Quantum Bell inequalities from Information Causality – tight for Macroscopic Locality

Mariami Gachechiladze1, Bartłomiej Bąk2, Marcin Pawłowski3, and Nikolai Miklin3

1Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany
2Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland
3International Centre for Theory of Quantum Technologies (ICTQT), University of Gdansk, 80-308 Gdańsk, Poland

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

Abstract

In a Bell test, the set of observed probability distributions complying with the principle of local realism is fully characterized by Bell inequalities. Quantum theory allows for a violation of these inequalities, which is famously regarded as Bell nonlocality. However, finding the maximal degree of this violation is, in general, an undecidable problem. Consequently, no algorithm can be used to derive quantum analogs of Bell inequalities, which would characterize the set of probability distributions allowed by quantum theory. Here we present a family of inequalities, which approximate the set of quantum correlations in Bell scenarios where the number of settings or outcomes can be arbitrary. We derive these inequalities from the principle of Information Causality, and thus, we do not assume the formalism of quantum mechanics. Moreover, we identify a subspace in the correlation space for which the derived inequalities give the necessary and sufficient conditions for the principle of Macroscopic Locality. As a result, we show that in this subspace, the principle of Information Causality is strictly stronger than the principle of Macroscopic Locality.

Quantum technologies have advanced rapidly in recent decades, both in academia and industry. The capabilities of quantum theory to boost computational speed, communication privacy, and measurement precision are dazzling and may appear unbounded. However, quantum theory has limitations, and understanding them is critical from a technological and fundamental standpoint. Constraints that do not allow for the infinite power of quantum correlations stem from the mathematical formalism of quantum mechanics, making them difficult to derive and give a physical interpretation. Another approach to the same problem is to constrain the set of quantum correlations by physical principles such as the ”no faster-than-light communication” principle. In this paper, we look at two seemingly unrelated physical principles and show that they lead to the same set of constraints, which we refer to as ”quantum Bell inequalities.” Rather than being coincidental, our constraints result from systematic methods that we develop for both principles, allowing us to derive them for correlations of any size.
The two principles that we consider are information causality and macroscopic locality. The former is a generalization of the ”no faster-than-light communication” principle and bounds the strength of correlations that two distant parties can establish if some amount of communication is allowed. The latter postulates that correlations obtained from measuring many microscopic systems at once must obey the laws of classical mechanics. We first derive an infinite family of quantum Bell inequalities from the information causality and subsequently show that, surprisingly, these inequalities constitute the necessary and sufficient conditions for the macroscopic locality.
Our inequalities are expected to be useful for security proofs in quantum cryptography. Furthermore, the techniques developed can help derive other inequalities from the considered principles.

► BibTeX data

► References

[1] J. S. Bell. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1: 195–200, Nov 1964. 10.1103/​PhysicsPhysiqueFizika.1.195. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysicsPhysiqueFizika.1.195.
https:/​/​doi.org/​10.1103/​PhysicsPhysiqueFizika.1.195

[2] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev., 47: 777–780, May 1935. 10.1103/​PhysRev.47.777. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRev.47.777.
https:/​/​doi.org/​10.1103/​PhysRev.47.777

[3] Antonio Acín, Nicolas Brunner, Nicolas Gisin, Serge Massar, Stefano Pironio, and Valerio Scarani. Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett., 98: 230501, Jun 2007. 10.1103/​PhysRevLett.98.230501. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.98.230501.
https:/​/​doi.org/​10.1103/​PhysRevLett.98.230501

[4] 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. Random numbers certified by Bell's theorem. Nature, 464 (7291): 1021–1024, Apr 2010. ISSN 1476-4687. 10.1038/​nature09008. URL https:/​/​doi.org/​10.1038/​nature09008.
https:/​/​doi.org/​10.1038/​nature09008

[5] Sergey Bravyi, David Gosset, and Robert König. Quantum advantage with shallow circuits. Science, 362 (6412): 308–311, 2018. ISSN 0036-8075. 10.1126/​science.aar3106.
https:/​/​doi.org/​10.1126/​science.aar3106

[6] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23: 880–884, Oct 1969. 10.1103/​PhysRevLett.23.880. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.23.880.
https:/​/​doi.org/​10.1103/​PhysRevLett.23.880

[7] Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar, and Sandu Popescu. Bell inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett., 88: 040404, Jan 2002. 10.1103/​PhysRevLett.88.040404. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.88.040404.
https:/​/​doi.org/​10.1103/​PhysRevLett.88.040404

[8] Daniel Collins and Nicolas Gisin. A relevant two qubit Bell inequality inequivalent to the CHSH inequality. Journal of Physics A: Mathematical and General, 37 (5): 1775–1787, jan 2004. 10.1088/​0305-4470/​37/​5/​021. URL https:/​/​doi.org/​10.1088/​0305-4470/​37/​5/​021.
https:/​/​doi.org/​10.1088/​0305-4470/​37/​5/​021

