Custom Bell inequalities from formal sums of squares

Custom Bell inequalities from formal sums of squares

Victor Barizien1, Pavel Sekatski2, and Jean-Daniel Bancal1

1Université Paris Saclay, CEA, CNRS, Institut de physique théorique, 91191 Gif-sur-Yvette, France
2Département de Physique Appliquée, Université de Genève, 1211 Genève, Suisse

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Abstract

Bell inequalities play a key role in certifying quantum properties for device-independent quantum information protocols. It is still a major challenge, however, to devise Bell inequalities tailored for an arbitrary given quantum state. Existing approaches based on sums of squares provide results in this direction, but they are restricted by the necessity of first choosing measurement settings suited to the state. Here, we show how the sum of square property can be enforced for an arbitrary target state by making an appropriate choice of nullifiers, which is made possible by leaving freedom in the choice of measurement. Using our method, we construct simple Bell inequalities for several families of quantum states, including partially entangled multipartite GHZ states and qutrit states. In most cases we are able to prove that the constructed Bell inequalities achieve self-testing of the target state. We also use the freedom in the choice of measurement to self-test partially entangled two-qubit states with a family of settings with two parameters. Finally, we show that some statistics can be self-tested with distinct Bell inequalities, hence obtaining new insight on the shape of the set of quantum correlations.

In a Bell experiment, the probability distributions generated by local hidden variable models are limited by Bell inequalities. Measurements on a quantum system can however violate these inequalities. In fact, these violations can provide insights on the quantum device used in the experiment. Ultimately, the maximal violation of a well-chosen Bell inequality can provide a full characterization of the states and measurements used in the experiment. This property, called self-testing, establishes a fruitful relationship between Bell inequalities and quantum objects which can be utilized in many quantum information protocols. However, it is generally not known how to best devise a Bell inequality that would be suited to a specific quantum state. In this paper, we consider the problem of finding a Bell inequality suited to self-test a specific target state and/or measurement. We propose a constructive method, based on Sum of Squares of polynomials, which grants candidates verifying the necessary condition that the inequality be maximally violated by the target realization.

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[1] 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

[2] Gilles Brassard, Anne Broadbent, and Alain Tapp. “Quantum Pseudo-Telepathy”. Found. Phys. 35, 1877–1907 (2005).
https:/​/​doi.org/​10.1007/​s10701-005-7353-4

[3] Reinhard F. Werner. “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model”. Phys. Rev. A 40, 4277–4281 (1989).
https:/​/​doi.org/​10.1103/​PhysRevA.40.4277

[4] Jean-Daniel Bancal, Nicolas Gisin, Yeong-Cherng Liang, and Stefano Pironio. “Device-Independent Witnesses of Genuine Multipartite Entanglement”. Phys. Rev. Lett. 106, 250404 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.106.250404

[5] Roger Colbeck. “Quantum and relativistic protocols for secure multi-party computation”. PhD thesis. University of Cambridge. (2006).
https:/​/​doi.org/​10.48550/​arXiv.0911.3814

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

[7] S. Pironio, A. Acín, S. Massar, A. Boyer de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe. “Random numbers certified by Bell’s theorem”. nature 464, 1021–1024 (2010).
https:/​/​doi.org/​10.1038/​nature09008

[8] Dominic Mayers and Andrew Yao. “Self testing quantum apparatus”. Quantum information & computation 4 (2003).
https:/​/​doi.org/​10.26421/​QIC4.4-3

[9] Antonio Acín, Nicolas Brunner, Nicolas Gisin, Serge Massar, Stefano Pironio, and Valerio Scarani. “Device-Independent Security of Quantum Cryptography against Collective Attacks”. Phys. Rev. Lett. 98, 230501 (2007).
https:/​/​doi.org/​10.1103/​PhysRevLett.98.230501

[10] Artur K. Ekert. “Quantum cryptography based on Bell’s theorem”. Phys. Rev. Lett. 67, 661–663 (1991).
https:/​/​doi.org/​10.1103/​PhysRevLett.67.661

[11] D. P. Nadlinger, P. Drmota, B. C. Nichol, G. Araneda, D. Main, R. Srinivas, D. M. Lucas, C. J. Ballance, K. Ivanov, E. Y.-Z. Tan, P. Sekatski, R. L. Urbanke, R. Renner, N. Sangouard, and J.-D. Bancal. “Experimental quantum key distribution certified by Bell’s theorem”. nature 607, 682–686 (2022).
https:/​/​doi.org/​10.1038/​s41586-022-04941-5

