Abstract
We study a class of generic quantum Markov semigroups on the algebra of all bounded operators on a Hilbert space \({\mathsf{h}}\) arising from the stochastic limit of a discrete system with generic Hamiltonian H S , acting on \({\mathsf{h}}\), interacting with a Gaussian, gauge invariant, reservoir. The selfadjoint operator H S determines a privileged orthonormal basis of \({\mathsf{h}}\). These semigroups leave invariant diagonal and off-diagonal bounded operators with respect to this basis. The action on diagonal operators describes a classical Markov jump process. We construct generic semigroups from their formal generators by the minimal semigroup method and discuss their conservativity (uniqueness). When the semigroup is irreducible we prove uniqueness of the equilibrium state and show that, starting from an arbitrary initial state, the semigroup converges towards this state. We also prove that the exponential speed of convergence of the quantum Markov semigroup coincides with the exponential speed of convergence of the classical (diagonal) semigroup towards its unique invariant measure. The exponential speed is computed or estimated in some examples.
Similar content being viewed by others
Bibliography
Accardi L., Fagnola F., Hachicha S. (2006). Generic q-Markov semigroups and speed of convergence of q-algorithms, Infinite Dimens. Anal. Quantum Probab. Rel. Topics 9: 567
Accardi L., Hachicha S., Ouerdiane H. (2005). Generic quantum Markov semigroups: the Fock case. Open Sys. Information Dyn. 12: 385
L. Accardi, S. Kozyrev, Lectures on Quantum Interacting Particle Systems, in: Quantum interacting particle systems (Trento, 2000), L. Accardi and F. Fagnola, eds., QP–PQ: Quantum Probab. White Noise Anal., 14, World Sci. Publishing, River Edge, NJ, 2002, pp. 1–195.
Accardi L., Lu Y.G., Volovich I. (2002). Quantum theory and its stochastic limit. Springer-Verlag, Berlin
Bahn C., Ko C.K., Park Y.M. (2005). Remarks on sufficient conditions for conservativity of minimal quantum dynamical semigroups. Rev. Math. Phys. 17: 745
Chebotarev A.M., Fagnola F. (1998). Sufficient conditions for conservativity of minimal quantum dynamical semigroups. J. Funct. Anal. 153: 382
Carbone R., Fagnola F. (2000). Exponential L 2-convergence of quantum Markov semigroups on \({\mathcal{B}}(h)\) . Math. Notes 68: 452
M. F. Chen, From Markov chains to non-equilibrium particle systems, World Scientific, 1992.
Fagnola F. (1999). Quantum Markov Semigroups and Quantum Markov Flows. Proyecciones 18: 1
Fagnola F., Quezada R. (2005). Two-photon absorption and emission process, Infinite Dimens. Anal. Quantum Probab. Rel. Topics 8: 573
Fagnola F., Rebolledo R. (1998). The approach to equilibrium of a class of quantum dynamical semigroups. Infinite Dimens. Anal. Quantum Probab. Rel. Topics 1: 561
F. Fagnola and R. Rebolledo, Lectures on the qualitative analysis of quantum Markov semigroups in: Quantum interacting particle systems (Trento, 2000), QP–PQ: Quantum Probab. White Noise Anal. 14, World Sci. Publishing, River Edge, NJ, 2002, pp. 197–239.
Frigerio A. (1978). Stationary states of quantum dynamical semigroups. Comm. Math. Phys. 63: 269
Gorini V., Kossakowski A., Sudarshan E.C.G. (1976). Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17: 821
Kossakowski A., Frigerio A., Gorini V., Verri M. (1977). Quantum detailed balance and KMS condition. Comm. Math. Phys. 57: 97
Liggett T. (1989). Exponential L 2 convergence of attractive reversible nearest particle systems. Ann. Prob. 17: 403
Lindblad G. (1976). On the generators of Quantum Dynamical Semigroups. Commun. Math. Phys. 48: 119
Majewski W.A., Streater R.F. (1998). Detailed balance and quantum dynamical maps. J. Phys. A 31: 7981
Miclo L. (1999). Relations entre isopérimétrie et trou spectral pour les chaînes the Markov finies. Probab. Theory Relat. Fields 114: 431
Pollett P.K. (1991). Invariant measures for Q-processes when Q is not regular. Adv. Appl. Prob. 23: 277
Pollett P.K. (1994). On the identification of continuous-time Markov chains with a given invariant measure. J. Appl. Prob. 31: 897
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Carbone, R., Fagnola, F. & Hachicha, S. Generic Quantum Markov Semigroups: the Gaussian Gauge Invariant Case. Open Syst Inf Dyn 14, 425–444 (2007). https://doi.org/10.1007/s11080-007-9066-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11080-007-9066-y