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Markovian quantum subsystems and stabilization design

  • Department of Automation, University of Science and Technology of China, Hefei 230027, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2014.10.003
  • Received Date: 13 December 2013
  • Accepted Date: 16 May 2014
  • Rev Recd Date: 16 May 2014
  • Publish Date: 30 October 2014
    Abstract

  • Abstract

    Quantum subsystems have been widely applied in the field of quantum information processing and control, so active control over quantum subsystems is of great importance. For Markovian open quantum dynamical model, some main theoretical results on invariant and attractive subsystems are summarized, and some important properties and their essential meanings are analyzed. Based on these, open-loop Hamiltonian design methods which make a subsystem become invariant and attractive were studied. In particular, the Hamiltonian design of invariance of a subsystem and attractiveness of an invariant subsystem achieved globally asymptotical stabilization of this subsystem. Finally, simulation experiments on a two-level open quantum system were conducted, and results show the validity of the proposed method.

    Abstract

    Quantum subsystems have been widely applied in the field of quantum information processing and control, so active control over quantum subsystems is of great importance. For Markovian open quantum dynamical model, some main theoretical results on invariant and attractive subsystems are summarized, and some important properties and their essential meanings are analyzed. Based on these, open-loop Hamiltonian design methods which make a subsystem become invariant and attractive were studied. In particular, the Hamiltonian design of invariance of a subsystem and attractiveness of an invariant subsystem achieved globally asymptotical stabilization of this subsystem. Finally, simulation experiments on a two-level open quantum system were conducted, and results show the validity of the proposed method.
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    • References(20)
  • [1]
    Nielsen M A, Chuang I L. Quantum Computation and Quantum Information [M]. Cambridge University Press, 2010.
    [2]
    Zanardi P, Rasetti M. Noiseless quantum codes [J]. Physical Review Letters, 1997, 79(17): 3 306-3 309.
    [3]
    Lidar D A, Chuang I L, Whaley K B. Decoherence-free subspaces for quantum computation [J]. Physical Review Letters, 1998, 81(12): 2 594-2 597.
    [4]
    Viola L, Fortunato E M, Pravia M A, et al. Experimental realization of noiseless subsystems for quantum information processing [J]. Science, 2001, 293(5537): 2 059-2 063.
    [5]
    Choi M D, Kribs D W. Method to find quantum noiseless subsystems [J]. Physical Review Letters, 2006, 96(5): 050501(1-5).
    [6]
    Xu G F, Zhang J, Tong D M, et al. Nonadiabatic holonomic quantum computation in decoherence-free subspaces [J]. Physical Review Letters, 2012, 109(17): 170501(1-4).
    [7]
    Kumagai H, Yamamoto T, Koashi M, et al. Robustness of quantum communication based on a decoherence-free subspace using a counter-propagating weak coherent light pulse [J]. Physical Review A, 2013, 87(5): 052325(1-8).
    [8]
    Shabani A, Lidar D A. Theory of initialization-free decoherence-free subspaces and subsystems [J]. Physical Review A, 2005, 72(4): 042303(1-15).
    [9]
    Viola L, Ticozzi F. Attractive quantum subsystems and feedback-stabilization problems [C]// Proceedings of the 3rd International conference on Physics and control. Potsdam, Germany: IEEE Press, 2007.
    [10]
    Ticozzi F, Viola L. Quantum Markovian subsystems: Invariance, attractivity, and control [J]. IEEE Transactions on Automatic Control, 2008, 53(9): 2 048-2 063.
    [11]
    Ticozzi F, Viola L. Analysis and synthesis of attractive quantum Markovian dynamics [J]. Automatica, 2009, 45(9): 2002-2009.
    [12]
    Nishio K, Kashima K, Imura J. Global feedback stabilization of quantum noiseless subsystems [C]// American Control Conference. St. Louis, USA: IEEE Press, 2009: 1 499-1 504.
    [13]
    Schirmer S G, Wang X T. Stabilizing open quantum systems by Markovian reservoir engineering [J]. Physical Review A, 2010, 81(6): 062306(1-14).
    [14]
    Ticozzi F, Lucchese R, Cappellaro P, et al. Hamiltonian control of quantum dynamical semigroups: Stabilization and convergence speed [J]. IEEE Transactions on Automatic Control, 2012, 57(8): 1 931-1 944.
