Asymptotic evolution of random unitary operations

Jaroslav Novotný, Gernot Alber 1 , and Igor Jex 2
  • 1 Institut für Angewandte Physik, Technische Universität Darmstadt, Hochschulstraße 4a, D-64289, Darmstadt, Germany
  • 2 Department of Physics, FJFI ČVUT v Praze, Břehová 7 Praha 1 - Staré Město, 115 19, Prague, Czech Republic


We analyze the asymptotic dynamics of quantum systems resulting from large numbers of iterations of random unitary operations. Although, in general, these quantum operations cannot be diagonalized it is shown that their resulting asymptotic dynamics is described by a diagonalizable superoperator. We prove that this asymptotic dynamics takes place in a typically low dimensional attractor space which is independent of the probability distribution of the unitary operations applied. This vector space is spanned by all eigenvectors of the unitary operations involved which are associated with eigenvalues of unit modulus. Implications for possible asymptotic dynamics of iterated random unitary operations are presented and exemplified in an example involving random controlled-not operations acting on two qubits.

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  • [1] S. Stenholm, K.-A. Suominen, Quantum Approach to Informatics (Wiley, New Jersey, 2005)

  • [2] M. E. J. Newman, SIAM Rev. 45, 167 (2003)

  • [3] R. Albert, A. L. Barabasi, Rev. Mod. Phys. 74, 47 (2002)

  • [4] D. T. Finkbeiner II, Introduction to Matrices and Linear Transformations (Freeman, San Francisco, 1978)

  • [5] A. Lozinski, K. Życzkowski, W. Słomczyński, Phys. Rev. E 68, 046110 (2003)

  • [6] A. Baraviera, C. F. Lardizabal, A. O. Lopes, M. T. Cunha, arXiv:0911.0182v2

  • [7] J. Novotný, G. Alber, I. Jex, J. Phys. A-Math. Theor. 42, 282003 (2009)

  • [8] A. S. Holevo, Statistical Structure of Quantum Theory (Springer, Berlin, 2001)

  • [9] R. Alicki, K. Lendi, Quantum Dynamical Semigroups and Applications (Springer-Verlag, Berlin, 1987)

  • [10] I. Bengtsson, K. Życzkowski, Geometry of Quantum States (Cambridge UP, Cambridge, 2006)

  • [11] R. Bhatia, Positive Definite Matrices (Princeton UP, Princeton, 2007)

  • [12] W. Bruzda, V. Cappellini, H.-J. Sommers, K. Życzkowski, Phys. Lett. A 373, 320 (2009)

  • [13] J. A. Holbrook, D. W. Kribs, R. Laflamme, Quantum Inf. Process 2, 381 (2004)

  • [14] D. W. Kribs, P. Edinburgh Math. Soc. (Series 2) 46, 421 (2003)

  • [15] E. Knill, R. Laflamme, L. Viola, Phys. Rev. Lett. 84, 2525 (2000)

  • [16] M. Hamermesh, Group Theory and Its Application to Physical problems (Dover Publications, New York, 1989)

  • [17] P. Facchi, D. A. Lidar, S. Pascazio, Phys. Rev. A 69, 032314 (2004)

  • [18] G. Toth, J. J. G.-Ripoll, arXiv:quant-ph/0609052v3


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