Skip to main content
SpringerLink
Log in

Entanglement of Multipartite Fermionic Coherent States for Pseudo-Hermitian Hamiltonians

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the entanglement of multiqubit fermionic pseudo-Hermitian coherent states (FPHCSs) described by anticommutative Grassmann numbers. We introduce pseudo-Hermitian versions of well-known maximally entangled pure states, such as Bell, GHZ, Werner, and biseparable states, by integrating over the tensor products of FPHCSs with a suitable choice of Grassmannian weight functions. As an illustration, we apply the proposed method to the tensor product of two- and three-qubit pseudo-Hermitian systems. For a quantitative characteristic of entanglement of such states, we use a measure of entanglement determined by the corresponding concurrence function and average entropy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, Cambridge (2000).

    MATH  Google Scholar 

  2. O. Gühne and G. Tóth, “Entanglement detection,” Phys. Rep., 474, 1–75 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  3. S. J. van Enk, N. Lütkenhaus, and H. J. Kimble, “Experimental procedures for entanglement verification,” Phys. Rev. A, 75, 052318 (2007).

    Article  ADS  Google Scholar 

  4. P. Horodecki, “Measuring quantum entanglement without prior state reconstruction,” Phys. Rev. Lett., 90, 167901 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. P. Horodecki and A. Ekert, “Method for direct detection of quantum entanglement,” Phys. Rev. Lett., 89, 127902 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich, F. Mintert, and A. Buchleitner, “Experimental determination of entanglement with a single measurement,” Nature, 440, 1022–1024 (2006).

    Article  ADS  Google Scholar 

  7. C. Schmid, N. Kiesel, W. Wieczorek, and H. Weinfurter, “Experimental direct observation of mixed state entanglement,” Phys. Rev. Lett., 101, 260505 (2008).

    Article  ADS  Google Scholar 

  8. G. Vidal, “Entanglement monotones,” J. Modern Opt., 47, 355–376 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  9. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge Univ. Press, Cambridge (1987).

    MATH  Google Scholar 

  10. S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett., 82, 1784–1787 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. S. J. van Enk, “Decoherence of multidimensional entangled coherent states,” Phys. Rev. A, 72, 022308 (2005).

    Article  ADS  Google Scholar 

  12. S. J. van Enk and O. Hirota, “Entangled coherent states: Teleportation and decoherence,” Phys. Rev. A, 64, 022313 (2001).

    Article  ADS  Google Scholar 

  13. H. Fu, X. Wang, and A. I. Solomon, “Maximal entanglement of nonorthogonal states: Classification,” Phys. Lett. A, 291, 73–76 (2001).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. X. Wang and B. C. Sanders, “Multipartite entangled coherent states,” Phys. Rev. A, 65, 012303 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  15. X. Wang, “Bipartite entangled non-orthogonal states,” J. Phys. A: Math. Gen., 35, 165–174 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. X. Wang, B. C. Sanders, and S. H. Pan, “Entangled coherent states for systems with SU(2) and SU(1, 1) symmetries,” J. Phys. A: Math. Gen., 33, 7451–7467 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. X. Wang, “Entanglement in the quantum Heisenberg XY model,” Phys. Rev. A, 64, 012313 (2001).

    Article  ADS  Google Scholar 

  18. L. Borsten, D. Dahanayake, M. J. Duff, and W. Rubens, “Superqubits,” Phys. Rev. D, 81, 105023 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  19. F. C. Khanna, J. M. C. Malbouisson, A. E. Santana, and E. S. Santos, “Maximum entanglement in squeezed boson and fermion states,” Phys. Rev. A, 76, 022109 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  20. G. Najarbashi and Y. Maleki, “Entanglement of Grassmannian coherent states for multi-partite n-level systems,” SIGMA, 7, 011 (2011); arXiv:1008.4836v2 [math-ph] (2010).

    MathSciNet  MATH  Google Scholar 

  21. G. Najarbashi and Y. Maleki, “Maximal entanglement of two-qubit states constructed by linearly independent coherent states,” Internat. J. Theor. Phys., 50, 2601–2608 (2011).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Y. Maleki, “Para-Grassmannian coherent and squeezed states for pseudo-Hermitian q-oscillator and their entanglement,” SIGMA, 7, 084 (2011); arXiv:1108.5005v1 [math-ph] (2011).

    MathSciNet  MATH  Google Scholar 

  23. S. Majid, “Random walk and the heat equation on superspace and anyspace,” J. Math. Phys., 35, 3753–3760 (1994).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. D. C. Cabra, E. F. Moreno, and A. Tanasă, “Para-Grassmann variables and coherent states,” SIGMA, 2, 087 (2006).

