Randomness versus nonlocality in the Mermin-Bell experiment with three parties

Erik Woodhead; Boris Bourdoncle; Antonio Acín

doi:10.22331/q-2018-08-17-82

Randomness versus nonlocality in the Mermin-Bell experiment with three parties

Erik Woodhead¹, Boris Bourdoncle¹, and Antonio Acín^1,2

¹ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
²ICREA - Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain

Published:	2018-08-17, volume 2, page 82
Eprint:	arXiv:1804.09733v2
Doi:	https://doi.org/10.22331/q-2018-08-17-82
Citation:	Quantum 2, 82 (2018).

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

Abstract

The detection of nonlocal correlations in a Bell experiment implies almost by definition some intrinsic randomness in the measurement outcomes. For given correlations, or for a given Bell violation, the amount of randomness predicted by quantum physics, quantified by the guessing probability, can generally be bounded numerically. However, currently only a few exact analytic solutions are known for violations of the bipartite Clauser-Horne-Shimony-Holt Bell inequality. Here, we study the randomness in a Bell experiment where three parties test the tripartite Mermin-Bell inequality. We give tight upper bounds on the guessing probabilities associated with one and two of the parties' measurement outcomes as a function of the Mermin inequality violation. Finally, we discuss the possibility of device-independent secret sharing based on the Mermin inequality and argue that the idea seems unlikely to work.

► BibTeX data

@article{Woodhead2018randomnessversus,
  doi = {10.22331/q-2018-08-17-82},
  url = {https://doi.org/10.22331/q-2018-08-17-82},
  title = {Randomness versus nonlocality in the {M}ermin-{B}ell experiment with three parties},
  author = {Woodhead, Erik and Bourdoncle, Boris and Ac{\'{i}}n, Antonio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {2},
  pages = {82},
  month = aug,
  year = {2018}
}

► References

[1] J. S. Bell, Physics 1, 195 (1964).
http://cds.cern.ch/record/111654/

[2] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Rev. Mod. Phys. 86, 419 (2014), arXiv:1303.2849 [quant-ph].
https://doi.org/10.1103/RevModPhys.86.419
arXiv:1303.2849

[3] D. Mayers and A. Yao, in Proceedings of the 39th Annual Symposium on Foundations of Computer Science (IEEE Computer Society, Los Alamitos, 1998) pp. 503–509, arXiv:quant-ph/9809039.
https://doi.org/10.1109/SFCS.1998.743501
arXiv:quant-ph/9809039

[4] A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Phys. Rev. Lett. 98, 230501 (2007), arXiv:quant-ph/0702152.
https://doi.org/10.1103/PhysRevLett.98.230501
arXiv:quant-ph/0702152

[5] R. Colbeck, Quantum And Relativistic Protocols For Secure Multi-Party Computation, Ph.D. thesis, University of Cambridge (2006), arXiv:0911.3814 [quant-ph].
arXiv:0911.3814

[6] 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, Nature 464, 1021 (2010), arXiv:0911.3427 [quant-ph].
https://doi.org/10.1038/nature09008
arXiv:0911.3427

[7] R. Colbeck and A. Kent, J. Phys. A: Math. Theor. 44, 095305 (2011), arXiv:1011.4474 [quant-ph].
https://doi.org/10.1088/1751-8113/44/9/095305
arXiv:1011.4474

[8] R. Renner, N. Gisin, and B. Kraus, Phys. Rev. A 72, 012332 (2005), arXiv:quant-ph/0502064.
https://doi.org/10.1103/PhysRevA.72.012332
arXiv:quant-ph/0502064

[9] R. Renner, Security of Quantum Key Distribution, Ph.D. thesis, ETH Zurich (2005), arXiv:quant-ph/0512258.
arXiv:quant-ph/0512258

[10] L. Masanes, S. Pironio, and A. Acín, Nat. Commun. 2, 238 (2011), arXiv:1009.1567 [quant-ph].
https://doi.org/10.1038/ncomms1244
arXiv:1009.1567

