Bell correlations at finite temperature

Matteo Fadel; Jordi Tura

doi:10.22331/q-2018-11-19-107

Bell correlations at finite temperature

¹Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
²Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany

Published:	2018-11-19, volume 2, page 107
Eprint:	arXiv:1805.00449v2
Doi:	https://doi.org/10.22331/q-2018-11-19-107
Citation:	Quantum 2, 107 (2018).

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

Abstract

We show that spin systems with infinite-range interactions can violate at thermal equilibrium a multipartite Bell inequality, up to a finite critical temperature $T_c$. Our framework can be applied to a wide class of spin systems and Bell inequalities, to study whether nonlocality occurs naturally in quantum many-body systems close to the ground state. Moreover, we also show that the low-energy spectrum of the Bell operator associated to such systems can be well approximated by the one of a quantum harmonic oscillator, and that spin-squeezed states are optimal in displaying Bell correlations for such Bell inequalities.

Popular summary

Measurements on quantum-mechanical systems can display statistics that escape our intuition, developed by our every-day interaction with a "classical" world. In 1964, John Bell developed a mathematical framework (Bell experiment) that allows to distinguish between statistics that are compatible with our intuitive view of the world (local realism) and those that go beyond, which we only know how to produce using intrinsically quantum effects (Bell correlations). Observing Bell correlations in systems that are composed of few particles is possible by performing conceptually simple, albeit technically challenging, Bell experiments. The case of multipartite systems naturally presents many more challenges, both theoretically and experimentally. Recently, however, important developments in the field allowed for the experimental detection of Bell correlations in many-body systems composed of hundreds and hundreds of thousands of atoms. Apart from these tailored experiments, where Bell correlations were prepared by artificially controlling the interactions among the atoms, we may ask whether Bell correlations can naturally occur in many-body systems. Theoretically, it has been proven that the ground state of some Hamiltonians show Bell correlations, but in practice a system can never be prepared exactly in its ground state, leaving the previous question open. Here, after showing that Bell correlations occur in the ground state of a physically relevant class of spin Hamiltonians, we show that they persist at finite temperature up to a critical value. For reasonable experimental parameters, we foresee the possibility of preparing Bell-correlated many-body spin systems by just cooling their spin degree of freedom to few pico Kelvins. As a result of our analysis, we also show that these low-energy states can be well-approximated by an important class of quantum-mechanical states, called spin-squeezed states.

► BibTeX data

@article{Fadel2018bellcorrelations,
  doi = {10.22331/q-2018-11-19-107},
  url = {https://doi.org/10.22331/q-2018-11-19-107},
  title = {Bell correlations at finite temperature},
  author = {Fadel, Matteo and Tura, Jordi},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {2},
  pages = {107},
  month = nov,
  year = {2018}
}

► References

[1] J. S. Bell, Physics 1, 195 (1964).

[2] A. Fine, Phys. Rev. Lett. 48, 291 (1982).
https://doi.org/10.1103/PhysRevLett.48.291

[3] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Rev. Mod. Phys. 86, 419 (2014).
https://doi.org/10.1103/RevModPhys.86.419

[4] A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Phys. Rev. Lett. 98, 230501 (2007).
https://doi.org/10.1103/PhysRevLett.98.230501

[5] S. Pironio, A. Acín, N. Brunner, N. Gisin, S. Massar, and V. Scarani, New Journal of Physics 11, 045021 (2009).
https://doi.org/10.1088/1367-2630/11/4/045021

[6] R. Colbeck and R. Renner, Nature Physics 8, 450 (2012).
https://doi.org/10.1038/nphys2300

[7] S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Nature 464, 1021 (2010).
https://doi.org/10.1038/nature09008

[8] D. Mayers and A. Yao, Quantum Information & Computation 4, 273 (2004).
http://www.rintonpress.com/xqic4/qic-4-4/273-286.pdf

[9] A. Coladangelo, K. T. Goh, and V. Scarani, Nature Communications 8, 15485 (2017), article.
https://doi.org/10.1038/ncomms15485

[10] A. Salavrakos, R. Augusiak, J. Tura, P. Wittek, A. Acín, and S. Pironio, Phys. Rev. Lett. 119, 040402 (2017).
https://doi.org/10.1103/PhysRevLett.119.040402

[11] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
https://doi.org/10.1103/RevModPhys.81.865

[12] R. F. Werner, Phys. Rev. A 40, 4277 (1989).
https://doi.org/10.1103/PhysRevA.40.4277

[13] R. Augusiak, M. Demianowicz, J. Tura, and A. Acín, Phys. Rev. Lett. 115, 030404 (2015).
https://doi.org/10.1103/PhysRevLett.115.030404

