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

Erik Woodhead1, Boris Bourdoncle1, and Antonio Acín1,2

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

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

► 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

[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

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