Abstract
We characterize generators ℒ of norm continuous quantum Markov semigroups satisfying the quantum detailed balance condition with respect to an antiunitary time reversal in terms of the operators H and L k in the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) representation ℒ(x) = i[H, x] − (1/2)gZ k (L* k L k x − 2L* k x L k + x L* k L k ).
Similar content being viewed by others
References
V. Gorini, A. Kossakowski and E. C. G. Sudarshan, “Completely positive dynamical semigroups of N-level systems,” J. Math. Phys. 17(5), 821–825 (1976).
G. Lindblad, “On the genarators of quantum dynamical semigroups,” Comm. Math. Phys. 48(2), 119–130 (1976).
R. Alicki, “On the detailed balance condition for non-hamiltonian systems,” Rep. Math. Phys. 10(2), 249–258 (1976).
R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, in Lecture Notes in Phys. (Springer-Verlag, Berlin, 1987), Vol. 286.
A. Frigerio and V. Gorini, “Markov dilations and quantum detailed balance,” Comm. Math. Phys. 93(4), 517–532 (1984).
A. Kossakowski, A. Frigerio, V. Gorini and M. Verri, “Quantum detailed balance and KMS condition,” Comm. Math. Phys. 57(2), 97–110 (1977).
F. Fagnola and V. Umanità, “Generators of detailed balance quantum Markov semigroups,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10(3), 335–363 (2007).
W. A. Majewski, “On the relationship between the reversibility of detailed balance conditions,” Ann. Inst. H. Poincaré Sec. A (N. S.) 39(1), 45–54 (1983).
W. A. Majewski, “The detailed balance condition in quantum statistical mechanics,” J. Math. Phys. 25(3), 614–616 (1984).
P. Talkner, “The failure of the quantum regression hypotesis,” Ann. Physics 167(2), 390–436 (1986).
G. S. Agarwal, “Open quantum Markovian systems and the microreversibility,” Z. Physik 258(5), 409–422 (1973).
L. Accardi and K. Imafuku, “Dynamical detailed balance and local KMS condition for non-equilibrium states,” Internat. J. Modern Phys. B 18(4–5), 435–467 (2004).
F. Fagnola and R. Quezada, “Two-photon absorption and emission process,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8(4), 573–591 (2005).
R. Carbone, F. Fagnola and S. Hachicha, “Generic quantum Markov semigroups: The Gaussian gauge invariant case,” Open Syst. Inf. Dyn. 14(4), 425–444 (2007).
L. Accardi, F. Fagnola and S. Hachicha, “Generic q-Markov semigroups and speed of convergence of q-algorithms,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9(4), 567–594 (2006).
K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, in Monogr. Math. (Birkhäuser-Verlag, Basel, 1992), Vol. 85.
S. Goldstein and J.M. Lindsay, “KMS-symmetric Markov semigroups,” Math. Z. 219(4), 591–608 (1995).
F. Cipriani, “Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebras,” J. Funct. Anal. 147(2), 259–300 (1997).
L. Accardi and A. Mohari, “Time Reflected Markov Processes,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2(3), 397–425 (1999).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Russian in Matematicheskie Zametki, 2008, Vol. 84, No. 1, pp. 108–116.
The text was submitted by the authors in English.
Rights and permissions
About this article
Cite this article
Fagnola, F., Umanità, V. Detailed balance, time reversal, and generators of quantum Markov semigroups. Math Notes 84, 108–115 (2008). https://doi.org/10.1134/S0001434608070092
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434608070092