  Correspondence in quantum chaotic systems is lost in short time scales.
Introducing some noise we study the spectrum of the resulting coarse grained
propagaor of density matrices. Some differen methods to compute the spectrum
are reviewed. Moreover, the relationship between the eigenvalues of the
coarse-grained superoperator and the classical Ruelle-Pollicott resonances is
remarked. As a concequence, classical decay rates in quantum time dependent
quantities appear.
