Maximally Entangled States in the Hydrogen Molecule: The Role of Spin and Correlation

ABSTRACT

Many electron calculations on a simplest realistic two electron system i.e.*H*_{2}molecule was applied and as the consequence correlation effects was reflected accurately in the wavefunctions of H_{2}. Zanardi’s entanglement measurement, demonstrated that the maximum of entanglement for the ground state happens when *U* =*J* and this resolved the controversial conclusion of *U* = 0 for maximum entanglement. It was shown that the ground and third excited states are maximally entangled. These maximally entangled states and also the minimally entangled states are correlated to their spin’s property. The wavefunctions of the not magnetic (S = 0) ground and excited states explicitly depend on correlation parameters whereas the first excited states which is magnetic (S^{2} = 2 and S_{z}≠0) is not entangled. The second excited state is not magnetic but its wavefunction does not depend on correlation parameters therefore it is a moderately entangled state. In any case, by switching on a magnetic field an entangled state with S_{z} = 0 can be extracted from a not entangled degenerate magnetic state. We suggest that in a realistic molecular scale system, there is two criteria for finding maximally entangled electronic states, first the system should be in moderately correlated regime and second the system should have a non-magnetic (S_{z} = 0) electronic state.

Many electron calculations on a simplest realistic two electron system i.e.

Cite this paper

nullM. Vesaghi, M. Babamoradi and M. Saani, "Maximally Entangled States in the Hydrogen Molecule: The Role of Spin and Correlation,"*Journal of Modern Physics*, Vol. 2 No. 7, 2011, pp. 664-668. doi: 10.4236/jmp.2011.27079.

nullM. Vesaghi, M. Babamoradi and M. Saani, "Maximally Entangled States in the Hydrogen Molecule: The Role of Spin and Correlation,"

References

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[19] H. Wang, and S. Kias, "Quantum Teleportation in One-Dimensional Quantum Dots System," Chemical Physics Letters, Vol. 421, 2006, pp. 338 (2006).

[20] S.-J. Gu, S.-S. Deng, Y.-Q. Li, and H.-Q. Lin, "Entanglement and Quantum Phase Transition in the Extended Hubbard Model," Physical Review Letters, Vol. 93, No. 8, 2004, pp. 086402-1-4.

[21] M. Heidari Saani, M. A. Vesaghi, K. Esfarjani, T. Ghods Elahi, M. Sayari, H. Hashemi, and N. Gorjizadeh, "Lattice Relaxation in Many-Electron States of the Diamond Vacancy," Physical Review B, Vol. 71, No. 3, 2005, pp. 035202-1-9.

[22] G. Chiappe, E. Louis, E. SanFabian, and J. A. Verges, "Hubbard Hamiltonian for the Hydrogen Molecule," Physical Review B, Vol. 75, No. 19, 2007, pp. 195104-1-6.

[23] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, "Concentrating Partial Entanglement by Local Operations," physical Review A, Vol. 53, No. 4, 1996, pp. 2046-2052.

[24] J. A. Vanwyk, O. D. Tucker, M. E. Newton, J. M. Baker, G. S. Woods, and P. Spear, "Magnetic-Resonance Measurements on the 5A2 Excited State of the Neutral Vacancy in Diamond" Physical Review B, Vol. 52, No. 17, 1995, pp. 12657-12667.

[1] A. Einstein, B. Podolsky, and N. Rosen, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?," Physical Review, Vol. 47, No. 10, 1935, pp. 777-780.

[2] E. Schr?dinger, "Discussion of Probability Relations Between Separated Systems," Proceedings of the Cambridge Philosophical Society, Vol. 31, 1935, pp. 555–563.

[3] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, "Quantum Entanglement," Reviews of Modern Physics, Vol. 81, No. 2, 2009, pp. 865-942.

[4] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, "Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels," Physical Review Letters, Vol. 70, No. 13, 1993, pp. 1895-1899.

[5] D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, "Experimental Quantum Teleportation," Nature, Vol. 390, 1997, pp. 575-579.

[6] C. H. Bennett, and S. J. Wiesner, "Communication via One- and Two-Particle Operators on Einstein-Podolsky-Rosen States," Physical Review Letters, Vol. 69, No. 20, 1992, pp. 2881-2884.