[9] Stefano Pironio. Lifting Bell inequalities. Journal of Mathematical Physics, 46 (6): 062112, 2005. 10.1063/​1.1928727. URL https:/​/​doi.org/​10.1063/​1.1928727.
https:/​/​doi.org/​10.1063/​1.1928727

[10] Denis Rosset, Jean-Daniel Bancal, and Nicolas Gisin. Classifying 50 years of Bell inequalities. Journal of Physics A: Mathematical and Theoretical, 47 (42): 424022, oct 2014. 10.1088/​1751-8113/​47/​42/​424022. URL https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424022.
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424022

[11] E. Zambrini Cruzeiro and N. Gisin. Complete list of tight Bell inequalities for two parties with four binary settings. Phys. Rev. A, 99: 022104, Feb 2019. 10.1103/​PhysRevA.99.022104. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.99.022104.
https:/​/​doi.org/​10.1103/​PhysRevA.99.022104

[12] Mariami Gachechiladze and Otfried Gühne. Completing the proof of ``Generic quantum nonlocality''. Physics Letters A, 381 (15): 1281–1285, 2017. ISSN 0375-9601. https:/​/​doi.org/​10.1016/​j.physleta.2016.10.001. URL https:/​/​www.sciencedirect.com/​science/​article/​pii/​S0375960116311355.
https:/​/​doi.org/​10.1016/​j.physleta.2016.10.001
https:/​/​www.sciencedirect.com/​science/​article/​pii/​S0375960116311355

[13] 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, Kevin J. Coakley, Shellee D. Dyer, Carson Hodge, Adriana E. Lita, Varun B. Verma, Camilla Lambrocco, Edward Tortorici, Alan L. Migdall, Yanbao Zhang, Daniel R. Kumor, William H. Farr, Francesco Marsili, Matthew D. Shaw, Jeffrey A. Stern, Carlos Abellán, Waldimar Amaya, Valerio Pruneri, Thomas Jennewein, Morgan W. Mitchell, Paul G. Kwiat, Joshua C. Bienfang, Richard P. Mirin, Emanuel Knill, and Sae Woo Nam. Strong loophole-free test of local realism. Phys. Rev. Lett., 115: 250402, Dec 2015. 10.1103/​PhysRevLett.115.250402. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.115.250402.
https:/​/​doi.org/​10.1103/​PhysRevLett.115.250402

[14] 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 (7575): 682–686, Oct 2015. ISSN 1476-4687. 10.1038/​nature15759. URL https:/​/​doi.org/​10.1038/​nature15759.
https:/​/​doi.org/​10.1038/​nature15759

[15] Marissa Giustina, Marijn A. M. Versteegh, Sören Wengerowsky, Johannes Handsteiner, Armin Hochrainer, Kevin Phelan, Fabian Steinlechner, Johannes Kofler, Jan-Åke Larsson, Carlos Abellán, Waldimar Amaya, Valerio Pruneri, Morgan W. Mitchell, Jörn Beyer, Thomas Gerrits, Adriana E. Lita, Lynden K. Shalm, Sae Woo Nam, Thomas Scheidl, Rupert Ursin, Bernhard Wittmann, and Anton Zeilinger. Significant-loophole-free test of Bell's theorem with entangled photons. Phys. Rev. Lett., 115: 250401, Dec 2015. 10.1103/​PhysRevLett.115.250401. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.115.250401.
https:/​/​doi.org/​10.1103/​PhysRevLett.115.250401

[16] Itamar Pitowsky. Correlation polytopes: Their geometry and complexity. Mathematical Programming, 50 (1): 395–414, Mar 1991. ISSN 1436-4646. 10.1007/​BF01594946. URL https:/​/​doi.org/​10.1007/​BF01594946.
https:/​/​doi.org/​10.1007/​BF01594946

[17] B. S. Tsirel'son. Quantum analogues of the Bell inequalities. The case of two spatially separated domains. Journal of Soviet Mathematics, 36 (4): 557–570, Feb 1987. ISSN 1573-8795. 10.1007/​BF01663472. URL https:/​/​doi.org/​10.1007/​BF01663472.
https:/​/​doi.org/​10.1007/​BF01663472

[18] Lawrence J. Landau. Empirical two-point correlation functions. Foundations of Physics, 18 (4): 449–460, Apr 1988. ISSN 1572-9516. 10.1007/​BF00732549. URL https:/​/​doi.org/​10.1007/​BF00732549.
https:/​/​doi.org/​10.1007/​BF00732549

