Papers
Topics
Authors
Recent
Search
2000 character limit reached

(Almost-)Quantum Bell Inequalities and Device-Independent Applications

Published 12 Sep 2023 in quant-ph | (2309.06304v4)

Abstract: Investigations of the boundary of the quantum correlation set through the derivation of quantum Bell inequalities have gained increased attention in recent years, which are related to Tsirelson's problem and have significant applications in DI information processing. However, determining quantum Bell inequalities is a notoriously difficult task and only isolated examples are known. In this paper, we present families of (almost-)quantum Bell inequalities and highlight three foundational and DI applications. Firstly, quantum correlations on the non-signaling boundary are crucial in the DI randomness extraction from weak sources. In the practical Bell scenario of two players with two k-outcome measurements, we derive quantum Bell inequalities that show a separation of the quantum boundary from certain portions of the no-signaling boundary of dimension up to 4k-8, extending previous results. As an immediate by-product of this, we give a general proof of Aumann's Agreement theorem for quantum systems and the almost-quantum correlations, which implies Aumann's agreement theorem is a reasonable physical principle in the context of epistemics to pick out both quantum theory and almost-quantum correlations from general no-signaling theories. Secondly, we present a family of quantum Bell inequalities in the two players with m binary measurements scenarios, that serve to self-test the two-qubit singlet and 2m measurements. Interestingly, this claim generalizes the result for m=2 discovered by Tsirelson-Landau-Masanes and shows an improvement over the state-of-the-art DIRA. Lastly, we use our quantum Bell inequalities to derive the general form of the principle of no advantage in nonlocal computation, which is an information-theoretic principle that serves to characterize the quantum correlation set. With this, we provide the most precise characterization of the quantum boundary known so far.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (56)
  1. J. S. Bell. The theory of local beables. Chapter 7 in Speakable and unspeakable in Quantum Mechanics. Cambridge University Press (2011).
  2. J. S. Bell. Free variables and local causality. Chapter 12 in Speakable and unspeakable in Quantum Mechanics. Cambridge University Press (2011).
  3. No Signaling and Quantum Key Distribution. Physical Review Letters 95(1): 010503 (2005).
  4. Device-Independent Security of Quantum Cryptography against Collective Attacks. Physical Review Letters 98(23): 230501 (2007).
  5. Random numbers certified by Bell’s theorem. Nature 464(7291): 1021 – 1024 (2010).
  6. Security of practical private randomness generation. Physical Review A 87(1): 012336 (2013).
  7. Quantum cryptography with imperfect apparatus. In IEEE Proceedings of the 39th Annual Symposium on Foundations of Computer Science (FOCS’98) 503 – 509 (1998).
  8. Self testing quantum apparatus. arXiv:quant-ph/0307205 (2003)
  9. Nonlocality and communication complexity. Reviews of Modern Physics 82(1): 665 (2010).
  10. I. Pitowsky. Correlation polytopes: Their geometry and complexity. Mathematical Programming 50: 395 – 414 (1991).
  11. Quantum nonlocality as an axiom. Foundations of Physics 24(3): 379 – 385 (1994).
  12. Geometry of the set of quantum correlations. Physical Review A 97(2): 022104 (2018).
  13. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New Journal of Physics 10(7): 073013 (2008).
  14. Bounding the Set of Quantum Correlations. Physical Review Letters 98(1): 010401 (2007).
  15. L. Masanes. Necessary and sufficient condition for quantum-generated correlations. quant-ph/0309137 (2003).
  16. S. Ishizaka. Necessary and sufficient criterion for extremal quantum correlations in the simplest bell scenario. Physical Review A, 97(5):050102 (2018).
  17. Extremal points of the quantum set in the CHSH scenario: conjectured analytical solution. arXiv:2302.10658 (2023).
  18. T. Fritz. Polyhedral duality in Bell scenarios with two binary observables. Journal of Mathematical Physics, 53(7) (2012).
  19. Extremal Tsirelson inequalities. arXiv:2401.12791 (2024).
  20. B. S. Cirel’son. Quantum generalizations of Bell’s inequality. Letters in Mathematical Physics 4: 93 – 100 (1980).
