Bell nonlocality with a single shot

Mateus Araújo1, Flavien Hirsch1, and Marco Túlio Quintino2,1,3

1Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
2Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
3Department of Physics, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan

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


In order to reject the local hidden variables hypothesis, the usefulness of a Bell inequality can be quantified by how small a $p$-value it will give for a physical experiment. Here we show that to obtain a small expected $p$-value it is sufficient to have a large gap between the local and Tsirelson bounds of the Bell inequality, when it is formulated as a nonlocal game. We develop an algorithm for transforming an arbitrary Bell inequality into an equivalent nonlocal game with the largest possible gap, and show its results for the CGLMP and $I_{nn22}$ inequalities.

We present explicit examples of Bell inequalities with gap arbitrarily close to one, and show that this makes it possible to reject local hidden variables with arbitrarily small $p$-value in a single shot, without needing to collect statistics. We also develop an algorithm for calculating local bounds of general Bell inequalities which is significantly faster than the naïve approach, which may be of independent interest.

Nonlocal games are cooperative games between two parties, Alice and Bob, that are not allowed to communicate. The maximal probability with which Alice and Bob can win the game depends on how the world fundamentally works: if it respects classical ideas about locality and determinism, this maximal probability is given by the local bound. On the other hand, if the world works according to quantum mechanics, the maximal probability is given by the Tsirelson bound, which is larger than the local bound. This makes it possible to experimentally falsify the classical worldview: let Alice and Bob play a nonlocal game with quantum devices for many rounds, and if they win more often than the local bound predicts, that's it.

The number of rounds it takes for a decisive rejection of the classical worldview depends on the statistical power of the nonlocal game: a more powerful game requires fewer rounds to reach a conclusion with the same degree of confidence. We show that in order to get a large statistical power, it is enough to have a large gap between the local bound and the Tsirelson bound of the nonlocal game. Moreover, we show that this gap depends on how precisely a nonlocal game is formulated, so we develop an algorithm to maximise the gap over all possible formulations of a nonlocal game. With this, we derive the most powerful version of several well-known nonlocal games, such as the CHSH game, the CGLMP games, and the Inn22 games.

A natural question to ask is how high can the statistical power of a nonlocal game get. We show that it can get arbitrarily high, by constructing two nonlocal games with gap between their local and Tsirelson bounds arbitrarily close to one. This makes it possible to conclusively falsify the classical worldview with a single round of the nonlocal game, without needing to collect statistics. Unfortunately, neither of these games is experimentally feasible, so the question of whether a single-shot falsification is possible in practice is still open.

► BibTeX data

► References

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

[2] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt, ``Proposed experiment to test local hidden-variable theories'' Physical Review Letters 23, 880–884 (1969).

[3] John S Bell ``The theory of local beables'' (1975).

[4] David Deutschand Patrick Hayden ``Information flow in entangled quantum systems'' Proceedings of the Royal Society A 456, 1759–1774 (2000).

[5] Harvey R. Brownand Christopher G. Timpson ``Bell on Bell's theorem: The changing face of nonlocality'' Cambridge University Press (2016).

[6] Alain Aspect, Jean Dalibard, and Gérard Roger, ``Experimental Test of Bell's Inequalities Using Time-Varying Analyzers'' Physical Review Letters 49, 1804–1807 (1982).

[7] Gregor Weihs, Thomas Jennewein, Christoph Simon, Harald Weinfurter, and Anton Zeilinger, ``Violation of Bell’s Inequality under Strict Einstein Locality Conditions'' Physical Review Letters 81, 5039–5043 (1998).

[8] M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, ``Experimental violation of a Bell's inequality with efficient detection'' Nature 409, 791–794 (2001).

[9] B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, ``Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres'' Nature 526, 682–686 (2015).

[10] Marissa Giustina, Marijn A. M. Versteegh, Sören Wengerowsky, Johannes Handsteiner, Armin Hochrainer, Kevin Phelan, Fabian Steinlechner, Johannes Kofler, Jan-à ke Larsson, Carlos Abellán, and et al., ``Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons'' Physical Review Letters 115, 250401 (2015).

