Investigación

Entanglement and control operations in Heisenberg 3–D interactions of two qubits

F. Delgado

Mathematics and Physics Department, Quantum Information Processing Group, Tecnológico de Monterrey, Campus Estado de México, Atizapán, Estado de México, 52926, México, e–mail: fdelgado@itesm.mx

Recibido el 18 de junio de 2009
Aceptado el 5 de noviembre de 2009

Abstract

Entanglement generated by the Heisenberg model has been studied by several authors in order to understand its relation to the magnetic properties of materials, using mainly particular cases in one or two dimensions for two or more particles. In this work, the evolution of the Heisenberg model is solved for two particles including an inhomogeneous magnetic field in three dimensions, giving a detailed study of the entanglement properties derived from this interaction. Some relations between entanglement and energy or spin are verified, based on known relations for these observables. Finally, some possible quantum control operations are suggested to drive bipartite qubits with an external magnetic field, controlling their evolution into a periodical behavior. These operations become useful to preserve the system properties as well as to transfer information between two parts which can be exploited in engineering applications (e.g. quantum computation or quantum information).

Keywords: Heisenberg model; entanglement; quantum control.

Resumen

El enmaranamiento generado por el modelo de Heisenberg ha sido estudiado por diversos autores con la finalidad de comprender su relación con las propiedades magnéticas de los materiales, usando para ello casos particulares para la interacción entre dos o más partículas en una y dos dimensiones. En este trabajo, la evolución del modelo de Heisenberg es resuelta para tres dimensiones y dos partículas, incluyendo además un campo magnético no homogéneo, dando un estudio detallado de las propiedades del enmarañamiento generado por esta interacción. Algunas relaciones del enmarañamiento con la energía o el espín son revisadas, de acuerdo a propiedades conocidas para estos observables. Finalmente, algunas operaciones de control son sugeridas para qubits bipartitas bajo la acción de campos magnéticos externos induciendo a la evolución hacia un comportamiento periódico. Estas operaciones resultan útiles para conservar las propiedades del sistema o bien para transferir información entre las dos partes que la conforman para aplicaciones útiles ingeniería cuántica (e.g. computo cuántico o información cuántica).

Descriptores: Modelo de Heisenberg; enmarañamiento; control cuántico.

PACS: 03.67.Bg; 03.65.Ud; 03.67.–a

DESCARGAR ARTÍCULO EN FORMATO PDF

Acknowledgements

I gratefully acknowledge the assistance Dr. Sergio Martinez–Casas in some fruitful discussions regarding the use of Ising and Heisenberg models in quantum cellular automata (in which this work was first inspired and for reviewing this manuscript) and of Dr. Bogdan Mielnik in offering comments regarding some basic quantum control operations, a heritage from other areas of quantum control in our past works.

References

1. E. Schrodinger, Proc. Cambridge Phil. Soc. 31 (1935) 555.        [ Links ]

2. E. Schrodinger, Naturwissenschften 23 (1935) 807.        [ Links ]

3. C.H. Benett, D.P. DiVicenzo, J.A. Smolin, and W.K. Wooters, Phys. Rev. A 54 (1996) 3824.        [ Links ]

4. C.H. Bennet and P. DiVicenzo, Nature (London) 404 (2000) 247.        [ Links ]

5. M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press 2000) p. 500.        [ Links ]

6. A.M. Branczyk, P.E.M.F. Mendonca, A. Gilchrist, A.C. Doherty, and S.D. Bartlett, Phys. Rev. A 75 (2007) 012329.        [ Links ]

7. Z. Xi and G. Jin, Int. J. Quant. Info. 5 (2007) 857.        [ Links ]

8. M.A. Nielsen, Ph. D. Thesis (University of New Mexico, 1998); see also LANL e–print: quant–ph/0011036.        [ Links ]