[7] A. K. Ekert, "Quantum Cryptography Based on Bell's Theorem," Physical Review Letters, Vol. 67, No. 6, 1991, pp. 661-663.

[8] C. A. Fuchs, "Nonorthogonal Quantum States Maximize Classical Information Capacity," Physical Review Letters, Vol. 79, No. 6, 1997, pp. 1162-1165.

[9] Z. Huang, and S. Kias, "Entanglement as Measure of Electron–Electron Correlation in Quantum Chemistry Calculations," Chemical Physics Letters, Vol. 413, 2005, pp. 1-5.

[10] T. A. C. Maiolo, F. della Sala, L. Martina, and G. Soliani, "Entanglement of Electrons in Interacting Molecules," Theoretical and Mathematical Physics, Vol. 152, No. 2, 2007, pp. 1146-1159.

[11] Y. S. Li, B. Zeng, X. S. Liu, and G. L. Long, "Entanglement in a Two-Identical-Particle System," Physical Review A, Vol. 64, No. 5, 2001, pp. 054302-1-4.

[12] W. K. Wootters, "Entanglement of Formation of an Arbitrary State of Two Qubits," Physical Review Letters, Vol. 80, No. 10, 1998, pp. 2245-2248.

[13] J. Schliemann, D. Loss, and A. H. MacDonald, "Double-Occupancy Errors, Adiabaticity, and Entanglement of Spin Qubits in Quantum Dots," Physical Review B, Vol. 63, No. 8, 2001, pp. 085311-1-8.

[14] J. Schliemann, J. Ignacio Cirac, M. Kus, M. Lewenstein, and D. Loss, "Quantum Correlations in Two-Fermion Systems," Physical Review A, Vol. 64, No. 2, 2001, pp. 022303-1-9.

[15] J. R. Gittings, and A. J. Fisher, "Describing Mixed Spin-Space Entanglement of Pure States of Indistinguishable Particles Using an Occupation-Number Basis," Physical Review A, Vol. 66, No. 3, 2002, pp. 032305-1-11.

[16] P. Zanardi, "Bipartite Mode Entanglement of Bosonic Condensates on Tunneling Graphs," Physical Review A, Vol. 67, No. 5, 2003, pp. 054301-1-4.

[17] P. Zanardi, "Quantum Entanglement in Fermionic Lattices," Physical Review A, Vol. 65, No. 4, 2002, pp. 042101-1-5.

[18] B. Alvarez-Fernández, and J. A. Blanco, "The Hubbard Model for the Hydrogen Molecule," European Journal of Physics, Vol. 23, No. 1, 2002, pp. 11-16.

[19] H. Wang, and S. Kias, "Quantum Teleportation in One-Dimensional Quantum Dots System," Chemical Physics Letters, Vol. 421, 2006, pp. 338 (2006).

[20] S.-J. Gu, S.-S. Deng, Y.-Q. Li, and H.-Q. Lin, "Entanglement and Quantum Phase Transition in the Extended Hubbard Model," Physical Review Letters, Vol. 93, No. 8, 2004, pp. 086402-1-4.

[21] M. Heidari Saani, M. A. Vesaghi, K. Esfarjani, T. Ghods Elahi, M. Sayari, H. Hashemi, and N. Gorjizadeh, "Lattice Relaxation in Many-Electron States of the Diamond Vacancy," Physical Review B, Vol. 71, No. 3, 2005, pp. 035202-1-9.

[22] G. Chiappe, E. Louis, E. SanFabian, and J. A. Verges, "Hubbard Hamiltonian for the Hydrogen Molecule," Physical Review B, Vol. 75, No. 19, 2007, pp. 195104-1-6.

[23] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, "Concentrating Partial Entanglement by Local Operations," physical Review A, Vol. 53, No. 4, 1996, pp. 2046-2052.

[24] J. A. Vanwyk, O. D. Tucker, M. E. Newton, J. M. Baker, G. S. Woods, and P. Spear, "Magnetic-Resonance Measurements on the 5A2 Excited State of the Neutral Vacancy in Diamond" Physical Review B, Vol. 52, No. 17, 1995, pp. 12657-12667.