[11] S. Pironio and S. Massar, Phys. Rev. A 87, 012336 (2013), arXiv:1111.6056 [quant-ph].
https://doi.org/10.1103/PhysRevA.87.012336
arXiv:1111.6056

[12] S. Pironio, L. Masanes, A. Leverrier, and A. Acín, Phys. Rev. X 3, 031007 (2013), arXiv:1211.1402 [quant-ph].
https://doi.org/10.1103/PhysRevX.3.031007
arXiv:1211.1402

[13] R. Arnon-Friedman, F. Dupuis, O. Fawzi, R. Renner, and T. Vidick, Nat. Commun. 9, 459 (2018).
https://doi.org/10.1038/s41467-017-02307-4

[14] M. Navascués, S. Pironio, and A. Acín, Phys. Rev. Lett. 98, 010401 (2007), arXiv:quant-ph/0607119.
https://doi.org/10.1103/PhysRevLett.98.010401
arXiv:quant-ph/0607119

[15] O. Nieto Silleras, S. Pironio, and J. Silman, New J. Phys. 16, 013035 (2014), arXiv:1309.3930 [quant-ph].
https://doi.org/10.1088/1367-2630/16/1/013035
arXiv:1309.3930

[16] J.-D. Bancal, L. Sheridan, and V. Scarani, New J. Phys. 16, 033011 (2014), arXiv:1309.3894 [quant-ph].
https://doi.org/10.1088/1367-2630/16/3/033011
arXiv:1309.3894

[17] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).
https://doi.org/10.1103/PhysRevLett.23.880

[18] J. Kaniewski and S. Wehner, New J. Phys. 18, 055004 (2016), arXiv:1601.06752 [quant-ph].
https://doi.org/10.1088/1367-2630/18/5/055004
arXiv:1601.06752

[19] J. Barrett, A. Kent, and S. Pironio, Phys. Rev. Lett. 97, 170409 (2006), arXiv:quant-ph/0605182.
https://doi.org/10.1103/PhysRevLett.97.170409
arXiv:quant-ph/0605182

[20] L. Aolita, R. Gallego, A. Cabello, and A. Acín, Phys. Rev. Lett. 108, 100401 (2012), arXiv:1109.3163 [quant-ph].
https://doi.org/10.1103/PhysRevLett.108.100401
arXiv:1109.3163

[21] N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990).
https://doi.org/10.1103/PhysRevLett.65.1838

[22] C. Bamps and S. Pironio, Phys. Rev. A 91, 052111 (2015), arXiv:1504.06960 [quant-ph].
https://doi.org/10.1103/PhysRevA.91.052111
arXiv:1504.06960

[23] M. Hillery, V. Bužek, and A. Berthiaume, Phys. Rev. A 59, 1829 (1999), arXiv:quant-ph/9806063.
https://doi.org/10.1103/PhysRevA.59.1829
arXiv:quant-ph/9806063

[24] D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, Am. J. Phys. 58, 1131 (1990).
https://doi.org/10.1119/1.16243

[25] J.-D. Bancal, N. Gisin, Y.-C. Liang, and S. Pironio, Phys. Rev. Lett. 106, 250404 (2011), arXiv:1102.0197 [quant-ph].
https://doi.org/10.1103/PhysRevLett.106.250404
arXiv:1102.0197

[26] G. Svetlichny, Phys. Rev. D 35, 3066 (1987).
https://doi.org/10.1103/PhysRevD.35.3066

[27] V. Scarani and N. Gisin, J. Phys. A: Math. Gen. 34, 6043 (2001), arXiv:quant-ph/0103068.
https://doi.org/10.1088/0305-4470/34/30/314
arXiv:quant-ph/0103068

[28] R. F. Werner and M. M. Wolf, Phys. Rev. A 64, 032112 (2001), arXiv:quant-ph/0102024.
https://doi.org/10.1103/PhysRevA.64.032112
arXiv:quant-ph/0102024

[29] ``SDPA official page'', http://sdpa.sourceforge.net/ (2011).
http://sdpa.sourceforge.net/