[19] Jos Uffink. Quadratic Bell inequalities as tests for multipartite entanglement. Phys. Rev. Lett., 88: 230406, May 2002. 10.1103/​PhysRevLett.88.230406. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.88.230406.
https:/​/​doi.org/​10.1103/​PhysRevLett.88.230406

[20] S. Zohren, P. Reska, R. D. Gill, and W. Westra. A tight Tsirelson inequality for infinitely many outcomes. EPL (Europhysics Letters), 90 (1): 10002, apr 2010. 10.1209/​0295-5075/​90/​10002. URL https:/​/​doi.org/​10.1209/​0295-5075/​90/​10002.
https:/​/​doi.org/​10.1209/​0295-5075/​90/​10002

[21] Tzyh Haur Yang, Miguel Navascués, Lana Sheridan, and Valerio Scarani. Quantum Bell inequalities from macroscopic locality. Phys. Rev. A, 83: 022105, Feb 2011. 10.1103/​PhysRevA.83.022105. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.83.022105.
https:/​/​doi.org/​10.1103/​PhysRevA.83.022105

[22] B. S. Cirel'son. Quantum generalizations of Bell's inequality. Letters in Mathematical Physics, 4 (2): 93–100, Mar 1980. ISSN 1573-0530. 10.1007/​BF00417500. URL https:/​/​doi.org/​10.1007/​BF00417500.
https:/​/​doi.org/​10.1007/​BF00417500

[23] Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. MIP* = RE. Commun. ACM, 64 (11): 131–138, oct 2021. ISSN 0001-0782. 10.1145/​3485628. URL https:/​/​doi.org/​10.1145/​3485628.
https:/​/​doi.org/​10.1145/​3485628

[24] Dominic Mayers and Andrew Yao. Self testing quantum apparatus. arXiv preprint quant-ph/​0307205, 2003. https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​0307205.
https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​0307205
arXiv:quant-ph/0307205

[25] Andrea Coladangelo, Koon Tong Goh, and Valerio Scarani. All pure bipartite entangled states can be self-tested. Nature Communications, 8 (1): 15485, May 2017. ISSN 2041-1723. 10.1038/​ncomms15485. URL https:/​/​doi.org/​10.1038/​ncomms15485.
https:/​/​doi.org/​10.1038/​ncomms15485

[26] Ivan Šupić and Joseph Bowles. Self-testing of quantum systems: a review. Quantum, 4: 337, September 2020. ISSN 2521-327X. 10.22331/​q-2020-09-30-337. URL https:/​/​doi.org/​10.22331/​q-2020-09-30-337.
https:/​/​doi.org/​10.22331/​q-2020-09-30-337

[27] Marcin Pawłowski, Tomasz Paterek, Dagomir Kaszlikowski, Valerio Scarani, Andreas Winter, and Marek Żukowski. Information causality as a physical principle. Nature, 461 (7267): 1101–1104, Oct 2009. ISSN 1476-4687. 10.1038/​nature08400. URL https:/​/​doi.org/​10.1038/​nature08400.
https:/​/​doi.org/​10.1038/​nature08400

[28] Miguel Navascués and Harald Wunderlich. A glance beyond the quantum model. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466 (2115): 881–890, 2010. 10.1098/​rspa.2009.0453. URL https:/​/​royalsocietypublishing.org/​doi/​abs/​10.1098/​rspa.2009.0453.
https:/​/​doi.org/​10.1098/​rspa.2009.0453

[29] Carlo Rovelli. Quantum Gravity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2004. 10.1017/​CBO9780511755804.
https:/​/​doi.org/​10.1017/​CBO9780511755804

[30] Sandu Popescu and Daniel Rohrlich. Quantum nonlocality as an axiom. Foundations of Physics, 24 (3): 379–385, Mar 1994. ISSN 1572-9516. 10.1007/​BF02058098. URL https:/​/​doi.org/​10.1007/​BF02058098.
https:/​/​doi.org/​10.1007/​BF02058098

[31] Marcin Pawłowski and Marek Żukowski. Entanglement-assisted random access codes. Phys. Rev. A, 81: 042326, Apr 2010. 10.1103/​PhysRevA.81.042326. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.81.042326.
https:/​/​doi.org/​10.1103/​PhysRevA.81.042326

[32] Miguel Navascués, Stefano Pironio, and Antonio Acín. Bounding the set of quantum correlations. Phys. Rev. Lett., 98: 010401, Jan 2007. 10.1103/​PhysRevLett.98.010401. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.98.010401.
https:/​/​doi.org/​10.1103/​PhysRevLett.98.010401

[33] Daniel Cavalcanti, Alejo Salles, and Valerio Scarani. Macroscopically local correlations can violate information causality. Nature Communications, 1 (1): 136, Dec 2010. ISSN 2041-1723. 10.1038/​ncomms1138. URL https:/​/​doi.org/​10.1038/​ncomms1138.
https:/​/​doi.org/​10.1038/​ncomms1138

[34] Ronald A Howard. Dynamic programming and Markov processes. John Wiley, 1960.

