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The Induced Semigroup of Schwarz Maps to the Space of Hilbert-Schmidt Operators

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Abstract

We prove that for every semigroup of Schwarz maps on the von Neumann algebra of all bounded linear operators on a Hilbert space which has a subinvariant faithful normal state there exists an associated semigroup of contractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak∗ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an orthonormal basis of the Hilbert space. We describe the form of the generator of a quantum Markov semigroup on the von Neumann algebra of all bounded linear operators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent.

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References

  1. Alazzawi, S., Baumgartner, B.: Generalized Kraus operators and generators of quantum dynamical semigroups. Rev. Math. Phys. 27(7), 1550016, 19 (2015)

    Article  MathSciNet  Google Scholar 

  2. Androulakis, G., Wiedemann, A.: GKSL generators and digraphs: Computing invariant states. J. Phys. A: Math. Theor. 52(30), 305201 (2019)

    Article  MathSciNet  Google Scholar 

  3. Androulakis, G., Ziemke, M.: Generators of quantum Markov semigroups. J. Math. Phys. 56(8), 083512, 16 (2015)

    Article  MathSciNet  Google Scholar 

  4. Attal, S.: Lecture 3: operator semigroups. In: Lectures in quantum noise theory. Retrieved from http://math.univ-lyon1.fr/attal/

  5. Bhatt, S.J.: Stinespring representability and Kadison’s Schwarz inequality in non-unital Banach star algebras and applications. Proc. Indian Acad. Sci. Math. Sci. 108(3), 283–303 (1998)

    Article  MathSciNet  Google Scholar 

  6. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. 1. In: Texts and Monographs in Physics. 2nd edn. C∗- and W∗-algebras, symmetry groups, decomposition of states. Springer, New York (1987)

  7. Carbone, R., Fagnola, F.: Exponential convergence in L2 of quantum markov semigroups on \(\mathcal B(\mathcal H)\). Mat. Zametki 68(4), 523–538 (2000)

    Article  MathSciNet  Google Scholar 

  8. Carbone, R., Sasso, E., Umanità, V.: Environment induced decoherence for Markovian evolutions. J. Math. Phys. 56(9), 092704, 22 (2015)

    Article  MathSciNet  Google Scholar 

  9. Chebotarev, A.M., Fagnola, F.: Sufficient conditions for conservativity of quantum dynamical semigroups. J. Funct. Anal. 118(1), 131–153 (1993)

    Article  MathSciNet  Google Scholar 

  10. Chebotarev, A.M., Fagnola, F.: Sufficient conditions for conservativity of minimal quantum dynamical semigroups. J. Funct. Anal. 153(2), 382–404 (1998)

    Article  MathSciNet  Google Scholar 

  11. Choi, M.D.: A Schwarz inequality for positive linear maps on C∗-algebras. Illinois J. Math. 18, 565–574 (1974)

    Article  MathSciNet  Google Scholar 

  12. Christensen, E., Evans, D.E.: Cohomology of operator algebras and quantum dynamical semigroups. J. London Math. Soc. (2) 20(2), 358–368 (1979)

    Article  MathSciNet  Google Scholar 

  13. Davies, E.B.: Quantum dynamical semigroups and the neutron diffusion equation. Rep. Math. Phys. 11(2), 169–188 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  14. Davies, E.B.: Generators of dynamical semigroups. J. Funct. Anal. 34(3), 421–432 (1979)

    Article  MathSciNet  Google Scholar 

  15. Dunford, N., Schwartz, J.T.: Linear Operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space. With the Assistance of William G. Bade and Robert G. Bartle. Interscience Publishers John Wiley & Sons, New York (1963)

    MATH  Google Scholar 

  16. Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194. Springer, New York (2000). With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt

    Google Scholar 

  17. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems Mathematics and its Applications, vol. 375. Kluwer Academic Publishers Group, Dordrecht (1996)

    Book  Google Scholar 

  18. Evans, D.E.: Conditionally completely positive maps on operator algebras. Quart. J. Math. Oxford Ser. (2) 28(111), 271–283 (1977)

    Article  MathSciNet  Google Scholar 

  19. Evans, D.E., Lewis, J.T.: Dilations of dynamical semi-groups. Comm. Math. Phys. 50(3), 219–227 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  20. Fagnola, F.: Quantum Markov semigroups and quantum flows. Proyecciones 18(3), 144 (1999)

    MathSciNet  MATH  Google Scholar 

  21. Fagnola, F., Rebolledo, R.: The approach to equilibrium of a class of quantum dynamical semigroups. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1(4), 561–572 (1998)

    Article  MathSciNet  Google Scholar 

  22. Fagnola, F., Rebolledo, R.: On the existence of stationary states for quantum dynamical semigroups. J. Math. Phys. 42(3), 1296–1308 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  23. Fagnola, F., Rebolledo, R.: Some results on invariant states for quantum Markov semigroups. In: Da Prato, G., Tubaro, L. (eds.) Stochastic Partial Differential Equations and Applications, pp 197–208. CRC Press, Boca Raton (2002)