[30] M. Nakata, in 2010 IEEE International Symposium on Computer-Aided Control System Design (IEEE, 2010) pp. 29–34.
https://doi.org/10.1109/CACSD.2010.5612693

[31] ``GitHub - ewoodhead/npa-hierarchy'', https://github.com/ewoodhead/npa-hierarchy (2018).
https://github.com/ewoodhead/npa-hierarchy

[32] A. Acín, S. Massar, and S. Pironio, Phys. Rev. Lett. 108, 100402 (2012), arXiv:1107.2754 [quant-ph].
https://doi.org/10.1103/PhysRevLett.108.100402
arXiv:1107.2754

[33] A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, PeerJ Computer Science 3, e103 (2017).
https://doi.org/10.7717/peerj-cs.103

[34] S. Gogioso and W. Zeng, ``Generalised Mermin-type non-locality arguments'', (2017), arXiv:1702.01772 [quant-ph].
arXiv:1702.01772

[35] R. Ramanathan and P. Mironowicz, ``Trade-offs in multi-party Bell inequality violations in qubit networks'', (2017), arXiv:1704.03790 [quant-ph].
arXiv:1704.03790

[36] A. Karlsson, M. Koashi, and N. Imoto, Phys. Rev. A 59, 162 (1999).
https://doi.org/10.1103/PhysRevA.59.162

[37] K. T. Goh, J. Kaniewski, E. Wolfe, T. Vértesi, X. Wu, Y. Cai, Y.-C. Liang, and V. Scarani, Phys. Rev. A 97, 022104 (2018), arXiv:1710.05892 [quant-ph].
https://doi.org/10.1103/PhysRevA.97.022104
arXiv:1710.05892

[38] J. Silman, S. Pironio, and S. Massar, Phys. Rev. Lett. 110, 100504 (2013), arXiv:1211.5921 [quant-ph].
https://doi.org/10.1103/PhysRevLett.110.100504
arXiv:1211.5921

[39] S. Pironio, J.-D. Bancal, and V. Scarani, J. Phys. A: Math. Theor. 44, 065303 (2011), arXiv:1101.2477 [quant-ph].
https://doi.org/10.1088/1751-8113/44/6/065303
arXiv:1101.2477

[1] J. S. Bell, Physics 1, 195 (1964).
http://cds.cern.ch/record/111654/

[2] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Rev. Mod. Phys. 86, 419 (2014), arXiv:1303.2849 [quant-ph].
https://doi.org/10.1103/RevModPhys.86.419
arXiv:1303.2849

[5] R. Colbeck, Quantum And Relativistic Protocols For Secure Multi-Party Computation, Ph.D. thesis, University of Cambridge (2006), arXiv:0911.3814 [quant-ph].
arXiv:0911.3814

[7] R. Colbeck and A. Kent, J. Phys. A: Math. Theor. 44, 095305 (2011), arXiv:1011.4474 [quant-ph].
https://doi.org/10.1088/1751-8113/44/9/095305
arXiv:1011.4474

[8] R. Renner, N. Gisin, and B. Kraus, Phys. Rev. A 72, 012332 (2005), arXiv:quant-ph/0502064.
https://doi.org/10.1103/PhysRevA.72.012332
arXiv:quant-ph/0502064

[9] R. Renner, Security of Quantum Key Distribution, Ph.D. thesis, ETH Zurich (2005), arXiv:quant-ph/0512258.
arXiv:quant-ph/0512258

[10] L. Masanes, S. Pironio, and A. Acín, Nat. Commun. 2, 238 (2011), arXiv:1009.1567 [quant-ph].
https://doi.org/10.1038/ncomms1244
arXiv:1009.1567

[11] S. Pironio and S. Massar, Phys. Rev. A 87, 012336 (2013), arXiv:1111.6056 [quant-ph].
https://doi.org/10.1103/PhysRevA.87.012336
arXiv:1111.6056

[12] S. Pironio, L. Masanes, A. Leverrier, and A. Acín, Phys. Rev. X 3, 031007 (2013), arXiv:1211.1402 [quant-ph].
https://doi.org/10.1103/PhysRevX.3.031007
arXiv:1211.1402