[35] Alan V Oppenheim, John R Buck, and Ronald W Schafer. Discrete-time signal processing. Vol. 2. Upper Saddle River, NJ: Prentice Hall, 2001.

[36] Wim Van Dam. Implausible consequences of superstrong nonlocality. arXiv preprint quant-ph/​0501159, 2005. https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​0501159.
https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​0501159
arXiv:quant-ph/0501159

[37] Wim van Dam. Implausible consequences of superstrong nonlocality. Natural Computing, 12 (1): 9–12, Mar 2013. ISSN 1572-9796. 10.1007/​s11047-012-9353-6. URL https:/​/​doi.org/​10.1007/​s11047-012-9353-6.
https:/​/​doi.org/​10.1007/​s11047-012-9353-6

[38] Marcin Pawłowski and Valerio Scarani. Information Causality, pages 423–438. Springer Netherlands, Dordrecht, 2016. ISBN 978-94-017-7303-4. 10.1007/​978-94-017-7303-4_12. URL https:/​/​doi.org/​10.1007/​978-94-017-7303-4_12.
https:/​/​doi.org/​10.1007/​978-94-017-7303-4_12

[39] Robert M Fano. Transmission of information: a statistical theory of communications. Mit Press, 1968.

[40] A. J. Leggett and Anupam Garg. Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? Phys. Rev. Lett., 54: 857–860, Mar 1985. 10.1103/​PhysRevLett.54.857. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.54.857.
https:/​/​doi.org/​10.1103/​PhysRevLett.54.857

[41] Johannes Kofler and Časlav Brukner. Classical world arising out of quantum physics under the restriction of coarse-grained measurements. Phys. Rev. Lett., 99: 180403, Nov 2007. 10.1103/​PhysRevLett.99.180403. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.99.180403.
https:/​/​doi.org/​10.1103/​PhysRevLett.99.180403

[42] George E Collins. Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In Automata theory and formal languages, pages 134–183. Springer, 1975. https:/​/​doi.org/​10.1007/​3-540-07407-4_17.
https:/​/​doi.org/​10.1007/​3-540-07407-4_17

[43] Karin Gatermann and Pablo A. Parrilo. Symmetry groups, semidefinite programs, and sums of squares. Journal of Pure and Applied Algebra, 192 (1): 95 – 128, 2004. ISSN 0022-4049. https:/​/​doi.org/​10.1016/​j.jpaa.2003.12.011. URL http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0022404904000131.
https:/​/​doi.org/​10.1016/​j.jpaa.2003.12.011
http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0022404904000131

[44] Denis Rosset. Characterization of correlations in quantum networks. 2015. https:/​/​doi.org/​10.13097/​archive-ouverte/​unige:77401.
https:/​/​doi.org/​10.13097/​archive-ouverte/​unige:77401

[45] Nikolai Miklin and Marcin Pawłowski. Information causality without concatenation. Phys. Rev. Lett., 126: 220403, Jun 2021. 10.1103/​PhysRevLett.126.220403. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.126.220403.
https:/​/​doi.org/​10.1103/​PhysRevLett.126.220403

[46] Miguel Navascués, Yelena Guryanova, Matty J Hoban, and Antonio Acín. Almost quantum correlations. Nature Communications, 6 (1): 6288, Feb 2015. ISSN 2041-1723. 10.1038/​ncomms7288. URL https:/​/​doi.org/​10.1038/​ncomms7288.
https:/​/​doi.org/​10.1038/​ncomms7288

[47] G Lejeune Dirichlet. Sur la convergence des séries trigonométriques qui servent à représenter une fon... Journal für die reine und angewandte Mathematik, 4, 1829.

[48] Daniel Zwillinger, Victor Moll, I.S. Gradshteyn, and I.M. Ryzhik, editors. 1 - Elementary Functions. Academic Press, Boston, eighth edition edition, 2014. ISBN 978-0-12-384933-5. https:/​/​doi.org/​10.1016/​B978-0-12-384933-5.00001-1. URL https:/​/​www.sciencedirect.com/​science/​article/​pii/​B9780123849335000011.
https:/​/​doi.org/​10.1016/​B978-0-12-384933-5.00001-1
https:/​/​www.sciencedirect.com/​science/​article/​pii/​B9780123849335000011

[49] Thomas M Cover. Elements of information theory. 1999. 10.1002/​047174882X.
https:/​/​doi.org/​10.1002/​047174882X

Cited by

[1] Wojciech Bruzda, "Structured Unitary Matrices and Quantum Entanglement", arXiv:2204.12470.

The above citations are from SAO/NASA ADS (last updated successfully 2022-07-05 06:49:36). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2022-07-05 06:49:34).