Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant $K_G(3)$

Flavien Hirsch1, Marco Túlio Quintino1,2, Tamás Vértesi3, Miguel Navascués4, and Nicolas Brunner1

1Groupe de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland
2Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan
3Institute for Nuclear Research, Hungarian Academy of Sciences, H-4001 Debrecen, P.O. Box 51, Hungary
4Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmangasse 3, 1090 Vienna, Austria

We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the two-qubit Werner state $\rho = v |\psi_- \rangle \langle\psi_- | + (1- v ) \frac{1}{4}$ via a local hidden variable (LHV) model, where $|\psi_- \rangle$ denotes the singlet state. We show analytically that these correlations are local for $ v = 999\times689\times{10^{-6}}$ $\cos^2(\pi/50) \simeq 0.6829$. In turn, as this problem is closely related to a purely mathematical one formulated by Grothendieck, our result implies a new bound on the Grothendieck constant $K_G(3) \leq 1/v \simeq 1.4644$. We also present a LHV model for reproducing the statistics of arbitrary POVMs on the Werner state for $v \simeq 0.4553$. The techniques we develop can be adapted to construct LHV models for other entangled states, as well as bounding other Grothendieck constants.

Share

► BibTeX data

► References

[1] J. S. Bell, ``On the Einstein-Poldolsky-Rosen paradox,'' Physics 1, 195-200 (1964). https:/​/​cds.cern.ch/​record/​111654/​files/​vol1p195-200_001.pdf.
https:/​/​cds.cern.ch/​record/​111654/​files/​vol1p195-200_001.pdf

[2] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, ``Bell nonlocality,'' Reviews of Modern Physics 86, 419-478 (2014), arXiv:1303.2849 [quant-ph].
https://doi.org/10.1103/RevModPhys.86.419
arXiv:1303.2849

[3] N. Gisin, ``Bell's inequality holds for all non-product states,'' Physics Letters A 154, 201 - 202 (1991).
https://doi.org/10.1016/0375-9601(91)90805-I

[4] R. 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

[5] J. Barrett, ``Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality,'' Phys. Rev. A 65, 042302 (2002), quant-ph/​0107045.
https://doi.org/10.1103/PhysRevA.65.042302
arXiv:quant-ph/0107045

[6] M. L. Almeida, S. Pironio, J. Barrett, G. Tóth, and A. Acín, ``Noise Robustness of the Nonlocality of Entangled Quantum States,'' Phys. Rev. Lett. 99, 040403 (2007), quant-ph/​0703018.
https://doi.org/10.1103/PhysRevLett.99.040403
arXiv:quant-ph/0703018

[7] H. M. Wiseman, S. J. Jones, and A. C. Doherty, ``Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox,'' Phys. Rev. Lett. 98, 140402 (2007), quant-ph/​0612147.
https://doi.org/10.1103/PhysRevLett.98.140402
arXiv:quant-ph/0612147

[8] F. Hirsch, M. T. Quintino, J. Bowles, and N. Brunner, ``Genuine Hidden Quantum Nonlocality,'' Phys. Rev. Lett. 111, 160402 (2013), arXiv:1307.4404 [quant-ph].
https://doi.org/10.1103/PhysRevLett.111.160402
arXiv:1307.4404

[9] J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, ``One-way Einstein-Podolsky-Rosen Steering,'' Phys. Rev. Lett. 112, 200402 (2014), arXiv:1402.3607 [quant-ph].
https://doi.org/10.1103/PhysRevLett.112.200402
arXiv:1402.3607

[10] S. Jevtic, M. J. W. Hall, M. R. Anderson, M. Zwierz, and H. M. Wiseman, ``Einstein-Podolsky-Rosen steering and the steering ellipsoid,'' Journal of the Optical Society of America B Optical Physics 32, A40 (2015), arXiv:1411.1517 [quant-ph].
https://doi.org/10.1364/JOSAB.32.000A40
arXiv:1411.1517

[11] J. Bowles, F. Hirsch, M. T. Quintino, and N. Brunner, ``Local Hidden Variable Models for Entangled Quantum States Using Finite Shared Randomness,'' Phys. Rev. Lett. 114, 120401 (2015), arXiv:1412.1416 [quant-ph].
https://doi.org/10.1103/PhysRevLett.114.120401
arXiv:1412.1416

[12] J. Bowles, F. Hirsch, M. T. Quintino, and N. Brunner, ``Sufficient criterion for guaranteeing that a two-qubit state is unsteerable,'' Phys. Rev. A 93, 022121 (2016), arXiv:1510.06721 [quant-ph].
https://doi.org/10.1103/PhysRevA.93.022121
arXiv:1510.06721

[13] H. Chau Nguyen and T. Vu, ``Necessary and sufficient condition for steerability of two-qubit states by the geometry of steering outcomes,'' EPL (Europhysics Letters) 115, 10003 (2016), arXiv:1604.03815 [quant-ph].
https://doi.org/10.1209/0295-5075/115/10003
arXiv:1604.03815

[14] G. Tóth and A. Acín, ``Genuine tripartite entangled states with a local hidden-variable model,'' Phys. Rev. A 74, 030306 (2006), quant-ph/​0512088.
https://doi.org/10.1103/PhysRevA.74.030306
arXiv:quant-ph/0512088

[15] J. Bowles, J. Francfort, M. Fillettaz, F. Hirsch, and N. Brunner, ``Genuinely Multipartite Entangled Quantum States with Fully Local Hidden Variable Models and Hidden Multipartite Nonlocality,'' Phys. Rev. Lett. 116, 130401 (2016), arXiv:1511.08401 [quant-ph].
https://doi.org/10.1103/PhysRevLett.116.130401
arXiv:1511.08401

