Unraveling open quantum systems: classical reductions and classical dilations of quantum markov semigroups
Confluentes Mathematici, Volume 1 (2009) no. 1, pp. 123-167.

A number of results connecting quantum and classical Markov semigroups, as well as their dilations is reported. The method presented here is based on the analysis of the structure of the semigroup generator. In particular, measure-valued processes appear as a combination of classical reduction and classical dilation of a given quantum Markov semigroup.

Published online:
DOI: 10.1142/S1793744209000055
Rolando Rebolledo 1

1
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Rolando Rebolledo. Unraveling open quantum systems: classical reductions and classical dilations of quantum markov semigroups. Confluentes Mathematici, Volume 1 (2009) no. 1, pp. 123-167. doi : 10.1142/S1793744209000055. https://cml.centre-mersenne.org/articles/10.1142/S1793744209000055/

[1] L. Accardi, A. Frigerio and Y. G. Lu, Quantum Probability and Applications (Springer, 1989) pp. 20.

[2] L. Accardi, A. Frigerio and Y. G. Lu, Comm. Math. Phys. 131, 537 (1990), DOI: 10.1007/BF02098275 .

[3] L. Accardi, Y. G. Lu and I. Volovich, Quantum Theory and Its Stochastic Limit (Springer-Verlag, 2002) p. 493.

[4] R. Alicki and K. Lendi , Quantum Dynamical Semigroups and Applications , Lect. Notes in Physics 286 ( Springer-Verlag , 1987 ) .

[5] G. Alli and G. L. Sewell, J. Math. Phys. 36, 5598 (1995), DOI: 10.1063/1.531279 .

[6] S Attal and A. Joye, J. Statist. Phys. 126, 1241 (2007), DOI: 10.1007/s10955-006-9085-z .

[7] O. Bratteli and D. W. Robinson , Operator Algebras and Quantum Statistical Mechanics, 2 , 2nd edn. ( Springer-Verlag , 1997 ) .

[8] O. Bratteli and D. W. Robinson , Operator Algebras and Quantum Statistical Mechanics, 1 , 2nd edn. ( Springer-Verlag , 1987 ) .

[9] H. J. Carmichael, J. Opt. Soc. Am. B 4, 1588 (1987), DOI: 10.1364/JOSAB.4.001588 .

[10] A. Chebotarev , Lectures on Quantum Probability , Aportaciones Matem. S.M.M 4 ( 2000 ) .

[11] A. Chebotarev and F. Fagnola, J. Funct. Anal. 153, 382 (1998), DOI: 10.1006/jfan.1997.3189 .

[12] E. Christensen and D. E. Evans, J. Lon. Math. Soc. 20, 358 (1979).

[13] E. B. Davies, Rep. Math. Phys. 11, 169 (1977), DOI: 10.1016/0034-4877(77)90059-3 .

[14] C. Dellacherie and P.-A. Meyer , Probabilités et Potentiel I-III ( Hermann , 1975 ) .

[15] L. M. Duan, Phys. Rev. Lett. 85, 3991 (2000), DOI: 10.1103/PhysRevLett.85.3991 .

[16] R. Rebolledo, F. Fagnola and C. Saavedra, J. Math. Phys. 35, 1 (1994).

[17] F. Fagnola, Proyecciones, J. Math. 18, 1 (1999).

[18] F. Fagnola, R. Rebolledo and C. Saavedra, Stochastic Analysis and Mathematical Physics (World Scientific, 1998) pp. 61-71.

[19] F. Fagnola and M. Skeide, Noncommutative Harmonic Analysis with Applications to Probability, Banach Center Publ 78 (Polish Acad. Sci., 2007) pp. 121-132.

[20] C. Gardiner, Phys. Rev. Lett. 56, 1917 (1986), DOI: 10.1103/PhysRevLett.56.1917 .

[21] G. Lindblad, Comm. Math. Phys. 48, 119 (1976), DOI: 10.1007/BF01608499 .

[22] C. M. Mora and R. Rebolledo, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10, 237 (2007), DOI: 10.1142/S0219025707002725 .

[23] C. M. Mora and R. Rebolledo, Ann. Appl. Probab. 18, 591 (2008), DOI: 10.1214/105051607000000311 .

[24] K. R. Parthasarathy , An Introduction to Quantum Stochastic Calculus , Monographs in Mathematics 85 ( Birkhaüser-Verlag , 1992 ) .

[25] G. Pedersen , Analysis Now , Graduate Texts in Mathematics 118 ( Springer-Verlag , 1989 ) .

[26] R. Rebolledo and D. Spehner, Stochastic Analysis in Mathematical Physics (World Scientific, 2008) pp. 94-108.

[27] R. Rebolledo, Ann. Inst. H. Poincaré Probab. Statist. 41, 349 (2005), DOI: 10.1016/j.anihpb.2004.12.003 .

[28] W. F. Stinespring, Proc. Am. Math. Soc. 6, 211 (1955), DOI: 10.2307/2032342 .

[29] A. Kossakowski, V. Gorini and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976).

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