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Multi-mode entangled states represented as Grassmannian polynomials

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Abstract

We introduce generalized Grassmannian representatives of multi-mode state vectors. By implementing the fundamental properties of Grassmann coherent states, we map the Hilbert space of the finite-dimensional multi-mode states to the space of some Grassmannian polynomial functions. These Grassmannian polynomials form a well-defined space in the framework of Grassmann variables; namely Grassmannian representative space. Therefore, a quantum state can be uniquely defined and determined by an element of Grassmannian representative space. Furthermore, the Grassmannian representatives of some maximally entangled states are considered, and it is shown that there is a tight connection between the entanglement of the states and their Grassmannian representatives.

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References

  1. Filippov, A.T., Isaev, A.P., Kurdikov, A.B.: Para-Grassmann differential calculus. Theor. Math. Phys. 94, 150–165 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Isaev, A.P.: Para-Grassmann integral, discrete systems and quantum groups. Int. J. Mod. Phys. A 12, 201–206 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Mansour, T., Schork, M.: On linear differential equations with variable coefficients involving a para-Grassmann variable. J. Math. Phys. 51, 043512 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Mansour, T., Schork, M.: On linear differential equations involving a paragrassmann variable. SIGMA 5, 073 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Cugliandolo, L.F., Lozano, G.S., Moreno, E.F., Schaposnik, F.A.: A note on Gaussian integrals over para-Grassmann variables. Int. J. Mod. Phys. A 19, 1705–1714 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Ilinski, K.N., Kalinin, G.V., Stepanenko, A.S.: q-Functional Wick’s theorems for particles with exotic statistics. J. Phys. A Math. Gen. 30, 5299–5310 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. El Baz M., Fresneda, R., Gazeau, J.P., Hassouni, Y.: Coherent state quantization of paragrassmann algebras. J. Phys. A Math. Theor. 43, 385202 (2010)

  8. Chaichian, M., Demichev, A.P.: Polynomial algebras and higher spins. Phys. Lett. A 222, 14–20 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Trifonov, D.A.: Nonlinear fermions and coherent states. J. Phys. A Math. Theor. 45, 244037 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Greenberg, O.W.: Conservation of statistics and generalized Grassmann numbers. Phys. Lett. A 209, 137–142 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Baz, M.E., Hassouni, Y., Madouri, F.: Z3-graded Grassmann variables, parafermions and their coherent states. Phys. Lett. B 536, 321 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Cahill, K.E., Glauber, R.J.: Density operators for fermions. Phys. Rev. A 59, 1538–1555 (1999)

    Article  ADS  Google Scholar 

  13. Cabra, D.C., Moreno, E.F., Tanasa, A.: Para-Grassmann variables and coherent states. SIGMA 2, 087 (2006)

    MathSciNet  MATH  Google Scholar 

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

  15. Chaichian, M., Demichev, A.P.: Path integrals with generalized Grassmann variables. Phys. Lett. A 207, 23–30 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Cherbal, O., Drir, M., Maamache, M., Trifonov, D.A.: Supersymmetric extension of non-Hermitian \(su(2)\) Hamiltonian and supercoherent states. SIGMA 6, 096 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Daoud, M., Hassouni, Y., Kibler, M.: On generalized super-coherent states. Phys. At. Nucl. 61, 1821–1824 (1998)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Daoud, M., Kibler, M.: Fractional supersymmetric quantum mechanics as a set of replicas of ordinary supersymmetric quantum mechanics. Phys. Lett. A 321, 147–151 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Maleki, Y.: Para-Grassmannian coherent and squeezed states for pseudo-Hermitian q-oscillator and their entanglement. SIGMA 7, 084 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Baz, M.E., Hassouni, Y.: On the construction of generalized Grassmann representatives of state vectors. J. Phys. A Math. Gen. 37, 4361 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Borsten, L., Dahanayake, D., Duff, M.J., Rubens, W.: Superqubits. Phys. Rev. D 81, 105023 (2010)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  24. Frydryszak, A.M.: Qubits, superqubits and squbits. J. Phys Conf. Ser. 411, 012015 (2013)

    Article  ADS  Google Scholar 

  25. Najarbashi, G., Maleki, Y.: Entanglement of Grassmannian coherent states for multi-partite n-level systems. SIGMA 7, 011 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Najarbashi, G., Maleki, Y.: Entanglement in multi-qubit pure fermionic coherent states. Rom. J. Phys. 58, 279–292 (2013)

    Google Scholar 

  27. Segal, I.: Mathematical characterization of the physical vacuum for a linear Bose–Einstein field. Ill. J. Math. 6, 500–523 (1962)

    MATH  Google Scholar 

  28. Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform part I. Commun. Pure Appl. Math. 14, 187–2014 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wiegmann, P.B.: Superconductivity in strongly correlated electronic systems and confinement versus deconfinement phenomenon. Phys. Rev. Lett. 60, 2445 (1988)

    Article  ADS  Google Scholar 

  30. Sarkar, S.: The supersymmetric t-J model in one dimension. J. Phys. A 24, 1137 (1991)

    Article  ADS  Google Scholar 

  31. Foerster, A., Karowski, M.: Completeness of the Bethe states for the supersymmetric t-J mode. Phys. Rev. B 46, 9234 (1992)

    Article  ADS  Google Scholar 

  32. Kitaev, A.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Kitaev, A.: Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Majid, S., Rodriguez-Plaza, M.J.: Random walk and the heat equation on superspace and anyspace. J. Math. Phys. 35, 3753–3760 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Kerner, R.: Z3-graded algebras and the cubic root of the supersymmetry translations. J. Math. Phys. 33, 403–411 (1992)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgments

The author would like to thank F. Khashami for valuable assistance on preparation of the draft of the manuscript.

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  1. Department of Physics, Texas A&M University, College Station, TX, 77843-4242, USA

    Y. Maleki

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  1. Y. Maleki
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Correspondence to Y. Maleki.

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Maleki, Y. Multi-mode entangled states represented as Grassmannian polynomials. Quantum Inf Process 15, 3893–3907 (2016). https://doi.org/10.1007/s11128-016-1357-1

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  • DOI: https://doi.org/10.1007/s11128-016-1357-1

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