[16] R. Augusiak, M. Demianowicz, J. Tura, and A. Acín, ``Entanglement and Nonlocality are Inequivalent for Any Number of Parties,'' Phys. Rev. Lett. 115, 030404 (2015), arXiv:1407.3114 [quant-ph].
https://doi.org/10.1103/PhysRevLett.115.030404
arXiv:1407.3114

[17] R. Augusiak, M. Demianowicz, and A. Acín, ``Local hidden-variable models for entangled quantum states,'' Journal of Physics A Mathematical General 47, 424002 (2014), arXiv:1405.7321 [quant-ph].
https://doi.org/10.1088/1751-8113/47/42/424002
arXiv:1405.7321

[18] A. Acín, N. Gisin, and B. Toner, ``Grothendieck's constant and local models for noisy entangled quantum states,'' Phys. Rev. A 73, 062105 (2006), quant-ph/​0606138.
https://doi.org/10.1103/PhysRevA.73.062105
arXiv:quant-ph/0606138

[19] B. S. Tsirelson, ``Some results and problems on quantum Bell-type inequalities,'' Hadronic Journal Supplement 8, 329-345 (1993). http:/​/​www.tau.ac.il/​ tsirel/​download/​hadron.pdf.
http:/​/​www.tau.ac.il/​~tsirel/​download/​hadron.pdf

[20] A. Grothendieck, ``Résumé de la théorie métrique des produits tensoriels topologiques,'' Bol. Soc. Mat. São Paulo 8, (1953). https:/​/​www.ime.usp.br/​acervovirtual/​textos/​estrangeiros/​grothendieck/​produits_tensoriels_topologiques/​files/​produits_tensoriels_topologiques.pdf.
https:/​/​www.ime.usp.br/​acervovirtual/​textos/​estrangeiros/​grothendieck/​produits_tensoriels_topologiques/​files/​produits_tensoriels_topologiques.pdf

[21] J. Krivine, ``Constantes de Grothendieck et fonctions de type positif sur les sphères,'' Advances in Mathematics 31, 16 - 30 (1979).
https://doi.org/10.1016/0001-8708(79)90017-3

[22] T. Vértesi, ``More efficient Bell inequalities for Werner states,'' Phys. Rev. A 78, 032112 (2008), arXiv:0806.0096 [quant-ph].
https://doi.org/10.1103/PhysRevA.78.032112
arXiv:0806.0096

[23] B. Hua, M. Li, T. Zhang, C. Zhou, X. Li-Jost, and S.-M. Fei, ``Towards Grothendieck constants and LHV models in quantum mechanics,'' Journal of Physics A Mathematical General 48, 065302 (2015), arXiv:1501.05507 [quant-ph].
https://doi.org/10.1088/1751-8113/48/6/065302
arXiv:1501.05507

[24] S. Brierley, M. Navascues, and T. Vertesi, ``Convex separation from convex optimization for large-scale problems,'' ArXiv e-prints (2016), arXiv:1609.05011 [quant-ph].
arXiv:1609.05011

[25] G. Pisier, ``Grothendieck's Theorem, past and present,'' Bull. Amer. Math. Soc. 49, 237-323 (2011), arXiv:1101.4195 [math.FA].
arXiv:1101.4195

[26] F. Hirsch, M. T. Quintino, T. Vértesi, M. F. Pusey, and N. Brunner, ``Algorithmic Construction of Local Hidden Variable Models for Entangled Quantum States,'' Phys. Rev. Lett. 117, 190402 (2016), arXiv:1512.00262 [quant-ph].
https://doi.org/10.1103/PhysRevLett.117.190402
arXiv:1512.00262

[27] D. Cavalcanti, L. Guerini, R. Rabelo, and P. Skrzypczyk, ``General Method for Constructing Local Hidden Variable Models for Entangled Quantum States,'' Phys. Rev. Lett. 117, 190401 (2016), arXiv:1512.00277 [quant-ph].
https://doi.org/10.1103/PhysRevLett.117.190401
arXiv:1512.00277

[28] S. R. Finch, Mathematical constants. Cambridge University Press, 2003. http:/​/​www.cambridge.org/​catalogue/​catalogue.asp?isbn=0521818052.
http:/​/​www.cambridge.org/​catalogue/​catalogue.asp?isbn=0521818052

[29] M. Braverman, K. Makarychev, Y. Makarychev, and A. Naor, ``The Grothendieck constant is strictly smaller than Krivine's bound,'' Forum of Mathematics, Pi (2013), arXiv:1103.6161 [math.FA].
https://doi.org/10.1017/fmp.2013.4
arXiv:1103.6161

[30] E. G. Gilbert, ``An iterative procedure for computing the minimum of a quadratic form on a convex set,'' SIAM Journal on Control 4, 61-80 (1966).
https://doi.org/10.1137/0304007

[31] https:/​/​cloud.atomki.hu/​s/​KG3m625.
https:/​/​cloud.atomki.hu/​s/​KG3m625

[32] M. Oszmaniec, L. Guerini, P. Wittek, and A. Acín, ``Simulating positive-operator-valued measures with projective measurements,'' ArXiv e-prints (2016), arXiv:1609.06139 [quant-ph].
arXiv:1609.06139

[33] G. Mauro D'Ariano, P. Lo Presti, and P. Perinotti, ``Classical randomness in quantum measurements,'' Journal of Physics A Mathematical General 38, 5979-5991 (2005), quant-ph/​0408115.
https://doi.org/10.1088/0305-4470/38/26/010
arXiv:quant-ph/0408115