[11] Lynden K. Shalm, Evan Meyer-Scott, Bradley G. Christensen, Peter Bierhorst, Michael A. Wayne, Martin J. Stevens, Thomas Gerrits, Scott Glancy, Deny R. Hamel, Michael S. Allman, and et al., ``Strong Loophole-Free Test of Local Realism'' Physical Review Letters 115, 250402 (2015).

[12] Richard Cleve, Peter Høyer, Ben Toner, and John Watrous, ``Consequences and Limits of Nonlocal Strategies'' (2004).

[13] Jonathan Barrett, Daniel Collins, Lucien Hardy, Adrian Kent, and Sandu Popescu, ``Quantum nonlocality, Bell inequalities, and the memory loophole'' Physical Review A 66, 042111 (2002).

[14] Richard D. Gilland Jan-Åke Larsson ``Accardi Contra Bell (Cum Mundi): The Impossible Coupling'' Lecture Notes-Monograph Series 42, 133–154 (2003).

[15] Anup Rao ``Parallel repetition in projection games and a concentration bound'' SIAM Journal on Computing 40, 1871–1891 (2011).

[16] Julia Kempe, Oded Regev, and Ben Toner, ``Unique Games with Entangled Provers are Easy'' (2007).

[17] M. Junge, C. Palazuelos, D. Pérez-García, I. Villanueva, and M. M. Wolf, ``Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory'' Communications in Mathematical Physics 300, 715–739 (2010).

[18] M. Junge, C. Palazuelos, D. Pérez-García, I. Villanueva, and M. M. Wolf, ``Operator Space Theory: A Natural Framework for Bell Inequalities'' Physical Review Letters 104, 170405 (2010).

[19] B. S. Cirel'son ``Quantum generalizations of Bell's inequality'' Letters in Mathematical Physics 4, 93–100 (1980).

[20] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, ``Bell nonlocality'' Reviews of Modern Physics 86, 419–478 (2014).

[21] Michael Ben-Or, Shafi Goldwasser, Joe Kilian, and Avi Wigderson, ``Multi-Prover Interactive Proofs: How to Remove Intractability Assumptions'' Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing 113–131 (1988).

[22] Boris Tsirelson ``Quantum information processing - Lecture notes'' (1997).

[23] Harry Buhrman, Oded Regev, Giannicola Scarpa, and Ronald de Wolf, ``Near-Optimal and Explicit Bell Inequality Violations'' 2011 IEEE 26th Annual Conference on Computational Complexity (2011).

[24] Carlos Palazuelosand Thomas Vidick ``Survey on nonlocal games and operator space theory'' Journal of Mathematical Physics 57, 015220 (2016).

[25] Marcel Froissart ``Constructive generalization of Bell’s inequalities'' Il Nuovo Cimento B (1971-1996) 64, 241–251 (1981).

[26] Yanbao Zhang, Scott Glancy, and Emanuel Knill, ``Asymptotically optimal data analysis for rejecting local realism'' Physical Review A 84, 062118 (2011).

[27] David Elkoussand Stephanie Wehner ``(Nearly) optimal P-values for all Bell inequalities'' npj Quantum Information 2 (2016).

[28] Richard D. Gill ``Time, Finite Statistics, and Bell's Fifth Position'' (2003).

[29] Yanbao Zhang, Scott Glancy, and Emanuel Knill, ``Efficient quantification of experimental evidence against local realism'' Physical Review A 88, 052119 (2013).

[30] Peter Bierhorst ``A Rigorous Analysis of the Clauser–Horne–Shimony–Holt Inequality Experiment When Trials Need Not be Independent'' Foundations of Physics 44, 736–761 (2014).

[31] Denis Rosset, Jean-Daniel Bancal, and Nicolas Gisin, ``Classifying 50 years of Bell inequalities'' Journal of Physics A Mathematical General 47, 424022 (2014).

[32] M. O. Renou, D. Rosset, A. Martin, and N. Gisin, ``On the inequivalence of the CH and CHSH inequalities due to finite statistics'' Journal of Physics A Mathematical General 50, 255301 (2017).

[33] Steven Diamondand Stephen Boyd ``CVXPY: A Python-embedded modeling language for convex optimization'' Journal of Machine Learning Research 17, 1–5 (2016).