[12] 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

[13] Wen-Zhao Liu, Yu-Zhe Zhang, Yi-Zheng Zhen, Ming-Han Li, Yang Liu, Jingyun Fan, Feihu Xu, Qiang Zhang, and Jian-Wei Pan. “Toward a Photonic Demonstration of Device-Independent Quantum Key Distribution”. Phys. Rev. Lett. 129, 050502 (2022).
https:/​/​doi.org/​10.1103/​PhysRevLett.129.050502

[14] Itamar Pitowsky. “Classical Correlation Polytopes and Propositional Logic”. In Quantum Probability — Quantum Logic. Pages 11–51. Springer Berlin Heidelberg, Berlin, Heidelberg (1989).
https:/​/​doi.org/​10.1007/​BFb0021188

[15] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. “Proposed Experiment to Test Local Hidden-Variable Theories”. Phys. Rev. Lett. 23, 880 (1969).
https:/​/​doi.org/​10.1103/​PhysRevLett.23.880

[16] Dagomir Kaszlikowski, L. C. Kwek, Jing-Ling Chen, Marek Żukowski, and C. H. Oh. “Clauser-Horne inequality for three-state systems”. Phys. Rev. A 65, 032118 (2002).
https:/​/​doi.org/​10.1103/​PhysRevA.65.032118

[17] Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar, and Sandu Popescu. “Bell Inequalities for Arbitrarily High-Dimensional Systems”. Phys. Rev. Lett. 88, 040404 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.88.040404

[18] A. Acín, T. Durt, N. Gisin, and J. I. Latorre. “Quantum nonlocality in two three-level systems”. Phys. Rev. A 65, 052325 (2002).
https:/​/​doi.org/​10.1103/​PhysRevA.65.052325

[19] Miguel Navascués, Stefano Pironio, and Antonio Acín. “Bounding the Set of Quantum Correlations”. Physical Review Letters 98, 010401 (2007).
https:/​/​doi.org/​10.1103/​PhysRevLett.98.010401

[20] Marie Ioannou and Denis Rosset. “Noncommutative polynomial optimization under symmetry” (2022). arXiv:2112.10803.
arXiv:2112.10803

[21] Se-Wan Ji, Jinhyoung Lee, James Lim, Koji Nagata, and Hai-Woong Lee. “Multisetting Bell Inequality for Qudits”. Physical Review A 78, 052103 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.78.052103

[22] Yeong-Cherng Liang, Chu-Wee Lim, and Dong-Ling Deng. “Reexamination of a multisetting Bell inequality for qudits”. Physical Review A 80, 052116 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.80.052116

[23] James Lim, Junghee Ryu, Seokwon Yoo, Changhyoup Lee, Jeongho Bang, and Jinhyoung Lee. “Genuinely High-Dimensional Nonlocality Optimized by Complementary Measurements”. New Journal of Physics 12, 103012 (2010).
https:/​/​doi.org/​10.1088/​1367-2630/​12/​10/​103012

[24] 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 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.040402

[25] Antonio Acín, Serge Massar, and Stefano Pironio. “Randomness versus Nonlocality and Entanglement”. Physical Review Letters 108, 100402 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.108.100402

[26] Stefano Pironio. “Violations of Bell Inequalities as Lower Bounds on the Communication Cost of Nonlocal Correlations”. Physical Review A 68, 062102 (2003).
https:/​/​doi.org/​10.1103/​PhysRevA.68.062102

[27] O. Nieto-Silleras, S. Pironio, and J. Silman. “Using Complete Measurement Statistics for Optimal Device-Independent Randomness Evaluation”. New Journal of Physics 16, 013035 (2014).
https:/​/​doi.org/​10.1088/​1367-2630/​16/​1/​013035

[28] Jean-Daniel Bancal, Lana Sheridan, and Valerio Scarani. “More Randomness from the Same Data”. New Journal of Physics 16, 033011 (2014).
https:/​/​doi.org/​10.1088/​1367-2630/​16/​3/​033011

[29] Peter Brown, Hamza Fawzi, and Omar Fawzi. “Device-independent lower bounds on the conditional von neumann entropy” (2023). arXiv:2106.13692.
arXiv:2106.13692

