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Bipartite entanglement of generalized Barut–Girardello nonlinear coherent states

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Abstract

We investigate the entanglement of bipartite nonlinear quantum systems in the context of generalized Heisenberg algebra and generalized su(1, 1) algebra. Particularly, we examine the entanglement properties of states prepared by Barut–Girardello nonlinear coherent states of the square-well potential. Furthermore, we show that the entanglement amount of these nonlinear coherent states depends on the corresponding algebraic structure and on the characteristic function of the associated algebras. Moreover, a large class of entangled and maximally entangled states can be recovered and constructed.

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References

  1. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete ? Phys. Rev. 47, 777 (1935)

    MATH  ADS  Google Scholar 

  2. Nielsen, M.A., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  3. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)

    MathSciNet  MATH  ADS  Google Scholar 

  4. Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996)

    MathSciNet  MATH  ADS  Google Scholar 

  5. Chruściński, D., Jurkowski, J., Kossakowski, A.: Quantum states with strong positive partial transpose. Phys. Rev. A 77, 022113 (2008)

    ADS  Google Scholar 

  6. Lilong, Q.: Separability of multipartite quantum states with strong positive partial transpose. Phys. Rev. A 98, 012307 (2018)

    Google Scholar 

  7. Wootters, W.K.: Entanglement of formation and concurrence. Quant. Inf. Comput 1, 27 (2001)

    MathSciNet  MATH  Google Scholar 

  8. Nielsen, M.A.: Conditions for a class of entanglement transformations. Phys. Rev. Lett. 83, 436 (1999)

    ADS  Google Scholar 

  9. Berrada, K., El Baz, M., Saif, F., Hassouni, Y., Mnia, S.: Entanglement generation from deformed spin coherent states using a beam splitter. J. Phys. A: Math. Theor 42, 285306 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046 (1996)

    ADS  Google Scholar 

  11. Popescu, S., Rohrlich, D.: Thermodynamics and the measure of entanglement. Phys. Rev. A 56, R3319 (1997)

    MathSciNet  ADS  Google Scholar 

  12. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1982)

    MATH  Google Scholar 

  13. Korepin, V.E., Bogoliubov, N.M., Izergin, A.G.: Quantum In-verse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  14. Messiah, A.: Quantum Mechanics, vol. I. Wiley, New York (1968)

    MATH  Google Scholar 

  15. Arik, M., Coon, D.D.: Hilbert spaces of analytic functions and generalized coherent states. J. Math. Phys 17, 524 (1976)

    MathSciNet  MATH  ADS  Google Scholar 

  16. Bonatsos, D., Daskaloyannis, C.: Quantum groups and their applications in nuclear physics. Prog. Part. Nucl. Phys. 43, 537 (1999)

    ADS  Google Scholar 

  17. Bonatsos, D., Raychev, P.P., Faessler, A.: Quantum algebraic description of vibrational molecular spectra. Chem. Phys. Lett. 178, 221 (1991)

    ADS  Google Scholar 

  18. Monteiro, M.R., Rodrigues, L.M.C.S., Wulck, S.: Quantum Algebraic Nature of the Phonon Spectrum in \(^4He\). Phys. Rev.Lett. 76, 1098 (1996)

    Google Scholar 

  19. Wybourne, B.G.: Classical Groups for Physicists. Wiley, New York (1974)

    MATH  Google Scholar 

  20. Gilmore, R.: Lie Groups, Lie Algebras and Some of Their Applications. (Dover Books on Mathematics), (1974)

  21. Man’ko, V.I., Marmo, G., Sudarshan, E.C.G., Zaccaria, F.: f-oscillators and nonlinear coherent states. Phys. Scr 55, 528 (1997)

    ADS  Google Scholar 

  22. Zelaya, K., Rosas-Ortiz, O., Blanco-Garcia, Z., Cruz y Cruz, S.: Completeness and nonclassicality of coherent states for generalized oscillator algebras. Adv. Math. Phys 2017, 7168592 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Fernández C, David J., Hussin, Véronique: Higher-order SUSY, linearized nonlinear Heisenberg algebras and coherent statess. J. Phys. A: Math. Gen 32, 3603 (1999)

    MathSciNet  MATH  ADS  Google Scholar 

  24. Lohe, M.A., Thilagam, A.: Weyl-ordered polynomials in fractional-dimensional quantum mechanics. J. Phys. A: Math. Gen 38, 461 (2005)

