Bell-type inequalities for systems of relativistic vector bosons

Alan J. Barr; Paweł Caban; Jakub Rembieliński

doi:10.22331/q-2023-07-27-1070

Bell-type inequalities for systems of relativistic vector bosons

Alan J. Barr¹, Paweł Caban², and Jakub Rembieliński²

¹Department of Physics, Keble Road, University of Oxford, OX1 3RH and Merton College, Merton Street, Oxford, OX1 4JD
²Department of Theoretical Physics, University of Łódź, Pomorska 149/153, PL-90-236 Łódź, Poland

Published:	2023-07-27, volume 7, page 1070
Eprint:	arXiv:2204.11063v2
Doi:	https://doi.org/10.22331/q-2023-07-27-1070
Citation:	Quantum 7, 1070 (2023).

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Abstract

We perform a detailed analysis of the possible violation of various Bell-type inequalities for systems of vector boson-antiboson pairs. Considering the general case of an overall scalar state of the bipartite system, we identify two distinct classes of such states, and determine the joint probabilities of spin measurement outcomes for each them. We calculate the expectation values of the CHSH, Mermin and CGLMP inequalities and find that while the generalised CHSH inequality is not expected to be violated for any of the scalar states, in the case of the Mermin and CGLMP inequalities the situation is different – these inequalities can be violated in certain scalar states while they cannot be violated in others. Moreover, the degree of violation depends on the relative speed of the two particles.

► BibTeX data

@article{Barr2023belltype,
  doi = {10.22331/q-2023-07-27-1070},
  url = {https://doi.org/10.22331/q-2023-07-27-1070},
  title = {Bell-type inequalities for systems of relativistic vector bosons},
  author = {Barr, Alan J. and Caban, Pawe{\l{}} and Rembieli{\'{n}}ski, Jakub},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1070},
  month = jul,
  year = {2023}
}

► References

[1] A. Einstein, B. Podolsky, and N. Rosen. ``Can quantum-mechanical description of physical reality be considered complete?''. Phys. Rev. 47, 777–780 (1935).
https://doi.org/10.1103/PhysRev.47.777

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

[3] Stuart J. Freedman and John F. Clauser. ``Experimental test of local hidden-variable theories''. Phys. Rev. Lett. 28, 938–941 (1972).
https://doi.org/10.1103/PhysRevLett.28.938

[4] Alain Aspect, Jean Dalibard, and Gérard Roger. ``Experimental test of Bell's inequalities using time-varying analyzers''. Phys. Rev. Lett. 49, 1804–1807 (1982).
https://doi.org/10.1103/PhysRevLett.49.1804

[5] M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland. ``Experimental violation of a Bell's inequality with efficient detection''. Nature 409, 791–794 (2001).
https://doi.org/10.1038/35057215

[6] Markus Ansmann et al. ``Violation of Bell's inequality in Josephson phase qubits''. Nature 461, 504–506 (2009).
https://doi.org/10.1038/nature08363

[7] Wolfgang Pfaff, Tim H. Taminiau, Lucio Robledo, Hannes Bernien, Matthew Markham, Daniel J. Twitchen, and Ronald Hanson. ``Demonstration of entanglement-by-measurement of solid-state qubits''. Nature Physics 9, 29–33 (2013).
https://doi.org/10.1038/nphys2444

[8] B. Hensen et al. ``Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres''. Nature 526, 682–686 (2015).
https://doi.org/10.1038/nature15759

[9] Marissa Giustina et al. ``Significant-loophole-free test of Bell's theorem with entangled photons''. Phys. Rev. Lett. 115, 250401 (2015).
https://doi.org/10.1103/PhysRevLett.115.250401

[10] Lynden K. Shalm et al. ``Strong loophole-free test of local realism''. Phys. Rev. Lett. 115, 250402 (2015).
https://doi.org/10.1103/PhysRevLett.115.250402

[11] Alipasha Vaziri, Gregor Weihs, and Anton Zeilinger. ``Experimental two-photon, three-dimensional entanglement for quantum communication''. Phys. Rev. Lett. 89, 240401 (2002).
https://doi.org/10.1103/PhysRevLett.89.240401

[12] Marek Czachor. ``Einstein-Podolsky-Rosen-Bohm experiment with relativistic massive particles''. Phys. Rev. A 55, 72–77 (1997).
https://doi.org/10.1103/PhysRevA.55.72

[13] Paul M. Alsing and Gerard J. Milburn. ``On entanglement and Lorentz transfotmations''. Quantum Info. Comput. 2, 487 (2002).
https://doi.org/10.26421/QIC2.6-4

[14] Robert M. Gingrich and Christoph Adami. ``Quantum entanglement of moving bodies''. Phys. Rev. Lett. 89, 270402 (2002).
https://doi.org/10.1103/PhysRevLett.89.270402