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

[31] Sandu Popescu and Daniel Rohrlich. “Which States Violate Bell’s Inequality Maximally?”. Physics Letters A 169, 411–414 (1992).
https:/​/​doi.org/​10.1016/​0375-9601(92)90819-8

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

[33] Ivan Šupić and Joseph Bowles. “Self-testing of quantum systems: a review”. Quantum 4, 337 (2020).
https:/​/​doi.org/​10.22331/​q-2020-09-30-337

[34] 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 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.052111

[35] ShubhayanAU Sarkar, Debashis Saha, Jędrzej Kaniewski, and Remigiusz Augusiak. “Self-testing quantum systems of arbitrary local dimension with minimal number of measurements”. npj Quantum Information 7, 151 (2021).
https:/​/​doi.org/​10.1038/​s41534-021-00490-3

[36] Andrea Coladangelo, Koon Tong Goh, and Valerio Scarani. “All pure bipartite entangled states can be self-tested”. nature comm 8, 15485 (2017).
https:/​/​doi.org/​10.1038/​ncomms15485

[37] 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 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.042339

[38] Károly F. Pál, Tamás Vértesi, and Miguel Navascués. “Device-independent tomography of multipartite quantum states”. Phys. Rev. A 90, 042340 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.042340

[39] Matthew McKague. “Self-Testing Graph States”. In Dave Bacon, Miguel Martin-Delgado, and Martin Roetteler, editors, Theory of Quantum Computation, Communication, and Cryptography. Pages 104–120. Berlin, Heidelberg (2014). Springer.
https:/​/​doi.org/​10.1007/​978-3-642-54429-3_7

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

[41] F. Baccari, R. Augusiak, I. Šupić, J. Tura, and A. Acín. “Scalable Bell Inequalities for Qubit Graph States and Robust Self-Testing”. Phys. Rev. Lett. 124, 020402 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.020402

[42] Ivan Šupić, Joseph Bowles, Marc-Olivier Renou, Antonio Acín, and Matty J. Hoban. “Quantum networks self-test all entangled states”. Nature Physics 19, 670–675 (2023).
https:/​/​doi.org/​10.1038/​s41567-023-01945-4

[43] Shubhayan Sarkar and Remigiusz Augusiak. “Self-testing of multipartite Greenberger-Horne-Zeilinger states of arbitrary local dimension with arbitrary number of measurements per party”. Phys. Rev. A 105, 032416 (2022).
https:/​/​doi.org/​10.1103/​PhysRevA.105.032416

[44] Tzyh Haur Yang, Tamás Vértesi, Jean-Daniel Bancal, Valerio Scarani, and Miguel Navascués. “Robust and Versatile Black-Box Certification of Quantum Devices”. Physical Review Letters 113, 040401 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.040401

[45] Ashutosh Rai, Matej Pivoluska, Souradeep Sasmal, Manik Banik, Sibasish Ghosh, and Martin Plesch. “Self-Testing Quantum States via Nonmaximal Violation in Hardy’s Test of Nonlocality”. Physical Review A 105, 052227 (2022).
https:/​/​doi.org/​10.1103/​PhysRevA.105.052227

[46] 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 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.022104

[47] Kai-Siang Chen, Gelo Noel M. Tabia, Chellasamy Jebarathinam, Shiladitya Mal, Jun-Yi Wu, and Yeong-Cherng Liang. “Quantum correlations on the no-signaling boundary: self-testing and more”. Quantum 7, 1054 (2023).
https:/​/​doi.org/​10.22331/​q-2023-07-11-1054

[48] Andrea Coladangelo. “Generalization of the Clauser-Horne-Shimony-Holt Inequality Self-Testing Maximally Entangled States of Any Local Dimension”. Physical Review A 98, 052115 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.052115

[49] Ana Belén Sainz, Yelena Guryanova, Will McCutcheon, and Paul Skrzypczyk. “Adjusting Inequalities for Detection-Loophole-Free Steering Experiments”. Physical Review A 94, 032122 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.032122

[50] M. Zwerger, W. Dür, J.-D. Bancal, and P. Sekatski. “Device-Independent Detection of Genuine Multipartite Entanglement for All Pure States”. Phys. Rev. Lett. 122, 060502 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.060502

