Remarks concerning an Entropy-like Quantity $S_q$ especially for Quantum   Systems with a Particle Picture; Model with pure Point Spectra
  For quantum mechanical systems an entropy-like quantity $S_q$ is defined. $S_q$ can differ from the usually defined entropy $S$ and $S_q$ may increase with time for an isolated system. The essential condition for the difference between $S$ and $S_q$ is the assumption that the set {\bf A} of observables which can be represented by a measurement is a proper subset of the set of selfadjoint operators. The underlying idea is made visible in the case of particle systems with non-trivial scattering. The model-character of the reasoning comes from the fact that continuous spectra are replaced by point-spectra. So it seems evident, that no direct connection exists between $S_q$ and the Sinai-Kolmogorov-Entropy at least in this model with pure point-spectra. 
S.Schlieder