Skip to main content
SpringerLink
Log in

Matrix representations for adjoint and anti-adjoint operators in multi-spin 1/2 systems

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

This paper focuses on investigating the problems of matrix representations of adjoint and anti-adjoint operators as well as computations for these matrices in multi-spin 1/2 systems. By introducing a multi-index transformation mapping, adjoint and anti-adjoint operators on tensor space as well as their matrix representations are defined to describe dynamics of multi-spin 1/2 systems. Formulas for computing these matrices of the adjoint and anti-adjoint operators in multi-spin 1/2 systems are given in terms of matrix representations of the adjoint and anti-adjoint operators in single-spin 1/2 systems.

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. Albertini F., D’Alessandro D.: Control of the evolution of Heisenberg spin systems. Eur. J. Control 10(5), 497–504 (2004)

    Article  MathSciNet  Google Scholar 

  2. Altafini C.: Representation multiqubit unitary evolution via Stokes tensors. Phys. Rev. A 70(3), 032331 (2004)

    Article  ADS  CAS  Google Scholar 

  3. Altafini C.: Feedback control of spin systems. Quantum Inf. Process. 6(1), 9–36 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Boscain U., Mason P.: Time minimal trajectories for a spin 1/2 particle in a magnetic field. J. Math. Phys. 47(6), 062101 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  5. D’Alessandro D.: Small time controllability of systems on compact Lie groups and spin angular momentum. J. Math Phys. 42(9), 4488–4496 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Daniel B.: Lie groups. Springer-Verlag, New York (2004)

    MATH  Google Scholar 

  7. Doherty A.C., Habib S., Jacobs K., Mabuchi H., Tan S.M.: Quantum feedback control and classical control theory. Phys. Rev. A 62(1), 012105 (2000)

    Article  ADS  Google Scholar 

  8. Geremia J.M., Stockton J.K., Mabuchi H.: Real-time quantum feedback control of atomic spin-squeezing. Science 304(5668), 270–273 (2004)

    Article  PubMed  CAS  ADS  Google Scholar 

  9. Hall B.C.: Lie Groups Lie Algebras and Represesntations: An Elementary Introduction. Springer-Verlag, New York (2003)

    MATH  Google Scholar 

  10. Horn R.A., Johnson C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  11. Khaneja N., Glaser S.J.: Cartan decomposition of SU(2n) and control of spin systems. Chem. Phys. 267(1–3), 11–23 (2001)

    Article  CAS  ADS  Google Scholar 

  12. Khaneja N., Brockett R., Glaser S.J.: Time optimal control in spin systems. Phys. Rev. A 63(3), 032308 (2001)

    Article  ADS  CAS  Google Scholar 

  13. Mabuchi H., Khaneja N.: Principles and applications of control in quantum systems. Int. J. Robust Nonlinear Control 15(15), 647–667 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Shafarevich I.R.: Basic Notions of Algebra. Springer-Verlag, Berlin, Heidelberg (2005)

    MATH  Google Scholar 

  15. Tsyplyatyev O., Loss D.: Dynamics of the inhomogeneous Dicke model for a single-boson mode coupled to a bath of nonidentical spin-1/2 systems. Phys. Rev. A 80(2), 023803 (2009)

    Article  ADS  CAS  Google Scholar 

  16. Yamamoto N., Tsumura K., Hara S.: Feedback control of quantum entanglement in a two-spin systems. Automatica 43(6), 981–992 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Yanagisawa M., Kimura H.: Transfer function approach to quantum control—Part I: Dynamics of quantum feedback systems. IEEE Trans. Autom. Control 48(12), 2107–2120 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Automation, Hangzhou Dianzi University, 310018, Hangzhou, Zhejiang, People’s Republic of China

    Shaosheng Zhou & Shizhou Fu

Authors
  1. Shaosheng Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shizhou Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shaosheng Zhou.

Additional information

This work was supported in part by the National Natural Science Foundation of P. R. China under Grants 60974006 and 60934009, and by the 973 program No 2009CB320600.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhou, S., Fu, S. Matrix representations for adjoint and anti-adjoint operators in multi-spin 1/2 systems. Quantum Inf Process 10, 379–394 (2011). https://doi.org/10.1007/s11128-010-0203-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-010-0203-0

Keywords

Search

Navigation