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

[52] Andrew C. Doherty, Yeong-Cherng Liang, Ben Toner, and Stephanie Wehner. “The quantum moment problem and bounds on entangled multi-prover games”. In Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity. Page 199–210. CCC ’08USA (2008). IEEE Computer Society.
https:/​/​doi.org/​10.1109/​CCC.2008.26

[53] R Augusiak, A Salavrakos, J Tura, and A Acín. “Bell inequalities tailored to the Greenberger–Horne–Zeilinger states of arbitrary local dimension”. New Journal of Physics 21, 113001 (2019).
https:/​/​doi.org/​10.1088/​1367-2630/​ab4d9f

[54] 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 (2019).
https:/​/​doi.org/​10.22331/​q-2019-10-24-198

[55] Rafael Santos, Debashis Saha, Flavio Baccari, and Remigiusz Augusiak. “Scalable Bell inequalities for graph states of arbitrary prime local dimension and self-testing”. New Journal of Physics 25, 063018 (2023).
https:/​/​doi.org/​10.1088/​1367-2630/​acd9e3

[56] Pavel Sekatski, Jean-Daniel Bancal, Sebastian Wagner, and Nicolas Sangouard. “Certifying the Building Blocks of Quantum Computers from Bell’s Theorem”. Physical Review Letters 121, 180505 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.180505

[57] Denis Rosset, Jean-Daniel Bancal, and Nicolas Gisin. “Classifying 50 years of Bell inequalities”. Journal of Physics A: Mathematical and Theoretical 47, 424022 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424022

[58] Valerio Scarani and Nicolas Gisin. “Spectral decomposition of Bell’s operators for qubits”. Journal of Physics A: Mathematical and General 34, 6043–6053 (2001).
https:/​/​doi.org/​10.1088/​0305-4470/​34/​30/​314

[59] Otfried Gühne and Géza Tóth. “Entanglement detection”. Physics Reports 474, 1–75 (2009).
https:/​/​doi.org/​10.1016/​j.physrep.2009.02.004

[60] Sebastian Wagner, Jean-Daniel Bancal, Nicolas Sangouard, and Pavel Sekatski. “Device-Independent Characterization of Quantum Instruments”. Quantum 4, 243 (2020).
https:/​/​doi.org/​10.22331/​q-2020-03-19-243

[61] “Eigenvalue perturbation, Wikipedia, The Free Encyclopedia, Wikimedia Foundation, 12 July”. url: https:/​/​en.wikipedia.org/​wiki/​Eigenvalue_perturbation.
https:/​/​en.wikipedia.org/​wiki/​Eigenvalue_perturbation

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

[63] Tobias Moroder, Jean-Daniel Bancal, Yeong-Cherng Liang, Martin Hofmann, and Otfried Gühne. “Device-independent entanglement quantification and related applications”. Phys. Rev. Lett. 111, 030501 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.030501

[64] Ben W. Reichardt, Falk Unger, and Umesh Vazirani. “Classical command of quantum systems”. Nature 496, 456–460 (2013).
https:/​/​doi.org/​10.1038/​nature12035

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

[66] Tim Coopmans, Jędrzej Kaniewski, and Christian Schaffner. “Robust self-testing of two-qubit states”. Phys. Rev. A 99, 052123 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.052123

[67] Xinhui Li, Yukun Wang, Yunguang Han, and Shi-Ning Zhu. “Self-testing of different entanglement resources via fixed measurement settings”. Physical Review A 106 (2022).
https:/​/​doi.org/​10.1103/​physreva.106.052418

[68] B. S. Tsirelson. “Some results and problems on quantum Bell-type inequalities”. Hadronic Journal Supplement 8, 329–345 (1993). url: https:/​/​ci.nii.ac.jp/​naid/​10026857475/​en/​.
https:/​/​ci.nii.ac.jp/​naid/​10026857475/​en/​

[69] Thinh P. Le, Chiara Meroni, Bernd Sturmfels, Reinhard F. Werner, and Timo Ziegler. “Quantum Correlations in the Minimal Scenario”. Quantum 7, 947 (2023).
https:/​/​doi.org/​10.22331/​q-2023-03-16-947

Cited by

[1] Lewis Wooltorton, Peter Brown, and Roger Colbeck, “Device-independent quantum key distribution with arbitrarily small nonlocality”, arXiv:2309.09650, (2023).

[2] Barizien Victor and Bancal Jean-Daniel, “Extremal Tsirelson inequalities”, arXiv:2401.12791, (2024).

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