[14] J. Bowles, J. Francfort, M. Fillettaz, F. Hirsch, and N. Brunner, Phys. Rev. Lett. 116, 130401 (2016).
https://doi.org/10.1103/PhysRevLett.116.130401

[15] R. Augusiak, M. Demianowicz, and J. Tura, Phys. Rev. A 98, 012321 (2018).
https://doi.org/10.1103/PhysRevA.98.012321

[16] R. Augusiak, M. Demianowicz, and A. Acín, Journal of Physics A: Mathematical and Theoretical 47, 424002 (2014).
https://doi.org/10.1088/1751-8113/47/42/424002

[17] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008).
https://doi.org/10.1103/RevModPhys.80.517

[18] M. Vojta, Reports on Progress in Physics 66, 2069 (2003).
https://doi.org/10.1088/0034-4885/66/12/R01

[19] D. R. Nelson, Phys. Rev. Lett. 60, 1973 (1988).
https://doi.org/10.1103/PhysRevLett.60.1973

[20] S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006).
https://doi.org/10.1103/PhysRevLett.96.181602

[21] B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, Nature Chemistry 2, 106 (2010), article.
https://doi.org/10.1038/nchem.483

[22] S. Szalay, M. Pfeffer, V. Murg, G. Barcza, F. Verstraete, R. Schneider, and Örs Legeza, International Journal of Quantum Chemistry 115, 1342 (2015).
https://doi.org/10.1002/qua.24898

[23] F. Verstraete, V. Murg, and J. Cirac, Advances in Physics 57, 143 (2008).
https://doi.org/10.1080/14789940801912366

[24] D. Liu, S.-J. Ran, P. Wittek, C. Peng, R. B. García, G. Su, and M. Lewenstein, ``Machine learning by two-dimensional hierarchical tensor networks: A quantum information theoretic perspective on deep architectures,'' (2017), arXiv:1710.04833.
arXiv:arXiv:1710.04833

[25] I. Pitowsky, Quantum Probability – Quantum Logic, Lecture Notes in Physics (Springer-Verlag Berlin Heidelberg, 1989).
https://doi.org/10.1007/BFb0021186

[26] L. Babai, L. Fortnow, and C. Lund, computational complexity 1, 3 (1991).
https://doi.org/10.1007/BF01200056

[27] N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990).
https://doi.org/10.1103/PhysRevLett.65.1838

[28] R. F. Werner and M. M. Wolf, Phys. Rev. A 64, 032112 (2001).
https://doi.org/10.1103/PhysRevA.64.032112

[29] M. Żukowski and i. c. v. Brukner, Phys. Rev. Lett. 88, 210401 (2002).
https://doi.org/10.1103/PhysRevLett.88.210401

[30] J.-D. Bancal, N. Brunner, N. Gisin, and Y.-C. Liang, Phys. Rev. Lett. 106, 020405 (2011).
https://doi.org/10.1103/PhysRevLett.106.020405

[31] J.-L. Chen, D.-L. Deng, H.-Y. Su, C. Wu, and C. H. Oh, Phys. Rev. A 83, 022316 (2011).
https://doi.org/10.1103/PhysRevA.83.022316

[32] O. Gühne, G. Tóth, P. Hyllus, and H. J. Briegel, Phys. Rev. Lett. 95, 120405 (2005).
https://doi.org/10.1103/PhysRevLett.95.120405

[33] G. Tóth, O. Gühne, and H. J. Briegel, Phys. Rev. A 73, 022303 (2006).
https://doi.org/10.1103/PhysRevA.73.022303

[34] E. G. Cavalcanti, C. J. Foster, M. D. Reid, and P. D. Drummond, Phys. Rev. Lett. 99, 210405 (2007).
https://doi.org/10.1103/PhysRevLett.99.210405

[35] Q. Y. He, E. G. Cavalcanti, M. D. Reid, and P. D. Drummond, Phys. Rev. Lett. 103, 180402 (2009).
https://doi.org/10.1103/PhysRevLett.103.180402

[36] Q. Y. He, E. G. Cavalcanti, M. D. Reid, and P. D. Drummond, Phys. Rev. A 81, 062106 (2010).
https://doi.org/10.1103/PhysRevA.81.062106

[37] A. Salles, D. Cavalcanti, A. Acin, D. Perez-Garcia, and M. M. Wolf, Quantum Information and Computation 10, 0703 (2010).
http://www.rintonpress.com/xxqic10/qic-10-78/0703-0719.pdf

[38] J. Tura, R. Augusiak, A. B. Sainz, T. Vértesi, M. Lewenstein, and A. Acín, Science 344, 1256 (2014a).
https://doi.org/10.1126/science.1247715

