Trade-off relations between measurement dependence and hidden information for factorizable hidden variable models

Ryo Takakura; Kei Morisue; Issei Watanabe; Gen Kimura

doi:10.22331/q-2025-03-14-1662

Trade-off relations between measurement dependence and hidden information for factorizable hidden variable models

Ryo Takakura¹, Kei Morisue², Issei Watanabe³, and Gen Kimura³

¹Center for Quantum Information and Quantum Biology, Osaka University, Osaka, Japan
²Department of Physics, Waseda University, Tokyo, Japan
³College of Systems Engineering and Science, Shibaura Institute of Technology, Saitama, Japan

Published:	2025-03-14, volume 9, page 1662
Editor:	Ion Nechita
Eprint:	arXiv:2208.13634v4
Doi:	https://doi.org/10.22331/q-2025-03-14-1662
Citation:	Quantum 9, 1662 (2025).

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

Updated after initial publication: This publication was updated to version v4 after the initial publication.

Abstract

The Bell theorem is explored in terms of a trade-off relation between underlying assumptions within the hidden variable model framework. In this paper, recognizing the incorporation of hidden variables as one of the fundamental assumptions, we propose a measure termed `hidden information' taking account of their distribution. This measure quantifies the number of hidden variables that essentially contribute to the empirical statistics. For factorizable models, hidden variable models that satisfy `locality' without adhering to the measurement independence criterion, we derive novel relaxed Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequalities. These inequalities elucidate trade-off relations between measurement dependence and hidden information in the CHSH scenario. It is also revealed that the relation gives a necessary and sufficient condition for the measures to be realized by a factorizable model.

► BibTeX data

@article{Takakura2025tradeoffrelations,
  doi = {10.22331/q-2025-03-14-1662},
  url = {https://doi.org/10.22331/q-2025-03-14-1662},
  title = {Trade-off relations between measurement dependence and hidden information for factorizable hidden variable models},
  author = {Takakura, Ryo and Morisue, Kei and Watanabe, Issei and Kimura, Gen},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1662},
  month = mar,
  year = {2025}
}

► References

[1] J. S. Bell. ``On the Einstein Podolsky Rosen paradox''. Physics Physique Fizika 1, 195–200 (1964).
https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195

[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).
https://doi.org/10.1103/PhysRevLett.23.880

[3] J. S. Bell and Alain Aspect. ``Speakable and unspeakable in quantum mechanics: Collected papers on quantum philosophy''. Cambridge University Press. (2004). 2 edition.
https://doi.org/10.1017/CBO9780511815676

[4] Wayne Myrvold, Marco Genovese, and Abner Shimony. ``Bell's theorem''. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University (2024). Spring 2024 edition. url: plato.stanford.edu/archives/spr2024/entries/bell-theorem/.
https://plato.stanford.edu/archives/spr2024/entries/bell-theorem/

[5] Mary Bell and Shan Gao, editors. ``Quantum nonlocality and reality: 50 years of Bell's theorem''. Cambridge University Press. (2016).
https://doi.org/10.1017/CBO9781316219393

[6] Nicolas Gisin. ``Quantum chance''. Springer International Publishing. (2014). 2014 edition. url: doi.org/10.1007/978-3-319-05473-5.
https://doi.org/10.1007/978-3-319-05473-5

[7] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. ``Bell nonlocality''. Reviews of Modern Physics 86, 419–478 (2014).
https://doi.org/10.1103/RevModPhys.86.419

[8] Manik Banik, Md. Rajjak Gazi, Sibasish Ghosh, and Guruprasad Kar. ``Degree of complementarity determines the nonlocality in quantum mechanics''. Physical Review A 87, 052125 (2013).
https://doi.org/10.1103/PhysRevA.87.052125

[9] Michael M. Wolf, David Perez-Garcia, and Carlos Fernandez. ``Measurements incompatible in quantum theory cannot be measured jointly in any other no-signaling theory''. Physical Review Letters 103, 230402 (2009).
https://doi.org/10.1103/PhysRevLett.103.230402

[10] Ludovico Lami. ``Non-classical correlations in quantum mechanics and beyond''. PhD thesis. Universitat Autònoma de Barcelona. (2017). url: ddd.uab.cat/record/187745.
https://ddd.uab.cat/record/187745

[11] Martin Plávala. ``General probabilistic theories: An introduction''. Physics Reports 1033, 1–64 (2023).
https://doi.org/10.1016/j.physrep.2023.09.001

