Ergodic Dilation of a Quantum Dynamical System

Carlo Pandiscia[1]

  • [1] Universitá degli Studi di Roma “Tor Vergata”, Dipartimento di Ingegneria Elettronica, via del Politecnico, 00133 Roma, Italia

Confluentes Mathematici (2014)

  • Volume: 6, Issue: 1, page 77-91
  • ISSN: 1793-7434

Abstract

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Using the Nagy dilation of linear contractions on Hilbert space and the Stinespring’s theorem for completely positive maps, we prove that any quantum dynamical system admits a dilation in the sense of Muhly and Solel which satisfies the same ergodic properties of the original quantum dynamical system.

How to cite

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Pandiscia, Carlo. "Ergodic Dilation of a Quantum Dynamical System." Confluentes Mathematici 6.1 (2014): 77-91. <http://eudml.org/doc/275495>.

@article{Pandiscia2014,
abstract = {Using the Nagy dilation of linear contractions on Hilbert space and the Stinespring’s theorem for completely positive maps, we prove that any quantum dynamical system admits a dilation in the sense of Muhly and Solel which satisfies the same ergodic properties of the original quantum dynamical system.},
affiliation = {Universitá degli Studi di Roma “Tor Vergata”, Dipartimento di Ingegneria Elettronica, via del Politecnico, 00133 Roma, Italia},
author = {Pandiscia, Carlo},
journal = {Confluentes Mathematici},
keywords = {Quantum Markov process; completely positive maps; Nagy dilation; ergodic state; quantum Markov process},
language = {eng},
number = {1},
pages = {77-91},
publisher = {Institut Camille Jordan},
title = {Ergodic Dilation of a Quantum Dynamical System},
url = {http://eudml.org/doc/275495},
volume = {6},
year = {2014},
}

TY - JOUR
AU - Pandiscia, Carlo
TI - Ergodic Dilation of a Quantum Dynamical System
JO - Confluentes Mathematici
PY - 2014
PB - Institut Camille Jordan
VL - 6
IS - 1
SP - 77
EP - 91
AB - Using the Nagy dilation of linear contractions on Hilbert space and the Stinespring’s theorem for completely positive maps, we prove that any quantum dynamical system admits a dilation in the sense of Muhly and Solel which satisfies the same ergodic properties of the original quantum dynamical system.
LA - eng
KW - Quantum Markov process; completely positive maps; Nagy dilation; ergodic state; quantum Markov process
UR - http://eudml.org/doc/275495
ER -

References

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  1. L. Accardi and C. Cecchini. Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Ana., 45:245–273, 1982. Zbl0483.46043MR647075
  2. W. Arveson. Non commutative dynamics and Eo-semigroups, Monograph in mathematics, Springer-Verlag, 2003. Zbl1032.46001
  3. B.V. Bath and K.R. Parthasarathy. Markov dilations of nonconservative dynamical semigroups and quantum boundary theory, Ann. I.H.P. sec. B, 31(4):601–651, 1995. Zbl0832.46060MR1355610
  4. D. E. Evans and J. T. Lewis. Dilations of dynamical semi-groups, Comm. Math. Phys., 50(3):219–227, 1976. Zbl0402.46039MR468878
  5. A. Frigerio, V.Gorini, A. Kossakowski and M. Verri. Quantum detailed balance and KMS condition, Commun. Math. Phys., 57:97–110, 1977. Zbl0374.46060MR468989
  6. B. Kümmerer. Markov dilations on W*-algebras, J. Funct. Ana., 63:139–177, 1985. Zbl0601.46062MR803091
  7. W.A. Majewski. On the relationship between the reversibility of dynamics and balance conditions, Ann. I. H. P. sec. A, 39(1):45–54, 1983. Zbl0519.46068MR715131
  8. P.S. Muhly and B. Solel. Quantum Markov Processes (correspondeces and dilations), Int. J. Math., 13(8):863–906, 2002. Zbl1057.46050MR1928802
  9. B.Sz. Nagy and C. Foiaş. Harmonic analysis of operators on Hilbert space, Regional Conf. Ser. Math., 19, 1971. Zbl0201.45003MR275190
  10. C. Niculescu, A. Ströh and L.Zsidó. Noncommutative extensions of classical and multiple recurrence theorems, J. Oper. Th., 50:3–52, 2002. Zbl1036.46053MR2015017
  11. V.I. Paulsen. Completely bounded maps and dilations, Pitman Res. Notes Math. 146, Longman Scientific & Technical, 1986. Zbl0614.47006MR868472
  12. M. Skeide. Dilation theory and continuous tensor product systems of Hilbert modules, in: PQ-QP: Quantum Probability and White Noise Analysis XV, World Scientific, 2003. Zbl1050.81043MR2010609
  13. F. Stinesring. Positive functions on C* algebras, Proc. Amer. Math. Soc., 6:211–216, 1955. Zbl0064.36703MR69403
  14. L. Zsido. Personal communication, 2008. 

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