Bell correlations at finite temperature

Matteo Fadel1 and Jordi Tura2

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

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.

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

► 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 (beta)

Crossref's cited-by service has no data on citing works. Unfortunately not all publishers provide suitable citation data.

1 thought on “Bell correlations at finite temperature

  1. Pingback: Weekly Papers on Quantum Foundations (47)