  We define correlational (von Neumann) entropy for an individual quantum state
of a system whose time-independent hamiltonian contains random parameters and
is treated as a member of a statistical ensemble. This entropy is
representation independent and can be calculated as a trace functional of the
density matrix which describes the system in its interaction with the noise
source. We analyze perturbation theory in order to show the evolution from the
pure state to the mixed one. Exactly solvable examples illustrate the use of
correlational entropy as a measure of the degree of complexity in comparison
with other available suggestions such as basis-dependent information entropy.
It is shown in particular that a harmonic oscillator in a uniform field of
random strength comes to a quasithermal equilibrium; we discuss the relation
between effective temperature and canonical equilibrium temperature. The notion
of correlational entropy is applied to a realistic numerical caculation in the
framework of the nuclear shell model. In this system, which reveals generic
signatures of quantum chaos, correlational entropy and information entropy
calculated in the mean field basis display similar qualitative behavior.