[39] J. Tura, R. Augusiak, A. Sainz, B. Lücke, C. Klempt, M. Lewenstein, and A. Acín, Annals of Physics 362, 370 (2015).
https://doi.org/10.1016/j.aop.2015.07.021

[40] M. Fadel and J. Tura, Phys. Rev. Lett. 119, 230402 (2017).
https://doi.org/10.1103/PhysRevLett.119.230402

[41] S. Wagner, R. Schmied, M. Fadel, P. Treutlein, N. Sangouard, and J.-D. Bancal, Phys. Rev. Lett. 119, 170403 (2017).
https://doi.org/10.1103/PhysRevLett.119.170403

[42] J. Tura, G. De las Cuevas, R. Augusiak, M. Lewenstein, A. Acín, and J. I. Cirac, Phys. Rev. X 7, 021005 (2017).
https://doi.org/10.1103/PhysRevX.7.021005

[43] J. Tura, A. B. Sainz, T. Vértesi, A. Acín, M. Lewenstein, and R. Augusiak, Journal of Physics A: Mathematical and Theoretical 47, 424024 (2014b).
https://doi.org/10.1088/1751-8113/47/42/424024

[44] Z. Wang, S. Singh, and M. Navascués, Phys. Rev. Lett. 118, 230401 (2017).
https://doi.org/10.1103/PhysRevLett.118.230401

[45] Z. Wang and M. Navascués, Proceedings of the Royal Society of London A 474 (2018).
https://doi.org/10.1098/rspa.2017.0822

[46] F. Baccari, D. Cavalcanti, P. Wittek, and A. Acín, Phys. Rev. X 7, 021042 (2017).
https://doi.org/10.1103/PhysRevX.7.021042

[47] A. Sorensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Nature 409, 63 (2001).
https://doi.org/10.1038/35051038

[48] K. Eckert, O. Romero-Isart, M. Rodriguez, M. Lewenstein, E. S. Polzik, and A. Sanpera, Nat Phys 4, 50 (2008).
https://doi.org/10.1038/nphys776

[49] R. Schmied, J.-D. Bancal, B. Allard, M. Fadel, V. Scarani, P. Treutlein, and N. Sangouard, Science 352, 441 (2016).
https://doi.org/10.1126/science.aad8665

[50] N. J. Engelsen, R. Krishnakumar, O. Hosten, and M. A. Kasevich, Phys. Rev. Lett. 118, 140401 (2017).
https://doi.org/10.1103/PhysRevLett.118.140401

[51] J. Estève, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, Nature 455, 1216 (2008).
https://doi.org/10.1038/nature07332

[52] C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, Nature 464, 1165 (2010).
https://doi.org/10.1038/nature08919

[53] M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, Nature 464, 1170 (2010).
https://doi.org/10.1038/nature08988

[54] I. D. Leroux, M. H. Schleier-Smith, and V. Vuletić, Phys. Rev. Lett. 104, 073602 (2010).
https://doi.org/10.1103/PhysRevLett.104.073602

[55] J. W. Britton, B. C. Sawyer, A. C. Keith, C. C. J. Wang, J. K. Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger, Nature 484, 489 (2012).
https://doi.org/10.1038/nature10981

[56] J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall, A. M. Rey, M. Foss-Feig, and J. J. Bollinger, Science 352, 1297 (2016).
https://doi.org/10.1126/science.aad9958

[57] T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940).
https://doi.org/10.1103/PhysRev.58.1098

[58] N. N. Bogoljubov, Il Nuovo Cimento (1955-1965) 7, 794 (1958).
https://doi.org/10.1007/BF02745585

[59] J. G. Valatin, Il Nuovo Cimento (1955-1965) 7, 843 (1958).
https://doi.org/10.1007/BF02745589

[60] T. Moroder, P. Hyllus, G. Tóth, C. Schwemmer, A. Niggebaum, S. Gaile, O. Gühne, and H. Weinfurter, New Journal of Physics 14, 105001 (2012).
https://doi.org/10.1088/1367-2630/14/10/105001

[61] J. T. i Brugués, Characterizing Entanglement and Quantum Correlations Constrained by Symmetry (Springer International Publishing, 2017).
https://doi.org/10.1007/978-3-319-49571-2

[62] M. Christandl, The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography, Ph.D. thesis, University of Cambridge (2006).
arXiv:quant-ph/0604183

[63] A. Harrow, Applications of coherent classical communication and the Schur transform to quantum information theory, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (2005).
https://dspace.mit.edu/handle/1721.1/34973

[64] K. Audenaert, ``A digest on representation theory of the symmetric group,'' (2006).
https://pdfs.semanticscholar.org/78bd/f436a0df9dd59bad52db9572de9c1aae008f.pdf

[65] B. Toner and F. Verstraete, ``Monogamy of bell correlations and tsirelson's bound,'' (2006), arXiv:quant-ph/0611001.
arXiv:arXiv:quant-ph/0611001

Cited by

[1] Dongkeun Lee, Adel Sohbi, and Wonmin Son, "Detection of a quantum phase transition in a spin-1 chain through multipartite high-order correlations", Physical Review A 106 4, 042432 (2022).