[34] Akshay Agrawal, Robin Verschueren, Steven Diamond, and Stephen Boyd, ``A rewriting system for convex optimization problems'' Journal of Control and Decision 5, 42–60 (2018).

[35] Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar, and Sandu Popescu, ``Bell Inequalities for Arbitrarily High-Dimensional Systems'' Physical Review Letters 88, 040404 (2002).

[36] Antonio Acín, Richard Gill, and Nicolas Gisin, ``Optimal Bell Tests Do Not Require Maximally Entangled States'' Physical Review Letters 95, 210402 (2005).

[37] Stefan Zohrenand Richard D. Gill ``Maximal Violation of the CGLMP Inequality for Infinite Dimensional States'' Physical Review Letters 100 (2008).

[38] A. Acín, T. Durt, N. Gisin, and J. I. Latorre, ``Quantum nonlocality in two three-level systems'' Physical Review A 65 (2002).

[39] Miguel Navascués, Stefano Pironio, and Antonio Acín, ``Bounding the Set of Quantum Correlations'' Physical Review Letters 98 (2007).

[40] S. Zohren, P. Reska, R. D. Gill, and W. Westra, ``A tight Tsirelson inequality for infinitely many outcomes'' EPL (Europhysics Letters) 90, 10002 (2010).

[41] Daniel Collinsand Nicolas Gisin ``A relevant two qubit Bell inequality inequivalent to the CHSH inequality'' Journal of Physics A Mathematical General 37, 1775–1787 (2004).

[42] David Avisand Tsuyoshi Ito ``New classes of facets of the cut polytope and tightness of $I_{mm22}$ Bell inequalities'' Discrete Applied Mathematics 155, 1689 –1699 (2007).

[43] Károly F. Páland Tamás Vértesi ``Maximal violation of the I3322 inequality using infinite-dimensional quantum systems'' Physical Review A 82 (2010).

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

[45] Lance Fortnow ``Complexity-theoretic aspects of interactive proof systems'' thesis (1989).

[46] Uriel Feigeand László Lovász ``Two-Prover One-Round Proof Systems: Their Power and Their Problems (Extended Abstract)'' Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing 733–744 (1992).

[47] Gilles Brassard, Anne Broadbent, and Alain Tapp, ``Quantum Pseudo-Telepathy'' Foundations of Physics 35, 1877–1907 (2005).

[48] P. K. Aravind ``A simple demonstration of Bell's theorem involving two observers and no probabilities or inequalities'' American Journal of Physics 72, 1303 (2004).

[49] Ran Raz ``A Parallel Repetition Theorem'' SIAM Journal on Computing 27, 763–803 (1998).

[50] Thomas Holenstein ``Parallel repetition: simplifications and the no-signaling case'' Theory of Computing 141–172 (2009).

[51] S. A. Khotand N. K. Vishnoi ``The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into $\ell_1$'' 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05) 53–62 (2005).

[52] Carlos Palazuelos ``Superactivation of Quantum Nonlocality'' Physical Review Letters 109, 190401 (2012).

[53] Mafalda L. Almeida, Stefano Pironio, Jonathan Barrett, Géza Tóth, and Antonio Acín, ``Noise Robustness of the Nonlocality of Entangled Quantum States'' Physical Review Letters 99, 040403 (2007).

[54] Carlos Palazuelos ``On the largest Bell violation attainable by a quantum state'' Journal of Functional Analysis 267, 1959–1985 (2014).

[55] Boris Tsirelson ``Quantum analogues of the Bell inequalities. The case of two spatially separated domains'' Journal of Soviet Mathematics 36, 557–570 (1987).

[56] Antonio Acín, Nicolas Gisin, and Benjamin Toner, ``Grothendieck's constant and local models for noisy entangled quantum states'' Physical Review A 73, 062105 (2006).

[57] Flavien Hirsch, Marco Túlio Quintino, Tamás Vértesi, Miguel Navascués, and Nicolas Brunner, ``Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant $K_G(3)$'' Quantum 1, 3 (2017).

[58] Yeong-Cherng Liang, Chu-Wee Lim, and Dong-Ling Deng, ``Reexamination of a multisetting Bell inequality for qudits'' Physical Review A 80 (2009).

[59] Stephen Brierley, Miguel Navascués, and Tamas Vértesi, ``Convex separation from convex optimization for large-scale problems'' (2016).

