2, 022426 (2021).

[26] Yu Cai, Baichu Yu, Pooja Jayachandran, Nicolas Brunner, Valerio Scarani, and Jean-Daniel Bancal, "Entanglement for any definition of two subsystems", Physical Review A 103 5, 052432 (2021).

[27] Adel Sohbi, Damian Markham, Jaewan Kim, and Marco Túlio Quintino, "Certifying dimension of quantum systems by sequential projective measurements", Quantum 5, 472 (2021).

[28] Jean-Daniel Bancal, Kai Redeker, Pavel Sekatski, Wenjamin Rosenfeld, and Nicolas Sangouard, "Self-testing with finite statistics enabling the certification of a quantum network link", Quantum 5, 401 (2021).

[29] David Cui, Arthur Mehta, Hamoon Mousavi, and Seyed Sajjad Nezhadi, "A generalization of CHSH and the algebraic structure of optimal strategies", Quantum 4, 346 (2020).

[30] Flavio Baccari, Remigiusz Augusiak, Ivan Šupić, and Antonio Acín, "Device-Independent Certification of Genuinely Entangled Subspaces", Physical Review Letters 125 26, 260507 (2020).

[31] Huangjun Zhu, "Zero uncertainty states in the presence of quantum memory", npj Quantum Information 7 1, 47 (2021).

[32] Harshank Shrotriya, Kishor Bharti, and Leong-Chuan Kwek, "Robust semi-device-independent certification of all pure bipartite maximally entangled states via quantum steering", Physical Review Research 3 3, 033093 (2021).

[33] Davide Rattacaso, Gianluca Passarelli, Antonio Mezzacapo, Procolo Lucignano, and Rosario Fazio, "Optimal parent Hamiltonians for time-dependent states", Physical Review A 104 2, 022611 (2021).

[34] Ananda G. Maity, Shiladitya Mal, Chellasamy Jebarathinam, and A. S. Majumdar, "Self-testing of binary Pauli measurements requiring neither entanglement nor any dimensional restriction", Physical Review A 103 6, 062604 (2021).

[35] Jens Eisert, Dominik Hangleiter, Nathan Walk, Ingo Roth, Damian Markham, Rhea Parekh, Ulysse Chabaud, and Elham Kashefi, "Quantum certification and benchmarking", Nature Reviews Physics 2 7, 382 (2020).

[36] David Schmid, Thomas C. Fraser, Ravi Kunjwal, Ana Belen Sainz, Elie Wolfe, and Robert W. Spekkens, "Understanding the interplay of entanglement and nonlocality: motivating and developing a new branch of entanglement theory", arXiv:2004.09194.

[37] Nikolai Miklin, Jakub J. Borkała, and Marcin Pawłowski, "Semi-device-independent self-testing of unsharp measurements", Physical Review Research 2 3, 033014 (2020).

[38] Huangjun Zhu and Masahito Hayashi, "General framework for verifying pure quantum states in the adversarial scenario", Physical Review A 100 6, 062335 (2019).

[39] Zihao Li, Yun-Guang Han, and Huangjun Zhu, "Optimal Verification of Greenberger-Horne-Zeilinger States", Physical Review Applied 13 5, 054002 (2020).

[40] Pei Zeng, You Zhou, and Zhenhuan Liu, "Quantum gate verification and its application in property testing", Physical Review Research 2 2, 023306 (2020).

[41] Kishor Bharti, Maharshi Ray, Antonios Varvitsiotis, Adán Cabello, and Leong-Chuan Kwek, "Local certification of programmable quantum devices of arbitrary high dimensionality", arXiv:1911.09448.

[42] M. Ho, P. Sekatski, E. Y. -Z. Tan, R. Renner, J. -D. Bancal, and N. Sangouard, "Noisy Preprocessing Facilitates a Photonic Realization of Device-Independent Quantum Key Distribution", Physical Review Letters 124 23, 230502 (2020).