[2] Marcin Płodzień, Jan Chwedeńczuk, and Maciej Lewenstein, "Inherent quantum resources in stationary spin chains", Physical Review A 111 1, 012417 (2025).

[3] Carlo Marconi, Andreu Riera-Campeny, Anna Sanpera, and Albert Aloy, "Robustness of nonlocality in many-body open quantum systems", Physical Review A 105 6, L060201 (2022).

[4] Patrick Emonts, Mengyao Hu, Albert Aloy, and Jordi Tura, "Effects of topological boundary conditions on Bell nonlocality", Physical Review A 110 3, 032201 (2024).

[5] Jiajie Guo, Jordi Tura, Qiongyi He, and Matteo Fadel, "Detecting Bell Correlations in Multipartite Non-Gaussian Spin States", Physical Review Letters 131 7, 070201 (2023).

[6] J. Tura, A. Aloy, F. Baccari, A. Acín, M. Lewenstein, and R. Augusiak, "Optimization of device-independent witnesses of entanglement depth from two-body correlators", Physical Review A 100 3, 032307 (2019).

[7] J Knörzer, X Liu, B F Schiffer, and J Tura, "Distributed quantum information processing: a review of recent progress", Reports on Progress in Physics 89 7, 074401 (2026).

[8] Carlo Marconi, Guillem Müller-Rigat, Jordi Romero-Pallejà, Jordi Tura, and Anna Sanpera, "Symmetric quantum states: a review of recent progress", Reports on Progress in Physics 89 2, 024001 (2026).

[9] Albert Aloy, Guillem Müller-Rigat, Maciej Lewenstein, Jordi Tura, and Matteo Fadel, "Nonlocality, integrability and quantum chaos in the spectrum of Bell operators", npj Quantum Information 12 1, 120 (2026).

[10] Bradley Longstaff and Jonatan Bohr Brask, "Persistent nonlocality in an ultracold-atom environment", Quantum 7, 907 (2023).

[11] Marcin Płodzień and Jan Chwedeńczuk, "Generation and Read-Out of Many-Body Bell Correlations with a Probe Qubit", Physical Review Letters 136 10, 100204 (2026).

[12] F. Baccari, J. Tura, M. Fadel, A. Aloy, J.-D. Bancal, N. Sangouard, M. Lewenstein, A. Acín, and R. Augusiak, "Bell correlation depth in many-body systems", Physical Review A 100 2, 022121 (2019).

[13] Albert Aloy, Guillem Müller-Rigat, Jordi Tura, and Matteo Fadel, "Deriving Three-Outcome Permutationally Invariant Bell Inequalities", Entropy 26 10, 816 (2024).

[14] Albert Aloy, Matteo Fadel, and Jordi Tura, "The quantum marginal problem for symmetric states: applications to variational optimization, nonlocality and self-testing", New Journal of Physics 23 3, 033026 (2021).

[15] Matteo Fadel, Quantum Science and Technology 151 (2021) ISBN:978-3-030-85471-3.

[16] Gurpahul Singh, Kelly Wurtz, and Eduardo Martín-Martínez, "Phase-space analysis of Bell inequalities for mixed Gaussian states", Physical Review A 111 4, 042405 (2025).

[17] A. Niezgoda, J. Chwedeńczuk, L. Pezzé, and A. Smerzi, "Detection of Bell correlations at finite temperature from matter-wave interference fringes", Physical Review A 99 6, 062115 (2019).

[18] Mengyao Hu, Eloïc Vallée, Tim Seynnaeve, Patrick Emonts, Fatemeh Mohammadi, and Jordi Tura, "Characterizing translation-invariant Bell inequalities using tropical algebra and graph polytopes", Physical Review A 113 3, 032421 (2026).

[19] Irénée Frérot, Matteo Fadel, and Maciej Lewenstein, "Probing quantum correlations in many-body systems: a review of scalable methods", Reports on Progress in Physics 86 11, 114001 (2023).

[20] Esteban Marulanda and Andrés Gómez, "Influence of Initial Entangled States on the Temperature-Dependent CHSH Inequality", arXiv:2309.07455, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-19 19:10:50) and SAO/NASA ADS (last updated successfully 2026-08-19 19:10:53). The list may be incomplete as not all publishers provide suitable and complete citation data.

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.

Bell correlations at finite temperature

Abstract

Popular summary

► BibTeX data

► References

Cited by

1 thought on “Bell correlations at finite temperature”