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

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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.

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[1] J. S. Bell, ``On the Einstein-Poldolsky-Rosen paradox,'' Physics 1, 195–200 (1964). https:/​/​​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].

[3] N. Gisin, ``Bell's inequality holds for all non-product states,'' Physics Letters A 154, 201 – 202 (1991).

[4] R. F. Werner, ``Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,'' Phys. Rev. A 40, 4277–4281 (1989).

[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.

[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.

[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.

[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].

[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].

[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].

[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].

[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].

[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].

[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.

[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].

[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].

[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].

[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.

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

[20] A. Grothendieck, ``Résumé de la théorie métrique des produits tensoriels topologiques,'' Bol. Soc. Mat. São Paulo 8, (1953). https:/​/​​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).

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

[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].

[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].

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

[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].

[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].

[28] S. R. Finch, Mathematical constants. Cambridge University Press, 2003. http:/​/​​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].

[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).

[31] https:/​/​​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].

[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.

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[2] Nicolas Gisin, Quanxin Mei, Armin Tavakoli, Marc Olivier Renou, and Nicolas Brunner, "All entangled pure quantum states violate the bilocality inequality", Physical Review A 96 2, 020304 (2017).

[3] Péter Diviánszky, Erika Bene, and Tamás Vértesi, "Qutrit witness from the Grothendieck constant of order four", Physical Review A 96 1, 012113 (2017).

[4] Olga Goulko and Adrian Kent, "The grasshopper problem", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473 2207, 20170494 (2017).

[5] Michał Oszmaniec, Leonardo Guerini, Peter Wittek, and Antonio Acín, "Simulating Positive-Operator-Valued Measures with Projective Measurements", Physical Review Letters 119 19, 190501 (2017).

[6] Kai Sun, Xiang-Jun Ye, Ya Xiao, Xiao-Ye Xu, Yu-Chun Wu, Jin-Shi Xu, Jing-Ling Chen, Chuan-Feng Li, and Guang-Can Guo, "Demonstration of Einstein–Podolsky–Rosen steering with enhanced subchannel discrimination", npj Quantum Information 4 1, 12 (2018).

[7] Akshata Shenoy H., Sébastien Designolle, Flavien Hirsch, Ralph Silva, Nicolas Gisin, and Nicolas Brunner, "Unbounded sequence of observers exhibiting Einstein-Podolsky-Rosen steering", Physical Review A 99 2, 022317 (2019).

[8] Matteo Fadel, Quantum Science and Technology 57 (2021) ISBN:978-3-030-85471-3.

[9] Yu Xiang, Shuming Cheng, Qihuang Gong, Zbigniew Ficek, and Qiongyi He, "Quantum Steering: Practical Challenges and Future Directions", PRX Quantum 3 3, 030102 (2022).

[10] H. Chau Nguyen and Otfried Gühne, "Quantum steering of Bell-diagonal states with generalized measurements", Physical Review A 101 4, 042125 (2020).

[11] Michał Oszmaniec, Filip B. Maciejewski, and Zbigniew Puchała, "Simulating all quantum measurements using only projective measurements and postselection", Physical Review A 100 1, 012351 (2019).

[12] Hong-Ming Wang, Huan-Yu Ku, Jie-Yien Lin, and Hong-Bin Chen, "Deep learning the hierarchy of steering measurement settings of qubit-pair states", Communications Physics 7 1, 72 (2024).

[13] Sébastien Designolle, Roope Uola, Kimmo Luoma, and Nicolas Brunner, "Set Coherence: Basis-Independent Quantification of Quantum Coherence", Physical Review Letters 126 22, 220404 (2021).

[14] Marcello Nery, Marco Túlio Quintino, Philippe Allard Guérin, Thiago O. Maciel, and Reinaldo O. Vianna, "Simple and maximally robust processes with no classical common-cause or direct-cause explanation", Quantum 5, 538 (2021).

[15] Roope Uola, Fabiano Lever, Otfried Gühne, and Juha-Pekka Pellonpää, "Unified picture for spatial, temporal, and channel steering", Physical Review A 97 3, 032301 (2018).

