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Mathematical Notes

, Volume 93, Issue 3–4, pp 351–359 | Cite as

On analogs of spectral decomposition of a quantum state

  • G. G. Amosov
  • V. Zh. Sakbaev
Article
  • 53 Downloads

Abstract

The set of quantum states in a Hilbert space is considered. The structure of the set of extreme points of the set of states is investigated and an arbitrary state is represented as the Pettis integral over a finitely additive measure on the set of vector states, which is a generalization of the spectral decomposition of a normal state.

Keywords

quantum state spectral decomposition finitely additive measure 

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References

  1. 1.
    A. N. Sherstnev, Methods of Bilinear Forms in Noncommutative Measure and Integral Theory (Fizmatlit, Moscow, 2008) [in Russian].Google Scholar
  2. 2.
    U. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics. C*- and W*-Algebras, Symmetry Groups, Decomposition of States (New York, Springer, 1979; Mir, Moscow, 1982).CrossRefGoogle Scholar
  3. 3.
    R. Glauber, Quantum Optics and Radiophysics (Mir, Moscow, 1966), pp. 93–279 [Russian transl.].Google Scholar
  4. 4.
    M. D. Srinivas, “Collapse postulate for observables with continuous spectra,” Comm. Math. Phys. 71(2), 131–158 (1980).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    A. S. Kholevo, Probability and Statistical Aspects of Quantum Theory (Inst. Computer Studies, Moscow-Izhevsk, 2003) [in Russian].Google Scholar
  6. 6.
    A. L. Keri and F. A. Sukochev, “Dixmier traces and some applications in non-commutative geometry,” UspekhiMat. Nauk 61(6), 45–110 (2006) [RussianMath. Surveys 61 (6), 1039–1099 (2006)].MathSciNetCrossRefGoogle Scholar
  7. 7.
    G. G. Amosov and V. Zh. Sakbaev, “Stochastic properties of dynamics of quantum systems,” Vestnik SamGU 8(1), 479–494 (2008).Google Scholar
  8. 8.
    V. Zh. Sakbaev, “On dynamics of a degenerate quantum system in the space of functions integrable over a finitely additive measure,” Trudy MFTI 1(4), 126–147 (2009).Google Scholar
  9. 9.
    V. Zh. Sakbaev, “Stochastic properties of degenerated quantum systems,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13(1), 65–85 (2010).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    V. Zh. Sakbaev, “The set of quantum states and its averaged dynamic transformations,” Izv. Vyssh. Uchebn. Zaved.Mat., No. 10, 48–58 (2011) [RussianMath. (Iz. VUZ) 55 (10), 41–50 (2011)].Google Scholar
  11. 11.
    V. I. Bogachev and O. G. Smolyanov, Real and Functional Analysis (URSS, Moscow, 2009) [in Russian].Google Scholar
  12. 12.
    N. Dunford and L. Schwartz, Linear Operators, Vol 1: General Theory (Wiley, New York, 1958; Inostr. Lit., Moscow, 1962).Google Scholar
  13. 13.
    K. Yosida and E. Hewitt, “Finitely additive measures,” Trans. Amer.Math. Soc. 72, 46–66 (1952).MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    V. S. Varadarain, “Measures on topological spaces,” Mat. Sb. 55(1), 35–100 (1961) [Amer. Math. Soc., Transl. II, Ser. 48, 161–228 (1965)].MathSciNetGoogle Scholar
  15. 15.
    A. G. Chentsov, Finitely-Additive Measures and Relaxations of Extremum Problems (UIF Nauka, Ekaterinburg, 1993) [in Russian].Google Scholar
  16. 16.
    R. T. Powers, “Representations of uniformly hyperfinite algebras and their associated von Neumann rings,” Ann. of Math. (2) 86(1), 138–171 (1967).MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    L. Accardi, Y. G. Lu, and I. V. Volovich, Quantum Theory and Its Stochastic Limit (Springer, New York, 2002).MATHCrossRefGoogle Scholar
  18. 18.
    G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Wiley, New York, 1972; Mir, Moscow, 1976).MATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of Sciences Moscow Institute of Physics and Technology (State University)MoscowRussia

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