The evolution (in time) of an open quantum system usually tends to
an ultimate steady state. In such a case the ``natural" (often
called ``von Neumann's") entropy does not, in general, offer any
``measure of distance from the equilibrium" (since it does not
depend monotonically on time) while the Jakob's and Stenholm's one
(cf. refs. [10 - 13]) does. The present authors amend this result
(1) by the rigorous proofs to some technicalities (e.g., of the
pseudo-Hermiticity (i.e., basically, Krein-space Hermiticity) of the
Liouvillean for any (finite-dimensional) Hilbert space of density
matrices, or of the necessary and sufficient conditions of
positivity of similar maps), (2) by the extension of attention to
the so called quasi-Hermitian Hamiltonians (which were recognized as
fully compatible with the standard quantum theory by Scholtz, Geyer
and Hahne in 1992, cf. ref. [16]; the present reviewer would prefer
calling them cryptohermitian: cf. SIGMA 5 (2009) 001) and, finally,
(3) by the introduction of the two new Lyapunov entropy-mimicking
functionals covering also the degenerate cases.


MR2735604 Masillo, F.; Solombrino, L.; Scolarici, G. Time evolution
of quasi-Hermitian open systems and generalized entropy functional.
Int. J. Geom. Methods Mod. Phys. 7 (2010), no. 6, 1001--1020. 82C10
(81S22)