[16] Junior R. Gonzales-Ureta, Ana Predojević, and Adán Cabello, "Optimal and tight Bell inequalities for state-independent contextuality sets", Physical Review Research 5 1, L012035 (2023).

[17] Flavien Hirsch, Marco Túlio Quintino, and Nicolas Brunner, "Quantum measurement incompatibility does not imply Bell nonlocality", Physical Review A 97 1, 012129 (2018).

[18] 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).

[19] Andrew J. P. Garner, Marius Krumm, and Markus P. Müller, "Semi-device-independent information processing with spatiotemporal degrees of freedom", Physical Review Research 2 1, 013112 (2020).

[20] Erika Bene and Tamás Vértesi, "Measurement incompatibility does not give rise to Bell violation in general", New Journal of Physics 20 1, 013021 (2018).

[21] Joseph Bowles, Ivan Šupić, Daniel Cavalcanti, and Antonio Acín, "Self-testing of Pauli observables for device-independent entanglement certification", Physical Review A 98 4, 042336 (2018).

[22] Ernest Y.-Z. Tan, Pavel Sekatski, Jean-Daniel Bancal, René Schwonnek, Renato Renner, Nicolas Sangouard, and Charles C.-W. Lim, "Improved DIQKD protocols with finite-size analysis", Quantum 6, 880 (2022).

[23] Armin Tavakoli, Alastair A. Abbott, Marc-Olivier Renou, Nicolas Gisin, and Nicolas Brunner, "Semi-device-independent characterization of multipartite entanglement of states and measurements", Physical Review A 98 5, 052333 (2018).

[24] Máté Farkas, Maria Balanzó-Juandó, Karol Łukanowski, Jan Kołodyński, and Antonio Acín, "Bell Nonlocality Is Not Sufficient for the Security of Standard Device-Independent Quantum Key Distribution Protocols", Physical Review Letters 127 5, 050503 (2021).

[25] Jeongho Bang, Junghee Ryu, and Dagomir Kaszlikowski, "Fidelity deviation in quantum teleportation", Journal of Physics A: Mathematical and Theoretical 51 13, 135302 (2018).

[26] Shmuel Friedland, Lek-Heng Lim, and Jinjie Zhang, "Grothendieck constant is norm of Strassen matrix multiplication tensor", Numerische Mathematik 143 4, 905 (2019).

[27] Roope Uola, Ana C. S. Costa, H. Chau Nguyen, and Otfried Gühne, "Quantum steering", Reviews of Modern Physics 92 1, 015001 (2020).

[28] Márcio M. Cunha, Alejandro Fonseca, and Edilberto O. Silva, "Tripartite Entanglement: Foundations and Applications", Universe 5 10, 209 (2019).

[29] Anna de Rosier, Jacek Gruca, Fernando Parisio, Tamás Vértesi, and Wiesław Laskowski, "Strength and typicality of nonlocality in multisetting and multipartite Bell scenarios", Physical Review A 101 1, 012116 (2020).

[30] Ratul Banerjee, Srijon Ghosh, Shiladitya Mal, and Aditi Sen(De), "Spreading nonlocality in a quantum network", Physical Review Research 2 4, 043355 (2020).

[31] Tanmay Singal, Filip B. Maciejewski, and Michał Oszmaniec, "Implementation of quantum measurements using classical resources and only a single ancillary qubit", npj Quantum Information 8 1, 82 (2022).

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[33] Marie Ioannou, Jonatan Bohr Brask, and Nicolas Brunner, "Upper bound on certifiable randomness from a quantum black-box device", Physical Review A 99 5, 052338 (2019).

[34] A. Vourdas, "Quantum quantities near the Grothendieck bound in a single quantum system", Journal of Physics: Conference Series 2667 1, 012009 (2023).

[35] István Márton, Erika Bene, and Tamás Vértesi, "Bounding the detection efficiency threshold in Bell tests using multiple copies of the maximally entangled two-qubit state carried by a single pair of particles", Physical Review A 107 2, 022205 (2023).

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