[60] H. M. Wiseman, S. J. Jones, and A. C. Doherty, ``Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox'' Physical Review Letters 98, 140402 (2007).

[61] Marco Túlio Quintino, Tamás Vértesi, Daniel Cavalcanti, Remigiusz Augusiak, Maciej Demianowicz, Antonio Acín, and Nicolas Brunner, ``Inequivalence of entanglement, steering, and Bell nonlocality for general measurements'' Physical Review A 92, 032107 (2015).

[62] L. Gurvitsand H. Barnum ``Largest separable balls around the maximally mixed bipartite quantum state'' Physical Review A 66, 062311 (2002).

[63] Stephen Boydand Lieven Vandenberghe ``Convex Optimization'' Cambridge University Press (2004).

Cited by

[1] Nikolai Miklin, Anubhav Chaturvedi, Mohamed Bourennane, Marcin Pawłowski, and Adán Cabello, "Exponentially Decreasing Critical Detection Efficiency for Any Bell Inequality", Physical Review Letters 129 23, 230403 (2022).

[2] 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).

[3] Martin J. Renner, Armin Tavakoli, and Marco Túlio Quintino, "Classical Cost of Transmitting a Qubit", Physical Review Letters 130 12, 120801 (2023).

[4] Marc-Olivier Renou, David Trillo, Mirjam Weilenmann, Thinh P. Le, Armin Tavakoli, Nicolas Gisin, Antonio Acín, and Miguel Navascués, "Quantum theory based on real numbers can be experimentally falsified", Nature 600 7890, 625 (2021).

[5] Dardo Goyeneche, Wojciech Bruzda, Ondřej Turek, Daniel Alsina, and Karol Życzkowski, "Local hidden variable values without optimization procedures", Quantum 7, 911 (2023).

[6] Dian Wu, Yang-Fan Jiang, Xue-Mei Gu, Liang Huang, Bing Bai, Qi-Chao Sun, Xingjian Zhang, Si-Qiu Gong, Yingqiu Mao, Han-Sen Zhong, Ming-Cheng Chen, Jun Zhang, Qiang Zhang, Chao-Yang Lu, and Jian-Wei Pan, "Experimental Refutation of Real-Valued Quantum Mechanics under Strict Locality Conditions", Physical Review Letters 129 14, 140401 (2022).

[7] Lucas Tendick, Martin Kliesch, Hermann Kampermann, and Dagmar Bruß, "Distance-based resource quantification for sets of quantum measurements", Quantum 7, 1003 (2023).

[8] Lucas Tendick, Hermann Kampermann, and Dagmar Bruß, "Distributed Quantum Incompatibility", Physical Review Letters 131 12, 120202 (2023).

[9] Joshua Morris, Valeria Saggio, Aleksandra Gočanin, and Borivoje Dakić, "Quantum Verification and Estimation with Few Copies", Advanced Quantum Technologies 5 5, 2100118 (2022).

[10] M. Weilenmann, E. A. Aguilar, and M. Navascués, "Analysis and optimization of quantum adaptive measurement protocols with the framework of preparation games", Nature Communications 12 1, 4553 (2021).

[11] Xiao-Min Hu, Wen-Bo Xing, Yu Guo, Mirjam Weilenmann, Edgar A. Aguilar, Xiaoqin Gao, Bi-Heng Liu, Yun-Feng Huang, Chuan-Feng Li, Guang-Can Guo, Zizhu Wang, and Miguel Navascués, "Optimized Detection of High-Dimensional Entanglement", Physical Review Letters 127 22, 220501 (2021).

[12] Yu. I. Ozhigov and I. R. Pluzhnikov, "Superimposition and Antagonism in Chain Synthesis Using Entangled Biphotonic Control", Computational Mathematics and Modeling 33 1, 24 (2022).

[13] Mirjam Weilenmann, Edgar A. Aguilar, and Miguel Navascues, "Quantum Preparation Games", arXiv:2011.02216, (2020).

The above citations are from Crossref's cited-by service (last updated successfully 2024-04-15 04:28:20) and SAO/NASA ADS (last updated successfully 2024-04-15 04:28:20). The list may be incomplete as not all publishers provide suitable and complete citation data.