[13] R. Arnon-Friedman, F. Dupuis, O. Fawzi, R. Renner, and T. Vidick, Nat. Commun. 9, 459 (2018).
https://doi.org/10.1038/s41467-017-02307-4

[14] M. Navascués, S. Pironio, and A. Acín, Phys. Rev. Lett. 98, 010401 (2007), arXiv:quant-ph/0607119.
https://doi.org/10.1103/PhysRevLett.98.010401
arXiv:quant-ph/0607119

[15] O. Nieto Silleras, S. Pironio, and J. Silman, New J. Phys. 16, 013035 (2014), arXiv:1309.3930 [quant-ph].
https://doi.org/10.1088/1367-2630/16/1/013035
arXiv:1309.3930

[16] J.-D. Bancal, L. Sheridan, and V. Scarani, New J. Phys. 16, 033011 (2014), arXiv:1309.3894 [quant-ph].
https://doi.org/10.1088/1367-2630/16/3/033011
arXiv:1309.3894

[17] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).
https://doi.org/10.1103/PhysRevLett.23.880

[18] J. Kaniewski and S. Wehner, New J. Phys. 18, 055004 (2016), arXiv:1601.06752 [quant-ph].
https://doi.org/10.1088/1367-2630/18/5/055004
arXiv:1601.06752

[19] J. Barrett, A. Kent, and S. Pironio, Phys. Rev. Lett. 97, 170409 (2006), arXiv:quant-ph/0605182.
https://doi.org/10.1103/PhysRevLett.97.170409
arXiv:quant-ph/0605182

[20] L. Aolita, R. Gallego, A. Cabello, and A. Acín, Phys. Rev. Lett. 108, 100401 (2012), arXiv:1109.3163 [quant-ph].
https://doi.org/10.1103/PhysRevLett.108.100401
arXiv:1109.3163

[21] N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990).
https://doi.org/10.1103/PhysRevLett.65.1838

[22] C. Bamps and S. Pironio, Phys. Rev. A 91, 052111 (2015), arXiv:1504.06960 [quant-ph].
https://doi.org/10.1103/PhysRevA.91.052111
arXiv:1504.06960

[23] M. Hillery, V. Bužek, and A. Berthiaume, Phys. Rev. A 59, 1829 (1999), arXiv:quant-ph/9806063.
https://doi.org/10.1103/PhysRevA.59.1829
arXiv:quant-ph/9806063

[24] D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, Am. J. Phys. 58, 1131 (1990).
https://doi.org/10.1119/1.16243

[25] J.-D. Bancal, N. Gisin, Y.-C. Liang, and S. Pironio, Phys. Rev. Lett. 106, 250404 (2011), arXiv:1102.0197 [quant-ph].
https://doi.org/10.1103/PhysRevLett.106.250404
arXiv:1102.0197

[26] G. Svetlichny, Phys. Rev. D 35, 3066 (1987).
https://doi.org/10.1103/PhysRevD.35.3066

[27] V. Scarani and N. Gisin, J. Phys. A: Math. Gen. 34, 6043 (2001), arXiv:quant-ph/0103068.
https://doi.org/10.1088/0305-4470/34/30/314
arXiv:quant-ph/0103068

[28] R. F. Werner and M. M. Wolf, Phys. Rev. A 64, 032112 (2001), arXiv:quant-ph/0102024.
https://doi.org/10.1103/PhysRevA.64.032112
arXiv:quant-ph/0102024

[29] ``SDPA official page'', http://sdpa.sourceforge.net/ (2011).
http://sdpa.sourceforge.net/

[30] M. Nakata, in 2010 IEEE International Symposium on Computer-Aided Control System Design (IEEE, 2010) pp. 29–34.
https://doi.org/10.1109/CACSD.2010.5612693

[31] ``GitHub - ewoodhead/npa-hierarchy'', https://github.com/ewoodhead/npa-hierarchy (2018).
https://github.com/ewoodhead/npa-hierarchy