    MathSciNet  MATH  ADS  Google Scholar 

  25. Andrianov, A.A., Ioffe, M.V.: Nonlinear supersymmetric quantum mechanics: concepts and realizations. J. Phys. A: Math. Gen 45, 503001 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Aoyama, H., Sato, M., Tanaka, T.: General forms of a \(N\)-fold supersymmetric family. Phys. Lett. B 503, 423 (2001)

    MathSciNet  MATH  Google Scholar 

  27. Rosas-Ortiz, O., Zelaya, K.: Bi-orthogonal approach to non-Hermitian Hamiltonians with the oscillator spectrum: generalized coherent states for nonlinear algebras. Ann. Phys 388, 26 (2018)

    MathSciNet  MATH  ADS  Google Scholar 

  28. Bagarello, F., Curado, E.M.F., Gazeau, J.P.: Generalized Heisenberg algebra and (non linear) pseudo-bosons. J. Phys. A: Math. Theor 51, 155201 (2018)

    MathSciNet  MATH  ADS  Google Scholar 

  29. Curado, E.M.F., Rego-Monteiro, M.A.: Thermodynamic properties of a solid exhibiting the energy spectrum given by the logistic map. Phys. Rev. E 61, 6255 (2000)

    ADS  Google Scholar 

  30. Curado, E.M.F., Rego-Monteiro, M.A.: Multi-parametric deformed Heisenberg algebras: a route to complexity. J. Phys. A: Math. Gen 34, 3253 (2001)

    MathSciNet  MATH  ADS  Google Scholar 

  31. Curado, E.M.F., Hassouni, Y., Rego-Monteiro, M.A., Rodrigues, Ligia M.C.S.: Generalized Heisenberg algebra and algebraic method: The example of an infinite square-well potential. Phys. Lett. A 372, 3350 (2008)

    MathSciNet  MATH  ADS  Google Scholar 

  32. Shcrödinger, E.: Der stetige Übergang von der Mikro- zur Makromechanik. Naturwissenschaften 14, 664 (1926)

    MATH  ADS  Google Scholar 

  33. Gazeau, J.P.: Coherent States in Quantum Physics. Wiley, Dewey (2009)

    Google Scholar 

  34. Zhang, W., Feng, D., Gilmore, R.: Coherent states: Theory and some applications. Rev. Mod. Phys 62, 867 (1990)

    MathSciNet  ADS  Google Scholar 

  35. Glauber, R.J.: Photon Correlations. Phys. Rev. Lett 10, 84 (1963)

    MathSciNet  ADS  Google Scholar 

  36. Klauder, J.R.: Continuous-representation theory. I. Postulates of continuous-representation theory. J. Math. Phys 4, 1055 (1963)

    MathSciNet  MATH  ADS  Google Scholar 

  37. Klauder, J.R.: Continuous-representation theory. II. generalized relation between quantum and classical dynamics. J. Math. Phys 4, 1058 (1963)

    MathSciNet  MATH  ADS  Google Scholar 

  38. Barut, A.O., Girardello, L.: New “coherent”states associated with non-compact groups. Commun. Math. Phys 21, 41 (1971)

    MathSciNet  MATH  ADS  Google Scholar 

  39. Perelomov, A.M.: Coherent states for arbitrary Lie group. Comm. Math. Phys 26, 222 (1972)

    MathSciNet  MATH  ADS  Google Scholar 

  40. Perelomov, A.: Generalized Coherent States and Their Applications, vol. 31. Springer, New York (1986)

    MATH  Google Scholar 

  41. Solomon, A.I.: Group theory of superfluidity. J. Math. Phys 12, 390 (1971)

    MathSciNet  ADS  Google Scholar 

  42. Hassouni, Y., Curado, E.M.F., Rego-Monteiro, M.A.: Construction of coherent states for physical algebraic systems. Phys. Rev. A 71, 022104 (2005)

    MathSciNet  MATH  ADS  Google Scholar 

  43. Rego-Monteiro, M.A., Curado, E.M.F., Rodrigues, Ligia M.C.S.: Time evolution of linear and generalized Heisenberg algebra nonlinear Pöschl-Teller coherent states. Phys. Rev. A 96, 052122 (2017)

    MathSciNet  ADS  Google Scholar 

  44. Curado, E.M.F., Rego-Monteiro, M.A., Rodrigues, Ligia M.C.S., Hassouni, Y.: Coherent states for a degenerate system: the hydrogen atom. Phys A: Stat. Mech. Appl. 371, 16 (2006)

