Abstract
In this paper we consider the set of quantum states and passages to the limit for sequences of quantum dynamic semigroups in the mentioned set. We study the structure of the set of extreme points of the quantum state set and represent an arbitrary state as an integral over the set of one-dimensional orthogonal projectors; the obtained representation is similar to the spectral decomposition of a normal state. We apply the obtained results to the analysis of sequences of quantum dynamic semigroups which occur in the regularization of a degenerate Hamiltonian.
Similar content being viewed by others
References
D. M. Gitman and I. V. Tyutin, Canonical Quantization of Fields with Constraints (Nauka, Moscow, 1986) [in Russian].
V. Zh. Sakbaev, “Spectral Aspects of Regularization of the Cauchy Problem for a Degenerate Equation,” Trudy MIAN im. V. A. Steklova 261, 258–267 (2008).
V. V. Kozlov, “Dynamics of Systems with Nonintegrable Constraints. I,” Vestn. Mosk. Univ. Ser. 1 Matem., Mekhan., No. 3, 92–100 (1982).
O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics (Mir, Moscow, 1982; Springer, 2002, 2nd Ed., Vol. 1, 2).
G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Wiley-Interscience, New York, 1972; Mir, Moscow, 1976).
L. Accardi, Y. G. Lu, and I. V. Volovich, Quantum Theory and Its Stochastic Limit, Texts and monographs in physics (Springer, 2001).
A. L. Carey and F. A. Sukochev, “Dixmier Traces and Some Applications in Noncommutative Geometry,” Usp. Mat. Nauk 61(6) 45–110 (2006).
M. D. Srinivas, “Collapse Postulate for Observables with Continuous Spectra,” Comm. Math. Phys. 71(2) 131–158 (1980).
A. N. Sherstnev, Methods of Bilinear Forms in the Noncommutative Measure and Integration Theory (Fizmatlit, Moscow, 2008) [in Russian].
R. Glauber, Optical Coherence and Photon Statistics, in Quantum Optics and Radio Physics, Ed. by C. DeWitt (University of Grenoble, Houches, New York-London-Paris, 1965; Mir, Moscow, 1966), pp.93–279.
A. S. Kholevo, Probabilistic and Statistical Aspects of Quantum Theory (Elsevier Science Ltd., 1982; Moscow-Izhevsk, 2003).
N. Dunford and J. T. Schwartz, Operator Theory (Interscience, 1958; URSS, Moscow, 2004), Vol. 1.
K. Iosida and E. Hewitt, “Finitely Additive Measures,” Trans. Amer.Math. Soc. 72 46–66 (1952).
A. G. Chentsov, Finitely Additive Measures and Relaxations of Extremal Problems (UIF, Nauka, Ekaterinburg, 1993; Plenum Publishing Corporation, London, Moscow, 1996).
V. Zh. Sakbaev, “Stochastic Properties of Degenerated Quantum Systems,” Inf. Dimens. Anal., Quantum Probab. Relat. Top. 13(1) 65–85 (2010).
V. Zh. Sakbaev, “Averaging of Quantum Dynamical Semigroups,” Teor.Mat. Fiz. 164(3) 455–463 (2010).
V. Zh. Sakbaev, “On the Dynamics of a Degenerated Quantum Systemin the Spaces of Functions Integrable with Respect to Finitely Additive Measure,” Trudy Mosk. Fiz.-Tekhn. Inst. 1(4) 126–147 (2009).
V. Zh. Sakbaev, “Set-Valued Mappings Specified by Regularization of the Schrödinger Equation with Degeneration,” Zhurn. Vychisl. Matem. I Matem. Fiz. 46(4), 683–699 (2006).
G. G. Amosov and V. Zh. Sakbaev, “Stochastic Properties of Dynamics of Quantum Systems,” Vestn. SamGU 8(1) 479–494 (2008).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.Zh. Sakbaev, 2011, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2011, No. 10, pp. 48–58.
About this article
Cite this article
Sakbaev, V.Z. The set of quantum states and its averaged dynamic transformations. Russ Math. 55, 41–50 (2011). https://doi.org/10.3103/S1066369X11100069
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X11100069