Abstract
The fundamental concept of relative entropy is extended to a functional that is regular-valued also on arbitrary pairs of nonfaithful states of open quantum systems. This regularized version preserves almost all important properties of ordinary relative entropy such as joint convexity and contractivity under completely positive quantum dynamical semigroup time evolution. On this basis a generalized formula for entropy production is proposed, the applicability of which is tested in models of irreversible processes. The dynamics of the latter is determined by either Markovian or non-Markovian master equations and involves all types of states.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
S. Kullback, Information Theory and Statistics (Wiley, New York, 1959).
M. Nagasawa, Schrödinger Equations and Diffusion Theory (Birkhäuser, Basel, 1993).
A. Wehrl, Rev. Mod. Phys. 50:221 (1978).
W. Thirring, Lehrbuch der Theoretischen Physik, Vol. 4 (Springer, Vienna, 1980).
M. Ohya and D. Petz, Quantum Entropy and Its Use (Springer, Berlin, 1993).
E. H. Lieb and M. B. Ruskai, Phys. Rev. Lett. 30:434 (1973); J. Math. Phys. 14:1938 (1973).
R. Alicki, Rep. Math. Phys. 10:249 (1976); J. Stat. Phys. 20:671 (1979).
K. Lendi, Phys. Rev. A34:662 (1986).
R. Alicki, and K. Lendi, Quantum Dynamical Semigroups and Applications, Lecture Notes in Physics, Vol. 286 (Springer, Berlin, 1987).
H. Spohn, J. Math. Phys. 19:1227 (1978).
H. Spohn and J. L. Lebowitz, in Advances in Chemical Physics, Vol. 38, S. A. Rice, ed. (Wiley, New York, 1978).
K. Lendi, J. Stat. Phys. 50:1103 (1988).
R. Jost, Quantenmechanik II (Verlag der Fachvereine an der ETH/Z, Zürich, 1973).
R. T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970).
R. Webster, Convexity (Oxford University Press, Oxford, 1994).
E. H. Lieb, Adv. Math. 11:267 (1973).
G. Lindblad, Commun. Math. Phys. 39:111 (1974).
W. F. Stinespring, Proc. Am. Math. Soc. 6:911 (1955).
G. Lindblad, Commun. Math. Phys. 48:119 (1976).
G. Lindblad, Commun. Math. Phys. 40:147 (1975).
O. Klein, Z. Phys. 72:767 (1931).
G. S. Agarwal, Quantum Statistical Theories of Spontaneous Emission and their Relation to Other Approaches, Springer Tracts in Modern Physics, Vol. 70 (Springer, Berlin, 1974).
V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17:821 (1976).
K. Lendi, J. Phys. A: Math. Gen. 20:15 (1987).
F. Haake, in Springer Tracts in Modern Physics, Vol. 66, G. Hohler, ed. (Springer, Berlin, 1973).
A. J. van Wonderen and K. Lendi, J. Stat. Phys. 80:273 (1995).
S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, Amsterdam, 1962).
G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (Wiley, New York, 1977).
R. Alicki, J. Stat. Phys. 20:671 (1979).
D. Ruelle, J. Stat. Phys. 85:1 (1996).
W. Breymann, T. Tel, and J. Vollmer, Phys. Rev. Lett. 77:2945 (1996).
R. Alicki, J. Phys. A: Math. Gen. 12:L103 (1979).
F. Farhadmotamed, A. J. van Wonderen, and K. Lendi, J. Phys. A: Math. Gen. 31:3395 (1998).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lendi, K., Farhadmotamed, F. & van Wonderen, A.J. Regularization of Quantum Relative Entropy in Finite Dimensions and Application to Entropy Production. Journal of Statistical Physics 92, 1115–1135 (1998). https://doi.org/10.1023/A:1023004929346
Issue Date:
DOI: https://doi.org/10.1023/A:1023004929346