    [15]
    Ticozzi F, Nishio K, Altafini C. Stabilization of stochastic quantum dynamics via open and closed loop control [J]. IEEE Transactions on Automatic Control, 2013, 58(1): 74-85.
    [16]
    Lindblad G. On the generators of quantum dynamical semigroups [J]. Communications in Mathematical Physics, 1976, 48(2): 119-130.
    [17]
    Gorini V, Kossakowski A, Sudarshan E C G. Completely positive dynamical semigroups of N-level systems [J]. Journal of Mathematical Physics, 1976, 17(5): 821-825.
    [18]
    Viola L, Knill E, Laflamme R. Constructing qubits in physical systems [J]. Journal of Physics A: Mathematical and General, 2001, 34(35): 7 067-7 079.
    [19]
    Knill E. Protected realizations of quantum information [J]. Physical Review A, 2006, 74(4): 042301(1-11).
    [20]
    Ticozzi F, Schirmer S G, Wang X T. Stabilizing quantum states by constructive design of open quantum dynamics [J]. IEEE Transactions on Automatic Control, 2010, 55(12): 2 901-2 905.
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    [1]
    Nielsen M A, Chuang I L. Quantum Computation and Quantum Information [M]. Cambridge University Press, 2010.
    [2]
    Zanardi P, Rasetti M. Noiseless quantum codes [J]. Physical Review Letters, 1997, 79(17): 3 306-3 309.
    [3]
    Lidar D A, Chuang I L, Whaley K B. Decoherence-free subspaces for quantum computation [J]. Physical Review Letters, 1998, 81(12): 2 594-2 597.
    [4]
    Viola L, Fortunato E M, Pravia M A, et al. Experimental realization of noiseless subsystems for quantum information processing [J]. Science, 2001, 293(5537): 2 059-2 063.
    [5]
    Choi M D, Kribs D W. Method to find quantum noiseless subsystems [J]. Physical Review Letters, 2006, 96(5): 050501(1-5).
    [6]
    Xu G F, Zhang J, Tong D M, et al. Nonadiabatic holonomic quantum computation in decoherence-free subspaces [J]. Physical Review Letters, 2012, 109(17): 170501(1-4).
    [7]
    Kumagai H, Yamamoto T, Koashi M, et al. Robustness of quantum communication based on a decoherence-free subspace using a counter-propagating weak coherent light pulse [J]. Physical Review A, 2013, 87(5): 052325(1-8).
    [8]
    Shabani A, Lidar D A. Theory of initialization-free decoherence-free subspaces and subsystems [J]. Physical Review A, 2005, 72(4): 042303(1-15).
    [9]
    Viola L, Ticozzi F. Attractive quantum subsystems and feedback-stabilization problems [C]// Proceedings of the 3rd International conference on Physics and control. Potsdam, Germany: IEEE Press, 2007.
    [10]
    Ticozzi F, Viola L. Quantum Markovian subsystems: Invariance, attractivity, and control [J]. IEEE Transactions on Automatic Control, 2008, 53(9): 2 048-2 063.
    [11]
    Ticozzi F, Viola L. Analysis and synthesis of attractive quantum Markovian dynamics [J]. Automatica, 2009, 45(9): 2002-2009.
    [12]
    Nishio K, Kashima K, Imura J. Global feedback stabilization of quantum noiseless subsystems [C]// American Control Conference. St. Louis, USA: IEEE Press, 2009: 1 499-1 504.
    [13]
    Schirmer S G, Wang X T. Stabilizing open quantum systems by Markovian reservoir engineering [J]. Physical Review A, 2010, 81(6): 062306(1-14).
    [14]
    Ticozzi F, Lucchese R, Cappellaro P, et al. Hamiltonian control of quantum dynamical semigroups: Stabilization and convergence speed [J]. IEEE Transactions on Automatic Control, 2012, 57(8): 1 931-1 944.
    [15]
    Ticozzi F, Nishio K, Altafini C. Stabilization of stochastic quantum dynamics via open and closed loop control [J]. IEEE Transactions on Automatic Control, 2013, 58(1): 74-85.
    [16]
    Lindblad G. On the generators of quantum dynamical semigroups [J]. Communications in Mathematical Physics, 1976, 48(2): 119-130.
    [17]
    Gorini V, Kossakowski A, Sudarshan E C G. Completely positive dynamical semigroups of N-level systems [J]. Journal of Mathematical Physics, 1976, 17(5): 821-825.
    [18]
    Viola L, Knill E, Laflamme R. Constructing qubits in physical systems [J]. Journal of Physics A: Mathematical and General, 2001, 34(35): 7 067-7 079.
    [19]
    Knill E. Protected realizations of quantum information [J]. Physical Review A, 2006, 74(4): 042301(1-11).
    [20]
    Ticozzi F, Schirmer S G, Wang X T. Stabilizing quantum states by constructive design of open quantum dynamics [J]. IEEE Transactions on Automatic Control, 2010, 55(12): 2 901-2 905.
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