  21. L. J Landau. Empirical two-point correlation functions. Foundations of Physics 18: 449 – 460 (1988).
  22. Quantum correlations in the minimal scenario. Quantum 7, 947 (2023).
  23. R. Ramanathan. Violation of all two-party facet Bell inequalities by almost-quantum correlations. Physical Review Research 3(3): 033100 (2021).
  24. Guess your neighbor’s input: A multipartite nonlocal game with no quantum advantage. Physical Review Letters 104(23): 230404 (2010).
  25. Geometry of the quantum set on no-signaling faces. Physical Review A 99(3): 032106 (2019).
  26. Quantum correlations on the no-signaling boundary: self-testing and more. Quantum 7, 1054 (2023).
  27. Characterising the Performance of XOR Games and the Shannon Capacity of Graphs. Physical Review Letters 113(24): 240401 (2014).
  28. Quantum Nonlocality and Beyond: Limits from Nonlocal Computation. Physical Review Letters 99(18): 180502 (2007).
  29. Uncertainty-complementarity balance as a general constraint on non-locality. arXiv:1808.06416 (2018).
  30. No Quantum Realization of Extremal No-signaling Boxes. Physical Review Letters 117(5): 050401 (2016).
  31. P. Botteron, A. Broadbent and M.-O. Proulx. Extending the Known Region of Nonlocal Boxes that Collapse Communication Complexity. Physical Review Letters 132, 070201 (2024).
  32. Closed sets of nonlocal correlations. Physical Review A, 80(6):062107 (2009).
  33. Monotone measures for non-local correlations. IEEE Transactions on Information Theory, 61(9):5185–5208 (2015).
  34. Nonlocality distillation and quantum voids. Physical Review A 100(1): 012102 (2019).
  35. Robust self-testing of the singlet. Journal of Physics A: Mathematical and Theoretical 45(45): 455304 (2012).
  36. Information causality as a physical principle. Nature 461(7267): 1101 – 1104 (2009).
  37. A glance beyond the quantum model. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466(2115): 881 – 890 (2010).
  38. Local orthogonality as a multipartite principle for quantum correlations. Nature Communications 4(1): 2263 (2013).
  39. Limit on nonlocality in any world in which communication complexity is not trivial. Physical Review Letters 96(25): 250401 (2006).
  40. Observers of quantum systems cannot agree to disagree. Nature Communications 12(1): 7021 (2021).
  41. Almost quantum correlations. Nature Communications 6(1): 6288 (2015).
  42. All the self-testings of the singlet for two binary measurements. New Journal of Physics 18(2): 025021 (2016).
  43. Self-testing protocols based on the chained Bell inequalities. New Journal of Physics 18(3): 035013 (2016).
  44. Exploring the local orthogonality principle. Physical Review A 89(3): 032117 (2014).
  45. (Non-)Contextuality of Physical Theories as an Axiom. arXiv: 1010.2163 (2010).
  46. Relaxations of vertex packing. Journal of Combinatorial Theory, Series B 40(3): 330 – 343 (1986).
  47. A Combinatorial Approach to Nonlocality and Contextuality. Communications in Mathematical Physics 334: 533 – 628 (2015).
  48. On Grötschel-Lovász-Schrijver’s relaxation of stable set polytopes. Journal of the Operations Research Society of Japan 45(3): 285 – 292 (2002).
  49. Quantum Bell inequalities from Information Causality – tight for Macroscopic Locality. Quantum 6, 717 (2022).
  50. Quantum Bell inequalities from macroscopic locality. Physical Review A 83(2): 022105 (2011).
  51. Nonlocal correlations as an information-theoretic resource. Physical Review A 71(2): 022101 (2005).
  52. Tilted Hardy paradoxes for device-independent randomness extraction. Quantum 7, 1114 (2023).
  53. B. S. Tsirel’son. Quantum analogues of Bell inequalities. The case of two spatially separated domains. Journal of Soviet Mathematics 36: 557 – 570 (1987).
  54. On the Facial Structure of the Set of Correlation Matrices. SIAM Journal on Matrix Analysis and Applications 17(3): 530 – 547 (1996).
  55. Practical No-signaling proof Randomness Amplification using Hardy paradoxes and its experimental implementation. arXiv: 1810.11648 (2018).
  56. No-signaling-proof randomness extraction from public weak sources. arXiv: 2108.08819 (2021).
Citations (1)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Summarize

Paper to Video (Beta)

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up

Tweets

Sign up for free to view the 1 tweet with 1 like about this paper.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up for Free