[12] Ryo Takakura. ``Convexity and uncertainty in operational quantum foundations''. PhD thesis. Kyoto University. (2022).
https://doi.org/10.48550/arXiv.2202.13834

[13] Peter Janotta, Christian Gogolin, Jonathan Barrett, and Nicolas Brunner. ``Limits on nonlocal correlations from the structure of the local state space''. New Journal of Physics 13, 063024 (2011).
https://doi.org/10.1088/1367-2630/13/6/063024

[14] Neil Stevens and Paul Busch. ``Steering, incompatibility, and Bell-inequality violations in a class of probabilistic theories''. Physical Review A 89, 022123 (2014).
https://doi.org/10.1103/PhysRevA.89.022123

[15] Ryo Takakura. ``Optimal CHSH values for regular polygon theories in generalized probabilistic theories''. Journal of Physics A: Mathematical and Theoretical 57, 375305 (2024).
https://doi.org/10.1088/1751-8121/ad7077

[16] Artur K. Ekert. ``Quantum cryptography based on Bell's theorem''. Physical Review Letters 67, 661–663 (1991).
https://doi.org/10.1103/PhysRevLett.67.661

[17] Antonio Acín, Nicolas Gisin, and Lluis Masanes. ``From Bell's theorem to secure quantum key distribution''. Physical Review Letters 97, 120405 (2006).
https://doi.org/10.1103/PhysRevLett.97.120405

[18] 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''. Cahiers De La Revue De Theologie Et De Philosophie 464, 1021–1024 (2010).
https://doi.org/10.1038/nature09008

[19] Jan Bouda, Marcin Pawłowski, Matej Pivoluska, and Martin Plesch. ``Device-independent randomness extraction from an arbitrarily weak min-entropy source''. Physical Review A 90, 032313 (2014).
https://doi.org/10.1103/PhysRevA.90.032313

[20] J. Bell, A. Shimony, M. Horne, and J. Clauser. ``An exchange on local beables''. Dialectica 39, 85 (1985).
https://doi.org/10.1111/j.1746-8361.1985.tb01249.x

[21] Michael J. W. Hall. ``Local deterministic model of singlet state correlations based on relaxing measurement independence''. Physical Review Letters 105, 250404 (2010).
https://doi.org/10.1103/PhysRevLett.105.250404

[22] Jonathan Barrett and Nicolas Gisin. ``How much measurement independence is needed to demonstrate nonlocality?''. Phys. Rev. Lett. 106, 100406 (2011).
https://doi.org/10.1103/PhysRevLett.106.100406

[23] Howard M. Wiseman and Eric G. Cavalcanti. ``Causarum investigatio and the two Bell’s theorems of John Bell''. In Quantum [Un]Speakables II: Half a Century of Bell's Theorem. Pages 119–142. Springer International Publishing, Cham (2017).

[24] Thomas Scheidl, Rupert Ursin, Johannes Kofler, Sven Ramelow, Xiao-Song Ma, Thomas Herbst, Lothar Ratschbacher, Alessandro Fedrizzi, Nathan K. Langford, Thomas Jennewein, and Anton Zeilinger. ``Violation of local realism with freedom of choice''. Proceedings of the National Academy of Sciences 107, 19708–19713 (2010). arXiv:https://www.pnas.org/doi/pdf/10.1073/pnas.1002780107.
https://doi.org/10.1073/pnas.1002780107
arXiv:https://www.pnas.org/doi/pdf/10.1073/pnas.1002780107

[25] Abner Shimony, Michael A. Horne, and John F. Clauser. ``Comment on "The Theory of Local Beables"''. Epistemological Letters 13, 1 (1976).

[26] Jan Åke Larsson. ``Loopholes in bell inequality tests of local realism''. Journal of Physics A: Mathematical and Theoretical 47, 424003 (2014).
https://doi.org/10.1088/1751-8113/47/42/424003

[27] John S. Bell. ``Free variables and local causality''. Epistemological Letters 15, 79 (1977).