[15] Asher Peres, Petra F. Scudo, and Daniel R. Terno. ``Quantum entropy and special relativity''. Phys. Rev. Lett. 88, 230402 (2002).
https://doi.org/10.1103/PhysRevLett.88.230402

[16] Doyeol Ahn, Hyuk-jae Lee, Young Hoon Moon, and Sung Woo Hwang. ``Relativistic entanglement and Bell's inequality''. Phys. Rev. A 67, 012103 (2003).
https://doi.org/10.1103/PhysRevA.67.012103

[17] Hui Li and Jiangfeng Du. ``Relativistic invariant quantum entanglement between the spins of moving bodies''. Phys. Rev. A 68, 022108 (2003).
https://doi.org/10.1103/PhysRevA.68.022108

[18] H. Terashima and M. Ueda. ``Relativistic Einstein–Podolsky–Rosen correlation and Bell's inequality''. Int. J. Quant. Inf. 1, 93–114 (2003).
https://doi.org/10.1142/S0219749903000061

[19] Paweł Caban and Jakub Rembieliński. ``Lorentz-covariant reduced spin density matrix and Einstein–Podolsky–Rosen–Bohm correlations''. Phys. Rev. A 72, 012103 (2005).
https://doi.org/10.1103/PhysRevA.72.012103

[20] Paweł Caban and Jakub Rembieliński. ``Einstein-Podolsky-Rosen correlations of Dirac particles: Quantum field theory approach''. Phys. Rev. A 74, 042103 (2006).
https://doi.org/10.1103/PhysRevA.74.042103

[21] Paweł Caban, Jakub Rembieliński, and Marta Włodarczyk. ``Einstein-Podolsky-Rosen correlations of vector bosons''. Phys. Rev. A 77, 012103 (2008).
https://doi.org/10.1103/PhysRevA.77.012103

[22] Nicolai Friis, Reinhold A. Bertlmann, Marcus Huber, and Beatrix C. Hiesmayr. ``Relativistic entanglement of two massive particles''. Phys. Rev. A 81, 042114 (2010).
https://doi.org/10.1103/PhysRevA.81.042114

[23] Paul M Alsing and Ivette Fuentes. ``Observer-dependent entanglement''. Classical and Quantum Gravity 29, 224001 (2012).
https://doi.org/10.1088/0264-9381/29/22/224001

[24] Pablo L. Saldanha and Vlatko Vedral. ``Spin quantum correlations of relativistic particles''. Phys. Rev. A 85, 062101 (2012).
https://doi.org/10.1103/PhysRevA.85.062101

[25] E. R. F. Taillebois and A. T. Avelar. ``Spin-reduced density matrices for relativistic particles''. Phys. Rev. A 88, 060302 (2013).
https://doi.org/10.1103/PhysRevA.88.060302

[26] Paweł Caban, Jakub Rembieliński, Patrycja Rybka, Kordian A. Smoliński, and Piotr Witas. ``Relativistic Einstein-Podolsky-Rosen correlations and localization''. Phys. Rev. A 89, 032107 (2014).
https://doi.org/10.1103/PhysRevA.89.032107

[27] Veiko Palge and Jacob Dunningham. ``Behavior of Werner states under relativistic boosts''. Ann. Phys. 363, 275–304 (2015).
https://doi.org/10.1016/j.aop.2015.09.028

[28] Victor A. S. V. Bittencourt, Alex E. Bernardini, and Massimo Blasone. ``Global Dirac bispinor entanglement under Lorentz boosts''. Phys. Rev. A 97, 032106 (2018).
https://doi.org/10.1103/PhysRevA.97.032106

[29] Lucas F. Streiter, Flaminia Giacomini, and Časlav Brukner. ``Relativistic Bell test within quantum reference frames''. Phys. Rev. Lett. 126, 230403 (2021).
https://doi.org/10.1103/PhysRevLett.126.230403

[30] Matthias Ondra and Beatrix C. Hiesmayr. ``Single particle entanglement in the mid- and ultra-relativistic regime''. J. Phys. A: Math. Theor. 54, 435301 (2021).
https://doi.org/10.1088/1751-8121/ac2548

[31] H. Bacry. ``Localizability and space in quantum physics''. Lecture Notes in Physics Vol. 308. Springer–Verlag. Berlin, Heidelberg (1988).
https://doi.org/10.1007/BFb0019319

[32] Alan J. Barr. ``Testing Bell inequalities in Higgs boson decays''. Phys. Lett. B 825, 136866 (2022).
https://doi.org/10.1016/j.physletb.2021.136866