    MathSciNet  MATH  Google Scholar 

  25. O. Cherbal, M. Drir, M. Maamache, and D. A. Trifonov, “Fermionic coherent states for pseudo-Hermitian two-level systems,” J. Phys. A: Math. Theor., 40, 1835–1844 (2007).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. G. Najarbashi, M. A. Fasihi, and H. Fakhri, “Generalized Grassmannian coherent states for pseudo-Hermitian n-level systems,” J. Phys. A: Math. Theor., 43, 325301 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  27. F. G. Scholtz, H. B. Geyer, and F. J. W. Hahne, “Quasi-Hermitian operators in quantum mechanics and the variational principle,” Ann. Phys., 213, 74–101 (1992).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett., 80, 5243–5246 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. C. M. Bender, S. Boettcher, and P. N. Meisenger, “PT-symmetric quantum mechanics,” J. Math. Phys., 40, 2201–2229 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. C. M. Bender and G. V. Dunne, “Large-order perturbation theory for a non-Hermitian PT-symmetric Hamiltonian,” J. Math. Phys., 40, 4616–4621 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. F. Cannata, G. Junker, and J. Trost, “Schrödinger operators with complex potential but real spectrum,” Phys. Lett. A, 246, 219–226 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, “Supersymmetry without Hermiticity within PT symmetric quantum mechanics,” Phys. Lett. B, 483, 284–289 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys., 43, 205–214 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. A. Mostafazadeh, “Pseudo-Hermiticity versus PT-symmetry: II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum,” J. Math. Phys., 43, 2814–2816 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. A. Mostafazadeh, “Pseudo-Hermitian representation of quantum mechanics,” Intenat. J. Geom. Meth. Modern Phys., 7, 1191–1306 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  36. N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett., 77, 570–573 (1996).

    Article  ADS  Google Scholar 

  37. N. Hatano and D. R. Nelson, “Vortex pinning and non-Hermitian quantum mechanics,” Phys. Rev. B, 56, 8651–8673 (1997).

    Article  ADS  Google Scholar 

  38. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wooters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett., 70, 1895–1899 (1993).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. D. Bouwmeester, A. K. Ekert, and A. Zeilinger, eds., The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation, Springer, 2001.

    MATH  Google Scholar 

  40. A. K. Pati, “Minimum classical bit for remote preparation and measurement of a qubit,” Phys. Rev. A, 63, 014302 (2001).

    Article  ADS  Google Scholar 

  41. A. V. Sergienko, ed., Quantum Communications and Cryptography, CRC Press Taylor and Francis Group, Boca Raton, Fla. (2006).

    MATH  Google Scholar 

  42. P. G. O. Anicich and H. Grinberg, “Grassmann coherent states for spin systems,” J. Molec. Struct., 621, 9–18 (2003).

    Article  Google Scholar 

  43. S. Abe, “Adiabatic holonomy and evolution of fermionic coherent state,” Phys. Rev. D, 39, 2327–2331 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  44. J. Ohnuki and T. Kashiwa, “Coherent states of Fermi operators and the path integral,” Prog. Theor. Phys., 60, 548–564 (1978).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. F. A. Berezin, The Method of Second Quantization [in Russian], Fizmatlit, Moscow (1965); English transl., Acad. Press, New York (1966).

    MATH  Google Scholar 

  46. K. E. Cahill and R. J. Glauber, “Density operators for fermions,” Phys. Rev. A, 59, 1538–1555 (1999).

    Article  ADS  Google Scholar 

  47. A. Acin, D. Bruβ, M. Lewenstein, and A. Sanpera, “Classification of mixed three-qubit states,” Phys. Rev. Lett., 87, 040401 (2001).

    Article  ADS  MathSciNet  Google Scholar 

  48. A. K. Pati, “Entanglement in non-Hermitian quantum theory,” Pramana, 73, 485–498 (2010).

    Article  ADS  Google Scholar 

  49. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett., 80, 2245–2248 (1998).

    Article  ADS  MATH  Google Scholar 

  50. S. Hill and W. K. Wootters, “Entanglement of a pair of quantum bits,” Phys. Rev. Lett., 78, 5022–5025 (1997).

    Article  ADS  Google Scholar 

  51. W. E. Lamb, R. R. Schlicher, and M. O. Scully, “Matter-field interaction in atomic physics and quantum optics,” Phys. Rev. A, 36, 2763–2772 (1987).

    Article  ADS  Google Scholar 

  52. J. C. Garrison and E. M. Wright, “Complex geometrical phases for dissipative systems,” Phys. Lett. A, 128, 177–181 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  53. A. K. Rajagopal and R.W. Rendell, “Nonextensive statistical mechanics: Implications to quantum information,” Europhys. News, 36, 221–224 (2005).

    Article  ADS  Google Scholar 

  54. M. B. Plenio and V. Vedral, “Teleportation, entanglement and thermodynamics in the quantum world,” Contemp. Phys., 39, 431–446 (2001).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Faculty of Sciences, Sahand University of Technology, Tabriz, Iran

    S. Mirzaei

  2. Department of Physics, University of Mohaghegh Ardabili, Ardabil, Iran

    G. Najarbashi & F. Mirmasoudi

  3. Department of Physics, Azarbayjan Shahid Madani University, Tabriz, Iran

    M. A. Fasihi

Authors
  1. S. Mirzaei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Najarbashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. A. Fasihi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. F. Mirmasoudi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Mirzaei.

Additional information

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 196, No. 1, pp. 99–116, July, 2018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mirzaei, S., Najarbashi, G., Fasihi, M.A. et al. Entanglement of Multipartite Fermionic Coherent States for Pseudo-Hermitian Hamiltonians. Theor Math Phys 196, 1028–1042 (2018). https://doi.org/10.1134/S0040577918070097

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577918070097

Keywords

Search

Navigation