  A proof of the quantum $H$-theorem taking into account nonextensive effects
on the quantum entropy $S^Q_q$ is shown. The positiveness of the time variation
of $S^Q_q$ combined with a duality transformation implies that the nonextensive
parameter $q$ lies in the interval [0,2]. It is also shown that the equilibrium
states are described by quantum $q$-power law extensions of the Fermi-Dirac and
Bose-Einstein distributions. Such results reduce to the standard ones in the
extensive limit, thereby showing that the nonextensive entropic framework can
be harmonized with the quantum distributions contained in the quantum
statistics theory.
