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.
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REFERENCES
J. C. Garcia, “On the structure of a cone of normal unbounded completely positive maps,” Mat. Zametki [Math. Notes], 65 (1999), no. 2, pp. 194–205.
E. B. Davies, “Quantum dynamical semigroups and neutron diffusion equation,” Rep. Math. Phys., 11 (1979), no. 2, pp. 169–188.
A. M. Chebotarev, “Necessary and sufficient conditions for conservativity of a dynamical semigroup,” J. Soviet Math., 56 (1991), no. 5, pp. 2697–2719.
A. M. Chebotarev, “Sufficient conditions of the conservatism of a minimal dynamical semigroup,” Mat. Zametki [Math. Notes], 52 (1992), no. 4, pp. 112–127.
A. S. Holevo, “Stochastic differential equations in Hilbert space and quantum Markovian evolutions,” in: Probability Theory and Mathematical Statistics. Proc. of the 7th Japan-Russia Symposium, World Sci., Singapore, 1996, pp. 122–131.
A. M. Chebotarev, J. C. Garcia, and R. B. Quezada, “A priori estimates and existence theorems for the Lindblad equation with unbounded time-dependent coefficients,” in: Recent Trends in Infinite-Dimensional Non-Commutative Analysis, vol. 1035, Res. Inst. Math. Sci., Kokyuroku, 1998, pp. 44–65.
A. M. Chebotarev and F. Fagnola, “Sufficient conditions for conservativity of minimal quantum dynamical semigroups,” J. Funct. Anal., 153 (1998), no. 2, pp. 382–404.
O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, vol. I, Springer-Verlag, Berlin, 1981.
K. Kraus, “General state changes in quantum theory,” Ann. Phys., 64 (1971), pp. 311–335.
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1976.
R. Schack, T. A. Brun, and I. C. Percival, “Quantum state diffusion with a moving basis: computing quantum optical spectra,” Phys. Rev. A, 53 (1996), pp. 2694–2711.
P. Zoller and C. W. Gardiner, “Quantum noise in quantum optics: The stochastic Schödinger equation,” in: Lecture Notes for the Les Houches Summer School LXIII on Quantum Fluctuations in (July 1995) (E. Giacobino and S. Reynaud, editors), Elsevier Sci. Publ. (B.V.), 1997.
H. M. Wiseman and J. A. Vaccaro, “Maximally robust unravelings of quantum master equations,” Phys. Lett. A, 250 (1998), pp. 241–248.
T. B. L. Kist, M. Orszag, T. A. Brun, and L. Davidovich, “Physical interpretation of stochastic Schrödinger equations in cavity QED,” in: LANL E-print quant-ph/9805027.
M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. 1. Functional Analysis, Acad. Press, New York, 1981.
R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York, 1978.
D. Kilin and M. Schreiber, “Influence of phase-sensitive interaction on the decoherence process in molecular systems,” in: LANL E-print quant-ph/9707054.
L. Lanz, O. Melsheimer, and B. Vacchini, “Subdynamics through Time Scales and Scattering Maps in Quantum Field Theory,” in: Proceedings of the Third International Conference on Quantum Communication and Measurement 1996 (Hakone, Japan), 1997.
A. M. Chebotarev, Lectures on Quantum Probability, SMM, Mexico, 2000.
A. M. Chebotarev and S. Yu. Shustikov, “On a condition sufficient for violation of unitality,” in: VINITI 09.06.2000 No 1645-B00.
H. P. McKean, Stochastic Integrals, Acad. Press, New York, 1969.
A. M. Chebotarev, “On the maximal C ?-algebra of zeros of completely positive mapping and on the boundary of a quantum dynamical semigroup,” Mat. Zametki [Math. Notes], 56 (1994), no. 6, pp. 88–105.
E. B. Dynkin, Markov Processes, Springer-Verlag, Heidelberg-Berlin, 1965.
A. Grigor?yan, “Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds,” Bull. Amer. Math. Soc., 36 (1999), no. 2, pp. 135–249.
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Chebotarev, A.M., Shustikov, S.Y. Conditions Sufficient for the Conservativity of a Minimal Quantum Dynamical Semigroup. Mathematical Notes 71, 692–710 (2002). https://doi.org/10.1023/A:1015896123403
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DOI: https://doi.org/10.1023/A:1015896123403