[28] Carl H. Brans. ``Bell's theorem does not eliminate fully causal hidden variables''. International Journal of Theoretical Physics 27, 219–226 (1988).
https://doi.org/10.1007/bf00670750

[29] J. P. Jarrett. ``On the physical significance of the locality conditions in the Bell arguments''. Noûs 18, 569 (1984).
https://doi.org/10.2307/2214878

[30] Michael J. W. Hall. ``Complementary contributions of indeterminism and signaling to quantum correlations''. Physical Review A 82, 062117 (2010).
https://doi.org/10.1103/PhysRevA.82.062117

[31] Michael J. W. Hall. ``Relaxed Bell inequalities and Kochen-Specker theorems''. Physical Review A 84, 022102 (2011).
https://doi.org/10.1103/PhysRevA.84.022102

[32] Manik Banik, MD. Rajjak Gazi, Subhadipa Das, Ashutosh Rai, and Samir Kunkri. ``Optimal free will on one side in reproducing the singlet correlation''. Journal of Physics A: Mathematical and Theoretical 45, 205301 (2012).
https://doi.org/10.1088/1751-8113/45/20/205301

[33] Andrew S. Friedman, Alan H. Guth, Michael J. W. Hall, David I. Kaiser, and Jason Gallicchio. ``Relaxed Bell inequalities with arbitrary measurement dependence for each observer''. Physical Review A 99, 012121 (2019).
https://doi.org/10.1103/PhysRevA.99.012121

[34] Moji Ghadimi. ``Parameter dependence and Bell nonlocality''. Physical Review A 104, 032205 (2021).
https://doi.org/10.1103/PhysRevA.104.032205

[35] Gilles Pütz, Denis Rosset, Tomer Jack Barnea, Yeong-Cherng Liang, and Nicolas Gisin. ``Arbitrarily small amount of measurement independence is sufficient to manifest quantum nonlocality''. Physical Review Letters 113, 190402 (2014).
https://doi.org/10.1103/physrevlett.113.190402

[36] Dax Enshan Koh, Michael J. W. Hall, Setiawan, James E. Pope, Chiara Marletto, Alastair Kay, Valerio Scarani, and Artur Ekert. ``Effects of reduced measurement independence on Bell-based randomness expansion''. Physical Review Letters 109, 160404 (2012).
https://doi.org/10.1103/PhysRevLett.109.160404

[37] Michael J. W. Hall and Cyril Branciard. ``Measurement-dependence cost for Bell nonlocality: Causal versus retrocausal models''. Physical Review A 102, 052228 (2020).
https://doi.org/10.1103/PhysRevA.102.052228

[38] Gen Kimura, Yugo Susuki, and Kei Morisue. ``Relaxed Bell inequality as a trade-off relation between measurement dependence and hiddenness''. Physical Review A 108, 022214 (2023).
https://doi.org/10.1103/PhysRevA.108.022214

[39] Gilles Pütz and Nicolas Gisin. ``Measurement dependent locality''. New Journal of Physics 18, 055006 (2016).
https://doi.org/10.1088/1367-2630/18/5/055006

[40] Sandu Popescu and Daniel Rohrlich. ``Quantum nonlocality as an axiom''. Foundations of Physics 24, 379–385 (1994).
https://doi.org/10.1007/BF02058098

[41] Boris. S. Tsirelson. ``Quantum generalizations of Bell's inequality''. Letters in Mathematical Physics 4, 93–100 (1980).
https://doi.org/10.1007/BF00417500

[42] José M. Amigó, Sámuel G. Balogh, and Sergio Hernández. ``A brief review of generalized entropies''. Entropy 20 (2018).
https://doi.org/10.3390/e20110813

[43] Hans Maassen and J. B. M. Uffink. ``Generalized entropic uncertainty relations''. Physical Review Letters 60, 1103–1106 (1988).
https://doi.org/10.1103/PhysRevLett.60.1103

[44] Takayuki Miyadera. ``Uncertainty relations for joint localizability and joint measurability in finite-dimensional systems''. Journal of Mathematical Physics 52, 072105 (2011).
https://doi.org/10.1063/1.3614503

[45] Kiyosi Ito, editor. ``Encyclopedic dictionary of mathematics''. Volume I. MIT Press. (1993). 2nd edition.

Cited by

[1] Gen Kimura, Yugo Susuki, and Kei Morisue, "Relaxed Bell inequality as a trade-off relation between measurement dependence and hiddenness", Physical Review A 108 2, 022214 (2023).

[2] Hayato Arai and Masahito Hayashi, "Derivation of standard quantum theory via state discrimination", New Journal of Physics 26 5, 053046 (2024).

[3] Hayato Arai, Baichu Yu, and Masahito Hayashi, "Detection of Beyond-Quantum Non-locality based on Standard Local Quantum Observables", arXiv:2301.04196, (2023).

[4] Hayato Arai, Baichu Yu, and Masahito Hayashi, "Detecting beyond-quantum nonlocality using standard local quantum observables", Physical Review A 110 1, L010201 (2024).

The above citations are from SAO/NASA ADS (last updated successfully 2026-08-12 04:49:10). 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 2026-08-12 04:49:06).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.