[32] A. Acín, S. Massar, and S. Pironio, Phys. Rev. Lett. 108, 100402 (2012), arXiv:1107.2754 [quant-ph].
https://doi.org/10.1103/PhysRevLett.108.100402
arXiv:1107.2754

[34] S. Gogioso and W. Zeng, ``Generalised Mermin-type non-locality arguments'', (2017), arXiv:1702.01772 [quant-ph].
arXiv:1702.01772

[35] R. Ramanathan and P. Mironowicz, ``Trade-offs in multi-party Bell inequality violations in qubit networks'', (2017), arXiv:1704.03790 [quant-ph].
arXiv:1704.03790

[36] A. Karlsson, M. Koashi, and N. Imoto, Phys. Rev. A 59, 162 (1999).
https://doi.org/10.1103/PhysRevA.59.162

[38] J. Silman, S. Pironio, and S. Massar, Phys. Rev. Lett. 110, 100504 (2013), arXiv:1211.5921 [quant-ph].
https://doi.org/10.1103/PhysRevLett.110.100504
arXiv:1211.5921

[39] S. Pironio, J.-D. Bancal, and V. Scarani, J. Phys. A: Math. Theor. 44, 065303 (2011), arXiv:1101.2477 [quant-ph].
https://doi.org/10.1088/1751-8113/44/6/065303
arXiv:1101.2477

Cited by

[1] Ya Xi, Mao-Sheng Li, Libin Fu, and Zhu-Jun Zheng, "Sharing tripartite nonlocality sequentially by arbitrarily many independent observers", Physical Review A 107 6, 062419 (2023).

[2] Marco Túlio Quintino, Costantino Budroni, Erik Woodhead, Adán Cabello, and Daniel Cavalcanti, "Device-Independent Tests of Structures of Measurement Incompatibility", Physical Review Letters 123 18, 180401 (2019).

[3] Federico Grasselli, Quantum Science and Technology 105 (2021) ISBN:978-3-030-64359-1.

[4] Federico Grasselli, Gláucia Murta, Hermann Kampermann, and Dagmar Bruß, "Entropy Bounds for Multiparty Device-Independent Cryptography", PRX Quantum 2 1, 010308 (2021).

[5] Cameron Foreman, Sherilyn Wright, Alec Edgington, Mario Berta, and Florian J. Curchod, "Practical randomness amplification and privatisation with implementations on quantum computers", Quantum 7, 969 (2023).

[6] Federico Grasselli, Gláucia Murta, Hermann Kampermann, and Dagmar Bruß, "Boosting device-independent cryptography with tripartite nonlocality", Quantum 7, 980 (2023).

[7] Lewis Wooltorton, Peter Brown, and Roger Colbeck, "Tight Analytic Bound on the Trade-Off between Device-Independent Randomness and Nonlocality", Physical Review Letters 129 15, 150403 (2022).

[8] M. G. M. Moreno, Samuraí Brito, Ranieri V. Nery, and Rafael Chaves, "Device-independent secret sharing and a stronger form of Bell nonlocality", Physical Review A 101 5, 052339 (2020).

[9] Cameron Foreman, Sherilyn Wright, Alec Edgington, Mario Berta, and Florian J. Curchod, "Practical randomness amplification and privatisation with implementations on quantum computers", arXiv:2009.06551, (2020).

[10] Matthew Coudron, Jalex Stark, and Thomas Vidick, "Trading Locality for Time: Certifiable Randomness from Low-Depth Circuits", Communications in Mathematical Physics 382 1, 49 (2021).

[11] Matthew Coudron, Jalex Stark, and Thomas Vidick, "Trading locality for time: certifiable randomness from low-depth circuits", arXiv:1810.04233, (2018).

[12] Marco Túlio Quintino, Costantino Budroni, Erik Woodhead, Adán Cabello, and Daniel Cavalcanti, "Device-independent tests of structures of measurement incompatibility", arXiv:1902.05841, (2019).

The above citations are from Crossref's cited-by service (last updated successfully 2023-11-29 17:46:15) and SAO/NASA ADS (last updated successfully 2023-11-29 17:46:16). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.