    Google Scholar 

  45. Berrada, K., El Baz, M., Hichem Eleuch, H., Hassouni, Y.: Bipartite entanglement of nonlinear quantum systems in the context of the q-Heisenberg Weyl algebra. Quantum Inf. Process. 11, 351 (2012)

    MathSciNet  MATH  Google Scholar 

  46. Berrada, K.: Bipartite entanglement within the framework of power-law potential systems. J. Russ. Laser Res. 36, 35 (2015)

    Google Scholar 

  47. Hussin, V., Marquette, I.: Generalized Heisenberg algebras, SUSYQM and degeneracies: infinite well and Morse potential. SIGMA 7, 024 (2011)

    MathSciNet  MATH  Google Scholar 

  48. Delisle-Doray, L., Hussin, V., Kuru, Ş., Negro, J.: Classical ladder functions for Rosen–Morse and curved Kepler–Coulomb systems. Ann. Phys 405, 69 (2019)

    MathSciNet  MATH  ADS  Google Scholar 

  49. Curado, E.M.F., Rego-Monteiro, M.A.: Hidden symmetries in generalized su(2) algebras. Physica A 295, 268 (2001)

    MathSciNet  MATH  ADS  Google Scholar 

  50. Curado, E.M.F., Rego-Monteiro, M.A.: Non-linear generalization of the sl(2) algebra. Phys. Lett. A 300, 205 (2002)

    MathSciNet  MATH  ADS  Google Scholar 

  51. Belfakir, Abdessamad, Hassouni, Yassine: Generalized \(su(1,1)\) algebra and the construction of nonlinear coherent states for Pöschl-Teller potential. Phys. Lett. A 384, 126603 (2020)

    MathSciNet  MATH  Google Scholar 

  52. Belfakir, Abdessamad, Belhaj, Adil, Hassouni, Yassine: Robustness of deformed catlike states under dissipative decoherence. Phys. Rev. D 102, 065003 (2020)

    MathSciNet  ADS  Google Scholar 

  53. Fu, H., Wang, X., Solomon, A.I.: Maximal entanglement of nonorthogonal states: classification. Phys. Lett. A 291, 273 (2001)

    MathSciNet  MATH  Google Scholar 

  54. Wang, X.: Bipartite entangled nonorthogonal states. J. Phys. A: Math. Gen 35, 165 (2002)

    MATH  ADS  Google Scholar 

  55. Hillery, M., Zubairy, M.S.: Entanglement conditions for two-mode states. Phys. Rev. Lett 96, 050503 (2006)

    MathSciNet  ADS  Google Scholar 

  56. Berrada, K., Chafik, A., Eleuch, H., Hassouni, Y.: Concurrence in the framwork of coherentstates. Quant. Inf. Process 9, 13 (2010)

    MATH  Google Scholar 

  57. Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett 80, 2245 (1998)

    MATH  ADS  Google Scholar 

  58. Berrada, K., Benmoussa, A., Hassouni, Y.: Entanglement generation with deformed Barut-Girardello coherent states as input states in an unitary beam splitter. Quant. Inf. Process 10, 575 (2011)

    MathSciNet  MATH  Google Scholar 

  59. Sanders, B.C., Rice, D.A.: Nonclassical fields and the nonlinear interferometer. Phys. Rev. Lett 61, 013805 (1999)

    Google Scholar 

  60. Markham, D., Vedral, V.: Classicality of spin-coherent states via entanglement and distinguishability. Phys. Rev. A 67, 042113 (2003)

    ADS  Google Scholar 

  61. Gerry, Christopher C., Benmoussa, Adil: Beam splitting and entanglement: generalized coherent states, group contraction, and the classical limit. Phys. Rev. A 71, 062319 (2005)

    ADS  Google Scholar 

  62. Berrada, K.: Quantum metrology with SU(1,1) coherent states in the presence of nonlinear phase shifts. Phys. Rev. A 88, 013817 (2013)

    ADS  Google Scholar 

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Acknowledgements

This work is partially supported by the ICTP through AF-14.

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  1. Equipe des Sciences de la Matière et du Rayonnement (ESMaR), Faculté des Sciences, Université Mohammed V de Rabat, Av. Ibn Battouta, Agdal, B.P. 1014, Rabat, Morocco

    Abdessamad Belfakir & Yassine Hassouni

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  1. Abdessamad Belfakir
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  2. Yassine Hassouni
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Correspondence to Abdessamad Belfakir.

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Belfakir, A., Hassouni, Y. Bipartite entanglement of generalized Barut–Girardello nonlinear coherent states. Quantum Inf Process 20, 8 (2021). https://doi.org/10.1007/s11128-020-02941-w

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