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Carlo Pandiscia
Reversible part of quantum dynamical systems: A review
Confluentes Mathematici, 10 no. 2 (2018), p. 51-74, doi: 10.5802/cml.50
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Class. Math.: 46L07, 46L55, 81S22
Mots clés: Quantum dynamical system, Multiplicative core, Algebra of effective observables.

Résumé - Abstract

In this work a quantum dynamical system $(\mathfrak{M},\Phi , \varphi )$ is constituted by a von Neumann algebra $\mathfrak{M}$, a unital Schwartz map $\Phi :\mathfrak{M\rightarrow M}$ and a $\Phi $-invariant normal faithful state $\varphi $ on $\mathfrak{M}$. We will prove that the ergodic properties of a quantum dynamical system are determined by its reversible part $(\mathfrak{D}_\infty ,\Phi _\infty , \varphi _\infty )$; i.e. by a von Neumann sub-algebra $\mathfrak{D}_\infty $ of $\mathfrak{M}$, with an automorphism $\Phi _\infty $ and a normal state $\varphi _\infty $, as the restrictions on $\mathfrak{D}_\infty $. Moreover, if $\mathfrak{D}_\infty $ is a trivial algebra, then the quantum dynamical system is ergodic. Furthermore, we will show some properties of reversible part of the quantum dynamical system, finally we will study its relations with the canonical decomposition of Nagy-Fojas of linear contraction related to a quantum dynamical system.

Bibliographie

[1] L. Accardi and C. Cecchini: Conditional expectations on von Neumann algebras, J. Funct. Anal. 45 (1982), 245-273.
[2] A. Attal and R. Rebolledo: Quantum stationary states and classical Markov semigroups, Unpublished preprints.
[3] O.E. Barndorff-Nielsen, R. Gill and P.E. Jump: On Quantum Statistical Inference, J.R. Stat. Soc. Ser. B 65 (2003) 775-816.  MR 2017871
[4] M. Blanchard and R. Olkiewiez: Decoherence induced transition from quantum to classical dynamics, Phys. Rev. Lett. 90, (2003), 010403.
[5] M. Blanchard and M. Hellmich: Decoherence in infinite quantum systems, Quantum Africa 2010: Theoretical and Experimental Foundations of Recent Quantum Technology, (2012) AIP Conf. Proc. 1469:2-15.
[6] B. Blackadar: Operator algebra, Springer (2006).
[7] O. Bratteli and D.W. Robinson: Operator algebras and quantum mechanics Vol.I, Springer-Verlag (1979).  MR 545651
[8] R. Carbone, E. Sasso and V. Umanitá: Decoherence for positive semigroups on $M_2(\mathbb{C})$, J. Math. Phys. 52 (2011), 032202.
[9] R. Carbone, E. Sasso and V. Umanitá: Ergodic quantum Markov semigroups and decoherence, J. Oper. Theory 72(2) (2014), 293–312.
[10] G.E. Emch: Algebraic methods in statistical mechanics and quantum field theory Wiley-Interscience (1972).
[11] G.E. Emch: Mathematical and conceptual foundations of 20Th-century physics, North Holland (1984).
[12] R.S. Doran and J.M. Fell: Representation of *-algebras , locally compact groups, and Banach *-algebras bundles, Vol. 1, Academic press Inc. (1988).
[13] F. Fidaleo: On strong ergodic properties of quantum dynamical systems, Infin. Dimens. Anal. Quantum Probab. Rel. Top. Vol. 12, No. 4 (2009).  MR 2590155
[14] A. Frigerio, Stationary states of quantum dynamical semigroups, Commun. Math. Phys. 63 (1978), 269-276.  MR 522826
[15] A. Frigerio and M. Verri: Long-Time asymptotic properties of dynamical semigroups on W*-algebras, Math Z. (1982). 180 275-286.  MR 661704
[16] U. Haagerup and M. Musat: Factorization and dilation problems for completely positive maps on von Neumann algebras, Commun. Math. Phys. 303 (2011) 555-594 .  MR 2782624
[17] M. Hellmich: Decoherence in infinite quantum systems, Phd thesis University of Bielefeld.
[18] R.S. Ingarden, A. Kossakowski and M. Ohya Information dynamics and open systems, Springer Science vol. 86 (1997).  MR 1448405
[19] A. Jencova and D. Petz: Sufficiency in quantum statistical Inference, Commun. Math. Phys. 263 (2006), 259-276.  MR 2207329
[20] R.V. Kadison: Transformations of states in operator theory and dynamics Topology Vol. 3, Suppl. 2 (1965) 177-198.
[21] U. Krengel: Ergodic theorems, De Gruyter (1985).
[22] B. Kummerer: Markov dilation on $ W^*$-algebras, J. Funct. Anal. 63 (1985), 139-177.
[23] S. H. Kye: Positive linear maps between matrix algebras which fix diagonals, Linear Algebra Appl. 216 (1995). 239-256  MR 1319988
[24] P. Lugiewicz and R. Olkiewicz: Classical properties of Infinite quantum open systems, Commun. Math. Phys. 239, (2003), 241–259.  MR 1997441
[25] A. Mohari: A mean ergodic theorem of an amenable group action, Infin. Dimens. Anal. Quantum Probab. Rel. Top. Vol. 17, No. 1 (2014).
[26] H. Moriyoshi and T. Natsume: Operator algebras and geometry, Amer. Math. Soc. Vol. 237 (2008).
[27] B. Sz-Nagy and C. Foiaş: Harmonic analysis of operators on Hilbert space, Regional Conference Series in Mathematics, n.19, (1971).
[28] C. Niculescu, A. Ströh and L. Zsidó: Non commutative extensions of classical and multiple recurrence theorems, Operator Theory 50 (2002), 3-52.  MR 2015017
[29] C. Pandiscia: Ergodic Dilation of a Quantum Dynamical System, Confluentes Mathematici 6 (2014), no. 1, 77-91.  MR 3266886
[30] V.I. Paulsen: Completely bounded maps and dilations, Pitman Research Notes in Mathematics 146, Longman Scientific Technical (1986).  MR 868472
[31] R. Rebolledo: Decoherence of quantum Markov semigroups, Ann. I. H. Poincaré – PR 41 (2005), 349–373.  MR 2139024
[32] D.W. Robinson: Strongly positive semigroups and faithful invariant states, Commun. Math. Phys. 85 (1982), 129-142.
[33] E. Stormer: Multiplicative properties of positive maps, Math. Scand. 100 (2007), 184-192.
[34] E. Stormer: Positive linear maps of operator algebras, Springer-Verlag (2013).  MR 3012443
[35] M. Takesaki, Conditional expectations in von Neumann algebras, J. Funct. Anal. 9 (1972), 306-321.  MR 303307
[36] J. Tomiyama: On the projection of norm one in $W^*$-algebras, Proc. Japan Acad. 33 (1957), 608-612.
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