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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Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete ? Phys. Rev. 47, 777 (1935)
Nielsen, M.A., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)
Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996)
Chruściński, D., Jurkowski, J., Kossakowski, A.: Quantum states with strong positive partial transpose. Phys. Rev. A 77, 022113 (2008)
Lilong, Q.: Separability of multipartite quantum states with strong positive partial transpose. Phys. Rev. A 98, 012307 (2018)
Wootters, W.K.: Entanglement of formation and concurrence. Quant. Inf. Comput 1, 27 (2001)
Nielsen, M.A.: Conditions for a class of entanglement transformations. Phys. Rev. Lett. 83, 436 (1999)
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)
Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046 (1996)
Popescu, S., Rohrlich, D.: Thermodynamics and the measure of entanglement. Phys. Rev. A 56, R3319 (1997)
Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1982)
Korepin, V.E., Bogoliubov, N.M., Izergin, A.G.: Quantum In-verse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge (1993)
Messiah, A.: Quantum Mechanics, vol. I. Wiley, New York (1968)
Arik, M., Coon, D.D.: Hilbert spaces of analytic functions and generalized coherent states. J. Math. Phys 17, 524 (1976)
Bonatsos, D., Daskaloyannis, C.: Quantum groups and their applications in nuclear physics. Prog. Part. Nucl. Phys. 43, 537 (1999)
Bonatsos, D., Raychev, P.P., Faessler, A.: Quantum algebraic description of vibrational molecular spectra. Chem. Phys. Lett. 178, 221 (1991)
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)
Wybourne, B.G.: Classical Groups for Physicists. Wiley, New York (1974)
Gilmore, R.: Lie Groups, Lie Algebras and Some of Their Applications. (Dover Books on Mathematics), (1974)
Man’ko, V.I., Marmo, G., Sudarshan, E.C.G., Zaccaria, F.: f-oscillators and nonlinear coherent states. Phys. Scr 55, 528 (1997)
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)
Fernández C, David J., Hussin, Véronique: Higher-order SUSY, linearized nonlinear Heisenberg algebras and coherent statess. J. Phys. A: Math. Gen 32, 3603 (1999)
Lohe, M.A., Thilagam, A.: Weyl-ordered polynomials in fractional-dimensional quantum mechanics. J. Phys. A: Math. Gen 38, 461 (2005)
Andrianov, A.A., Ioffe, M.V.: Nonlinear supersymmetric quantum mechanics: concepts and realizations. J. Phys. A: Math. Gen 45, 503001 (2012)
Aoyama, H., Sato, M., Tanaka, T.: General forms of a \(N\)-fold supersymmetric family. Phys. Lett. B 503, 423 (2001)
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)
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)
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)
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)
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)
Shcrödinger, E.: Der stetige Übergang von der Mikro- zur Makromechanik. Naturwissenschaften 14, 664 (1926)
Gazeau, J.P.: Coherent States in Quantum Physics. Wiley, Dewey (2009)
Zhang, W., Feng, D., Gilmore, R.: Coherent states: Theory and some applications. Rev. Mod. Phys 62, 867 (1990)
Glauber, R.J.: Photon Correlations. Phys. Rev. Lett 10, 84 (1963)
Klauder, J.R.: Continuous-representation theory. I. Postulates of continuous-representation theory. J. Math. Phys 4, 1055 (1963)
Klauder, J.R.: Continuous-representation theory. II. generalized relation between quantum and classical dynamics. J. Math. Phys 4, 1058 (1963)
Barut, A.O., Girardello, L.: New “coherent”states associated with non-compact groups. Commun. Math. Phys 21, 41 (1971)
Perelomov, A.M.: Coherent states for arbitrary Lie group. Comm. Math. Phys 26, 222 (1972)
Perelomov, A.: Generalized Coherent States and Their Applications, vol. 31. Springer, New York (1986)
Solomon, A.I.: Group theory of superfluidity. J. Math. Phys 12, 390 (1971)
Hassouni, Y., Curado, E.M.F., Rego-Monteiro, M.A.: Construction of coherent states for physical algebraic systems. Phys. Rev. A 71, 022104 (2005)
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)
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)
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)
Berrada, K.: Bipartite entanglement within the framework of power-law potential systems. J. Russ. Laser Res. 36, 35 (2015)
Hussin, V., Marquette, I.: Generalized Heisenberg algebras, SUSYQM and degeneracies: infinite well and Morse potential. SIGMA 7, 024 (2011)
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)
Curado, E.M.F., Rego-Monteiro, M.A.: Hidden symmetries in generalized su(2) algebras. Physica A 295, 268 (2001)
Curado, E.M.F., Rego-Monteiro, M.A.: Non-linear generalization of the sl(2) algebra. Phys. Lett. A 300, 205 (2002)
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)
Belfakir, Abdessamad, Belhaj, Adil, Hassouni, Yassine: Robustness of deformed catlike states under dissipative decoherence. Phys. Rev. D 102, 065003 (2020)
Fu, H., Wang, X., Solomon, A.I.: Maximal entanglement of nonorthogonal states: classification. Phys. Lett. A 291, 273 (2001)
Wang, X.: Bipartite entangled nonorthogonal states. J. Phys. A: Math. Gen 35, 165 (2002)
Hillery, M., Zubairy, M.S.: Entanglement conditions for two-mode states. Phys. Rev. Lett 96, 050503 (2006)
Berrada, K., Chafik, A., Eleuch, H., Hassouni, Y.: Concurrence in the framwork of coherentstates. Quant. Inf. Process 9, 13 (2010)
Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett 80, 2245 (1998)
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)
Sanders, B.C., Rice, D.A.: Nonclassical fields and the nonlinear interferometer. Phys. Rev. Lett 61, 013805 (1999)
Markham, D., Vedral, V.: Classicality of spin-coherent states via entanglement and distinguishability. Phys. Rev. A 67, 042113 (2003)
Gerry, Christopher C., Benmoussa, Adil: Beam splitting and entanglement: generalized coherent states, group contraction, and the classical limit. Phys. Rev. A 71, 062319 (2005)
Berrada, K.: Quantum metrology with SU(1,1) coherent states in the presence of nonlinear phase shifts. Phys. Rev. A 88, 013817 (2013)
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This work is partially supported by the ICTP through AF-14.
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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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DOI: https://doi.org/10.1007/s11128-020-02941-w