  24. Fagnola, F., Rebolledo, R.: Quantum Markov semigroups and their stationary states. In: Stochastic Analysis and Mathematical Physics II, Trends Math., pp. 77–128. Basel, Birkhäuser (2003)

  25. Fagnola, F., Rebolledo, R.: Algebraic conditions for convergence of a quantum Markov semigroup to a steady state. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11(3), 467–474 (2008)

    Article  MathSciNet  Google Scholar 

  26. Frigerio, A.: Quantum dynamical semigroups and approach to equilibrium. Lett. Math. Phys. 2(2), 79–87 (1977/78)

    Article  ADS  MathSciNet  Google Scholar 

  27. Frigerio, A.: Stationary states of quantum dynamical semigroups. Comm. Math. Phys. 63(3), 269–276 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  28. Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of N-level systems. J. Mathematical Phys. 17(5), 821–825 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  29. Holevo, A.S.: On conservativity of covariant dynamical semigroups. In: Proceedings of the XXV Symposium on Mathematical Physics (Toruń, 1992), vol. 33, pp. 95–110 (1993)

  30. Holevo, A.S.: On the structure of covariant dynamical semigroups. J. Funct. Anal. 131(2), 255–278 (1995)

    Article  MathSciNet  Google Scholar 

  31. Holevo, A.S.: Statistical Structure of Quantum Theory, Lecture Notes in Physics Monographs, vol. 67. Springer, Berlin (2001)

    Google Scholar 

  32. Kadison, R.V.: A generalized Schwarz inequality and algebraic invariants for operator algebras. Ann. Math. (2) 56, 494–503 (1952)

    Article  MathSciNet  Google Scholar 

  33. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras. Vol. I, Graduate Studies in Mathematics. Elementary theory, Reprint of the 1983 original, vol. 15. American Mathematical Society, Providence (1997)

  34. Kossakowski, A.: On quantum statistical mechanics of non-Hamiltonian systems. Rep. Mathematical Phys. 3(4), 247–274 (1972)

    Article  ADS  MathSciNet  Google Scholar 

  35. Kossakowski, A., Frigerio, A., Gorini, V., Verri, M.: Quantum detailed balance and KMS condition. Comm. Math. Phys. 57(2), 97–110 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  36. Lindblad, G.: On the generators of quantum dynamical semigroups. Comm. Math. Phys. 48(2), 119–130 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  37. Ohya, M., Petz, D.: Quantum Entropy and its Use. Texts and Monographs in Physics. Springer-Verlag, Berlin (1993)

    Book  Google Scholar 

  38. Siemon, I., Holevo, A.S., Werner, R.F.: Unbounded generators of dynamical semigroups. Open Syst. Inf. Dyn. 24(4), 1740015, 24 (2017)

    Article  MathSciNet  Google Scholar 

  39. Sinha, K.B.: Quantum dynamical semigroups. In: Mathematical Results in Quantum Mechanics, Oper. Theory Adv. Appl., vol. 70, pp 161–169. Basel, Birkhäuser (1994)

  40. Størmer, E.: Positive linear maps of operator algebras. Acta Math. 110, 233–278 (1963)

    Article  MathSciNet  Google Scholar 

  41. Størmer, E.: Positive Linear Maps of Operator Algebras. Springer Monographs in Mathematics. Springer, Heidelberg (2013)

    Book  Google Scholar 

  42. Umanità, V.: Classification and decomposition of quantum Markov semigroups. Probab. Theory Related Fields 134(4), 603–623 (2006)

    Article  MathSciNet  Google Scholar 

  43. Wiedemann, A.: On the Generators of Quantum Dynamical Semigroups. (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/5231 (2019)

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Acknowledgements

We would like to thank Franco Fagnola. His contributions to the results of this work were vital from its conception to the final touches. Without his help the existence of this work would not be possible. We would also like to thank the referee for pointing our attention to a mistake in the original version of this article.

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  1. Alexander Wiedemann

    Present address: Department of Mathematics and Computer Science, Davidson College, Davidson, NC, USA

Authors and Affiliations

  1. Department of Mathematics, University of South Carolina, Columbia, SC, USA

    George Androulakis & Alexander Wiedemann

  2. Department of Mathematics, Drexel University, Philadelphia, PA, USA

    Matthew Ziemke

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  1. George Androulakis
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  2. Alexander Wiedemann
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Correspondence to George Androulakis.

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Androulakis, G., Wiedemann, A. & Ziemke, M. The Induced Semigroup of Schwarz Maps to the Space of Hilbert-Schmidt Operators. Math Phys Anal Geom 23, 10 (2020). https://doi.org/10.1007/s11040-020-09334-6

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