International Journal of Theoretical Physics

, Volume 57, Issue 5, pp 1272–1284 | Cite as

Algebraic Method in the Analysis of Decoherence-Free Subspaces in Open Quantum Systems

  • Takeo KamizawaEmail author


In open quantum systems, a subspace which is not affected by the environmental noise is called a decoherence-free subspace. Such a subspace plays an important role in applications such as quantum information transmissions. In the literature, several “definitions” of decoherence-free subspaces were proposed, but they are model-dependent and slightly different. In this paper, we will study a general framework of decoherence-free subspaces and provide a criterion for the existence of a decoherence-free subspace in open quantum systems.


Open quantum systems Decoherence-free subspace Generalised shemesh criterion Common reducing subspace 



This research is a part of the author’s PhD research, which was supported by Japan Student Services Organisation. The author would like to express his gratitude to Prof. Andrzej Jamiołkowski for valuable advices on this research.


  1. 1.
    Choi, M.-D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10(3), 285–290 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cirillo, G.I., Ticozzi, F.: Decompositions of Hilbert spaces, stability analysis and convergence probabilities for discrete-time quantum dynamical semigroups. J. Phys. A 48(8), 085302 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cong, S., Yang, F.: Control of quantum states in decoherence-free subspaces. J. Phys. A 46(7), 075305 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Jamiołkowski, A., Pastuszak, G.: Generalized shemesh criterion, common invariant subspaces and irreducible completely positive superoperators. Linear Multilinear Algebra 63(2), 314–325 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kamizawa, T.: On functionally commutative quantum systems. Open Syst. Inf. Dyn. 22(03), 1550020 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Karasik, R.I., Marzlin, K.-P., Sanders, B.C., Whaley, K.B.: Criteria for dynamically stable decoherence-free subspaces and incoherently generated coherences. Phys. Rev. A 77, 052301 (2008)ADSCrossRefGoogle Scholar
  7. 7.
    Lidar, D.A.: Review of decoherence free subspaces, noiseless subsystems, and dynamical decoupling. arXiv:1208.5791 (2012)
  8. 8.
    Lidar, D.A., Chuang, I.L., Whaley, K.B.: Decoherence-free subspaces for quantum computation. Phys. Rev Lett. 81(12), 2594 (1998)ADSCrossRefGoogle Scholar
  9. 9.
    Lidar, D.A., Whaley, K.B.: Decoherence-free subspaces and subsystems. In: Benatti, F., Floreanini, R. (eds.) Irreversible Quantum Dynamics, pp 83–120. Springer (2003)Google Scholar
  10. 10.
    Mundarain, D., Orszag, M.: Decoherence-free subspace and entanglement by interaction with a common squeezed bath. Phys. Rev. A 75, 040303 (2007)ADSCrossRefGoogle Scholar
  11. 11.
    Pastuszak, G., Jamiołkowski, A.: Common reducing unitary subspaces and decoherence in quantum systems. Electron. J Linear Algebra 30(1), 17 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Radjavi, H., Nohel, J.A.: Invariant subspaces. Dover Publications, USA (2003)Google Scholar
  13. 13.
    Shabani, A., Lidar, D.A.: Theory of initialization-free decoherence-free subspaces and subsystems. Phys. Rev. A 042303, 72 (2005)Google Scholar
  14. 14.
    Wilde, M.M.: From classical to quantum Shannon theory. arXiv:1106.1445 (2011)
  15. 15.
    Wu, S., Wang, L., Yi, X.: Time-dependent decoherence-free subspace. J. Phys. A: Math. Theor. 45(40), 405305 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yang, J., Cong, S., Yang, F.: State transfer based on decoherence-free target state by Lyapunov-based control. J. Control Theory Appl. 10(4), 549–553 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zanardi, P., Rasetti, M.: Noiseless quantum codes. Phys. Rev. Lett., 79, 3306–3309 (1997)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Physics, Astronomy and InformaticsNicolaus Copernicus UniversityToruńPoland

Personalised recommendations