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Mat. Zametki, 2002, Volume 71, Issue 5, Pages 761–781 (Mi mz384)  

This article is cited in 5 scientific papers (total in 5 papers)

Conditions Sufficient for the Conservativity of a Minimal Quantum Dynamical Semigroup

A. M. Chebotareva, S. Yu. Shustikovb

a M. V. Lomonosov Moscow State University, Faculty of Physics
b M. V. Lomonosov Moscow State University

Abstract: Conditions sufficient for a minimal quantum dynamical semigroup (QDS) to be conservative are proved for the class of problems in quantum optics under the assumption that the self-adjoint Hamiltonian of the QDS is a finite degree polynomial in the creation and annihilation operators. The degree of the Hamiltonian may be greater than the degree of the completely positive part of the generator of the QDS. The conservativity (or the unital property) of a minimal QDS implies the uniqueness of the solution of the corresponding master Markov equation, i.e., in the unital case, the formal generator determines the QDS uniquely; moreover, in the Heisenberg representation, the QDS preserves the unit observable, and in the Schrödinger representation, it preserves the trace of the initial state. The analogs of the conservativity condition for classical Markov evolution equations (such as the heat and the Kolmogorov–Feller equations) are known as nonexplosion conditions or conditions excluding the escape of trajectories to infinity.

DOI: https://doi.org/10.4213/mzm384

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English version:
Mathematical Notes, 2002, 71:5, 692–710

Bibliographic databases:

UDC: 517.983.51
Received: 12.10.1999
Revised: 12.01.2002

Citation: A. M. Chebotarev, S. Yu. Shustikov, “Conditions Sufficient for the Conservativity of a Minimal Quantum Dynamical Semigroup”, Mat. Zametki, 71:5 (2002), 761–781; Math. Notes, 71:5 (2002), 692–710

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. García J. C., Quezada R., “Hille-Yosida estimate and nonconservativity criteria for quantum dynamical semigroups”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 7:3 (2004), 383–394  crossref  mathscinet  zmath  isi  scopus
    2. Bahn Changsoo, Ko Chul Ki, “Conservative minimal quantum dynamical semigroups generated by noncommutative elliptic operators”, J. Korean Math. Soc., 42:6 (2005), 1231–1249  crossref  mathscinet  zmath  isi  elib  scopus
    3. Bahn Changsoo, Ko Chul Ki, Park Yong Moon, “Remarks on sufficient conditions for conservativity of minimal quantum dynamical semigroups”, Rev. Math. Phys., 17:7 (2005), 745–768  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    4. Mokhtar-Kharroubi M., “On perturbed positive semigroups on the Banach space of trace class operators”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 11:3 (2008), 405–425  crossref  mathscinet  zmath  isi  elib  scopus
    5. Arlotti L., Lods B., Mokhtar-Kharroubi M., “On Perturbed Substochastic Semigroups in Abstract State Spaces”, Z. Anal. ihre. Anwend., 30:4 (2011), 457–495  crossref  mathscinet  zmath  isi  elib  scopus  scopus
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