ERGODIC THEOREM FOR QUANTUM DYNAMICAL SEMIGROUP ON DECOHERENCE-FREE SUBALGEBRA OF A QUANTUM MARKOV SEMIGROUP
Keywords:
Cesaro means, decoherence-free subalgebra, quantum dynamical semigroup, quantum Markov semigroup, completely positive maps
Abstract
In this work, we study ergodic theorem for quantum dynamical semigroup on decoherence-free subalgebra of quantum Markov semigroup. A quantum dynamical semigroup of non-expansive maps which has at least one stationary point was established on decoherence-free subalgebra.The measurability of the semigroup was then used to ensure the weak convergence of the Cesaro means to a stationary point.
References
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[25] Reed, M. and Simon, B. .Methods of Modern Mathematical Physics II , Fourier Analysis, Self- Adjointness, Academic Press.(1975).
[2] J. Agredo, F. Fagnola and R. Rebolledo, Decoherence free subspaces of a quantum Markov semigroup, J. Math. Phys. 4 1343-1362 (2014).
[3] Alicki, R. and Lendi, K. . Quantum Dynamical Semigroups and Applications. Lect. Notes Phys. 286, Springer- Verlag (1987).
[4] Blanchard, P. and Olkiewicz, R. . Decoherence as iireversible dynamical process in open quantum systems. Open quantum systems, Recent developments Vol. III, Springer-Verlag Berlin, (2006), 117-159.
[5] Bratelli, O. and Robinson, D. W. . Operator Algebras and Quantum Statistical Mechanics, Vol. 1, Springer-Verlag, 2nd edition (1987).
[6] Chebotarev, A.M. and Fagnola, F. .Sufficient conditions for conservativity of minimal quantum dynamical semigroups J. Funct. Anal. 153, (1998), 382-404.
[7] Carbone, R., Sasso, E. and Umanita, V. . infin. Dimens. Anal. Quantum Probab. Relat. Top. 20, (2017), 1750012.
[8] Davies, E. B.. Generators of dynamical semigroups J. Funct. Anal., 34, (1979), 421-432.
[9] K. Engel, R. Nagel, One parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.
[10] Erik A. . Metric Methods in Ergodic Theory. U.U.D.M Project Report 2023, 7, (2023), 1-14.
[11] Fagnola, F. and Rebolledo, R. . Notes on the Qualitative Behaviour of Quantum Markov Semigroups Open quantum systems, Recent developments. Vol. III, Spinger Berlin, (2006), 161-205.
[12] Fagnola, F. . Quantum Markov semigroups and quantum flows, Proyecciones, J. Math., 18,(3), (1999), 1-144.
[13] Fagnola, F., Rebolledo, R. and Saavedra, C. . Quantum flows associated to master equation in quantum optics,J. Math. Phys., 35, no 1, (1994), 1-12.
[14] Fagnola, F., Sasso, E. and Umanita, V. . Structure of uniformly continuous Quantum Markov Semigroups with Atomic Decoherence-free subalgebra. Open system and information Dynamics. 24 , (2017), 3.
[15] Frigerio, A. and Verri, M. .Long-time asymptotic properties of dynamical semigroups on w*-algebras,Math. Zeitschrift, 180, no. 2, (1982), 275-286.
[16] M. Haase, The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, vol. 169, Birkhauser Verlag, Basel, 2006.
[17] Holevo, A. S. . On the structure of covariant dynamical semigroups,J. Funct. Anal. 131 (2), (1995), 255-278.
[18] Lidar, D.A. and Whaley, K.A. . Decoherence-free subspaces and subsystems, Lect. Notes phys. 622, 83, (2003).
[19] Ogundiran M. O., On the mild solutions of Quantum stochastic evolution inclusions, Communications in Appl. Anal., 19(2015),2,307-318.
[20] Ogundiran, M. O. . Ergodic-Type theorem for Quantum Dynamical semigroup.Evolutionary Processes and applications. Nova science Journal, (2019), 37-46.
[21] Parthasarathy K. R., An introduction to Quantum stochastic calculus,Monographs in Mathematics Vol. 85, Birkhauser-Verlag, Basel-Boston-Berlin, 1992.
[22] Pazy, P. . Semigroups of linear operators and applications to partial differential equation. Springer-Verlag New York, inc. (1983).
[23] Palma, G.M. and Olkiewicz, R. . Decoherence as ireversible dynamical process in open quantum systems. Open quantum systems, Recent developments Vol. 111, Springer-Verlag Berlin, (2003), 106-117.
[24] Ticozzi, F. and Viola, L. . IEEE Transaction on Automatic control 53,2048, (2008).
[25] Reed, M. and Simon, B. .Methods of Modern Mathematical Physics II , Fourier Analysis, Self- Adjointness, Academic Press.(1975).
Published
2024-02-24
How to Cite
Oluwafemi, E. A., Ogundiran, M. O., & Fabelurin, O. (2024). ERGODIC THEOREM FOR QUANTUM DYNAMICAL SEMIGROUP ON DECOHERENCE-FREE SUBALGEBRA OF A QUANTUM MARKOV SEMIGROUP. Unilag Journal of Mathematics and Applications, 3, 25-34. Retrieved from http://lagjma.unilag.edu.ng/article/view/2047
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Copyright (c) 2023 Ezekiel Abiodun Oluwafemi, Micheal Oluniyi Ogundiran, Olanrewaju Fabelurin
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