9. M.C. Arnesen, S. Bose, and V. Vedral, Phys. Rev. Lett. 87 (2001) 017901.        [ Links ]

10. X. Wang, Phys. Rev. A 64 (2001) 012313.        [ Links ]

11. G.P. Berman, G. D. Doolen, G.V. Lopez, and V.I. Tsifrinovich, Generalized Quantum Control–Not Gate in Two–Spin Ising System, quant–ph/9802013v1.        [ Links ]

12. X. Wang, Phys. Lett. A 281 (2001) 101.        [ Links ]

13. G.L. Kamta and A.F. Starace, Phys. Rev. Lett. 87 (2001) 017901.        [ Links ]

14. Y. Sun, Y. Chen, and H. Chen, Phys. Rev. A 68 (2003) 044301.        [ Links ]

15. L. Zhou, H.S. Song, Y.Q. Guo, and C. Li, Phys. Rev. A 64 (2001) 042302.        [ Links ]

16. D. Gunlycke, V.M. Kendon, V. Vedral, and S. Bose, Phys. Rev. A 64 (2001) 042302.        [ Links ]

17. D. D'alessandro, Introduction to Quantum Control and Dynamics, (Chapman Hall Applied Mathematics Nonlinear Science 2007) p. 261.        [ Links ]

18. F. Delgado, Quantum control on entangled bipartite qubits, quant–ph/08102110.        [ Links ]

19. E. Ising, Z. Phys. 31 (1925) 253.        [ Links ]

20. S.G. Brush, Rev. Mod. Phys. 39 (1967) 883.        [ Links ]

21. R.J. Baxter, Exactly solved models in statistical mechanics, (Acad. Press 1982) p. 32.        [ Links ]

22. A.F. Terzis and E. Paspalakis, Phys. Lett. A 333 (2004) 438.        [ Links ]

23. P. Stelmachovic and V. Buzek, Phys. Rev. A 70 (2004) 032313.        [ Links ]

24. J. Novotny, M. Stefanak, T. Kiss, and I. Jex, J. Phys. A: Math. Gen. 38 (2005) 9087.        [ Links ]

25. J.M. Cai, Z.W. Zhou, and G.C. Guo, Phys. Lett. A 363 (2007) 392.        [ Links ]

26. B. Theral, Phys. Lett. A 271 (2000) 319.        [ Links ]

27. A. Peres, Quantum Theory, Concepts and Methods, (Kluwer 1993) p. 179.        [ Links ]

28. S. Sachdev, Quantum Phase Transitions, (Cambridge University Press 2000) p. 367.        [ Links ]

29. S. Dusuel and J. Vidal, Phys. Rev. D 71 (2005) 224420.        [ Links ]

30. S.M. Gianpaolo, G. Adesso, and F. Illuminati, Phys. Rev. B 79 (2009) 224434.        [ Links ]

31. R. Rossignoli, N. Canosa, and J.M. Matera, Phys. Rev. A 77 (2008) 052322.        [ Links ]

32. I.L. Kirilyuk and S.V. Prants, Control and chaos of atomic Rabi oscillations in a cavity, (Proceedings of 2nd International Conference in Control of Oscillations and Chaos, vol. 2 2000) p. 369.        [ Links ]

33. Q.T. Meng, G.H. Yang, and K.L. Han, Int. J. Quant. Chem. 95 (2003) 30.        [ Links ]

34. B. Mielnik, J. Math. Phys. 27 (1986) 2290.        [ Links ]

35. D.J. Fernandez C., Int. J. Theor. Phys. 33 (1994) 2037.        [ Links ]

36. F. Delgado and B. Mielnik, J. Phys. A 31 (1997) 309.        [ Links ]

37. F. Delgado and B. Mielnik, Phys. Lett. A 249 (1998) 359.        [ Links ]

38. D. Burgarth, Quantum State transfer with Spin Chains, Ph. D. thesis, (University College London 2007) p. 142.        [ Links ]

39. C. Di Franco, M. Paternostro, and G.M. Palma, Int. J. Quant. Comp. 6 (2008) 659.        [ Links ]