[33] J. A. Aguilar-Saavedra, A. Bernal, J. A. Casas, and J. M. Moreno. ``Testing entanglement and Bell inequalities in ${H}{\rightarrow}{ZZ}$''. Phys. Rev. D 107, 016012 (2023).
https://doi.org/10.1103/PhysRevD.107.016012

[34] Rachel Ashby-Pickering, Alan J. Barr, and Agnieszka Wierzchucka. ``Quantum state tomography, entanglement detection and Bell violation prospects in weak decays of massive particles''. J. High Energ. Phys. 2023, 20 (2023).
https://doi.org/10.1007/JHEP05(2023)020

[35] J. A. Aguilar-Saavedra. ``Laboratory-frame tests of quantum entanglement in $H{\rightarrow}WW$''. Phys. Rev. D 107, 076016 (2023).
https://doi.org/10.1103/PhysRevD.107.076016

[36] M. Fabbrichesi, R. Floreanini, E. Gabrielli, and L. Marzola. ``Bell inequalities and quantum entanglement in weak gauge bosons production at the LHC and future colliders'' (2023). arXiv:2302.00683.
arXiv:2302.00683

[37] Paweł Caban. ``Helicity correlations of vector bosons''. Phys. Rev. A 77, 062101 (2008).
https://doi.org/10.1103/PhysRevA.77.062101

[38] T. D. Newton and E. P. Wigner. ``Localized states for elementary systems''. Rev. Mod. Phys. 21, 400–406 (1949).
https://doi.org/10.1103/RevModPhys.21.400

[39] N. N. Bogolubov, A. A. Logunov, and I. T. Todorov. ``Introduction to axiomatic quantum field theory''. W. A. Benjamin. Reading, Mass. (1975).

[40] Paweł Caban, Jakub Rembieliński, and Marta Włodarczyk. ``A spin observable for a Dirac particle''. Ann. of Phys. 330, 263–272 (2013).
https://doi.org/10.1016/j.aop.2012.12.001

[41] Paweł Caban, Jakub Rembieliński, and Marta Włodarczyk. ``Strange behavior of the relativistic Einstein-Podolsky-Rosen correlations''. Phys. Rev. A 79, 014102 (2009).
https://doi.org/10.1103/PhysRevA.79.014102

[42] Daniel R. Terno. ``Two roles of relativistic spin operators''. Phys. Rev. A 67, 014102 (2003).
https://doi.org/10.1103/PhysRevA.67.014102

[43] Pablo L Saldanha and Vlatko Vedral. ``Physical interpretation of the Wigner rotations and its implications for relativistic quantum information''. New J. Phys. 14, 023041 (2012).
https://doi.org/10.1088/1367-2630/14/2/023041

[44] Heiko Bauke, Sven Ahrens, Christoph H. Keitel, and Rainer Grobe. ``What is the relativistic spin operator?''. New J. Phys. 16, 043012 (2014).
https://doi.org/10.1088/1367-2630/16/4/043012

[45] Lucas C. Céleri, Vasilis Kiosses, and Daniel R. Terno. ``Spin and localization of relativistic fermions and uncertainty relations''. Phys. Rev. A 94, 062115 (2016).
https://doi.org/10.1103/PhysRevA.94.062115

[46] Liping Zou, Pengming Zhang, and Alexander J. Silenko. ``Position and spin in relativistic quantum mechanics''. Phys. Rev. A 101, 032117 (2020).
https://doi.org/10.1103/PhysRevA.101.032117

[47] E.R.F. Taillebois and A.T. Avelar. ``Relativistic spin operator must be intrinsic''. Phys. Lett. A 392, 127166 (2021).
https://doi.org/10.1016/j.physleta.2021.127166

[48] Heon Lee. ``Relativistic massive particle with spin-1/2: A vector bundle point of view''. J. Math. Phys. 63, 012201 (2022).
https://doi.org/10.1063/5.0064409

[49] Leslie E Ballentine. ``Quantum mechanics: A modern development''. World Scientific. (2014). 2nd edition.
https://doi.org/10.1142/9038

[50] 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 (1969).
https://doi.org/10.1103/PhysRevLett.23.880

[51] N. D. Mermin. ``Quantum mechanics vs local realism near the classical limit: A Bell inequality for spin $s$''. Phys. Rev. D 22, 356–361 (1980).
https://doi.org/10.1103/PhysRevD.22.356

[52] Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar, and Sandu Popescu. ``Bell inequalities for arbitrarily high-dimensional systems''. Phys. Rev. Lett. 88, 040404 (2002).
https://doi.org/10.1103/PhysRevLett.88.040404

[53] A Barut and R Raczka. ``Theory of group representations and applications''. World Scientific. (1986).
https://doi.org/10.1142/0352

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