[43] Alexey A. Melnikov, Pavel Sekatski, and Nicolas Sangouard, "Setting Up Experimental Bell Tests with Reinforcement Learning", Physical Review Letters 125 16, 160401 (2020).

[44] David Schmid, Denis Rosset, and Francesco Buscemi, "The type-independent resource theory of local operations and shared randomness", arXiv:1909.04065.

[45] Llorenç Escolà, John Calsamiglia, and Andreas Winter, "All tight correlation Bell inequalities have quantum violations", Physical Review Research 2 1, 012044 (2020).

[46] C. Jebarathinam, Jui-Chen Hung, Shin-Liang Chen, and Yeong-Cherng Liang, "Maximal violation of a broad class of Bell inequalities and its implication on self-testing", Physical Review Research 1 3, 033073 (2019).

[47] Tony Metger, Yfke Dulek, Andrea Coladangelo, and Rotem Arnon-Friedman, "Device-independent quantum key distribution from computational assumptions", arXiv:2010.04175.

[48] Jedrzej Kaniewski, "Weak form of self-testing", arXiv:1910.00706, Physical Review Research 2 3, 033420 (2020).

[49] Maharshi Ray, Naresh Goud Boddu, Kishor Bharti, Leong-Chuan Kwek, and Adán Cabello, "Graph-theoretic approach to dimension witnessing", arXiv:2007.10746, New Journal of Physics 23 3, 033006 (2021).

[50] Flavien Hirsch and Marcus Huber, "The Schmidt number of a quantum state cannot always be device-independently certified", arXiv:2003.14189.

[51] Dominik Hangleiter, "Sampling and the complexity of nature", arXiv:2012.07905.

[52] Martin Kliesch and Ingo Roth, "Theory of quantum system certification: a tutorial", arXiv:2010.05925.

[53] Satoshi Ishizaka, "Geometrical self-testing of partially entangled two-qubit states", New Journal of Physics 22 2, 023022 (2020).

[54] Kishor Bharti, Maharshi Ray, Zhen-Peng Xu, Masahito Hayashi, Leong-Chuan Kwek, and Adán Cabello, "Graph-Theoretic Framework for Self-Testing in Bell Scenarios", arXiv:2104.13035.

[55] Davide Orsucci, Jean-Daniel Bancal, Nicolas Sangouard, and Pavel Sekatski, "How post-selection affects device-independent claims under the fair sampling assumption", arXiv:1908.11123.

[56] Xinhui Li, Yukun Wang, Yunguang Han, Fei Gao, and Qiaoyan Wen, "Self-testing of symmetric three-qubit states", arXiv:1907.06397.

[57] Shuquan Ma, Changhua Zhu, Min Nie, and Dongxiao Quan, "Efficient self-testing system for quantum computations based on permutations", Chinese Physics B 30 4, 040305 (2021).

[58] Hiroyuki Ozeki and Satoshi Ishizaka, "Quantum limit of genuine tripartite correlations by bipartite extremality", International Journal of Quantum Information 18 4, 2050014 (2020).

[59] Marcel Dall'Agnol, Tom Gur, Subhayan Roy Moulik, and Justin Thaler, "Quantum Proofs of Proximity", arXiv:2105.03697.

[60] Xavier Valcarce, Pavel Sekatski, Davide Orsucci, Enky Oudot, Jean-Daniel Bancal, and Nicolas Sangouard, "What is the minimum CHSH score certifying that a state resembles the singlet?", arXiv:1910.04606.

[61] Andrea Coladangelo and Jalex Stark, "An inherently infinite-dimensional quantum correlation", Nature Communications 11, 3335 (2020).

[62] Borja Requena, Gorka Muñoz-Gil, Maciej Lewenstein, Vedran Dunjko, and Jordi Tura, "Certificates of quantum many-body properties assisted by machine learning", arXiv:2103.03830.

The above citations are from Crossref's cited-by service (last updated successfully 2021-10-20 01:41:42) and SAO/NASA ADS (last updated successfully 2021-10-20 01:41:43). The list may be incomplete as not all publishers provide suitable and complete citation data.