  Fundamental properties for the Tsallis relative entropy in both classical and
quantum systems are studied. As one of our main results, we give the parametric
extension of the trace inequality between the quantum relative entropy and the
minus of the trace of the relative operator entropy given by Hiai and Petz. The
monotonicity of the quantum Tsallis relative entropy for the trace preserving
completely positive linear map is also shown without the assumption that the
density operators are invertible.
  The generalized Tsallis relative entropy is defined and its subadditivity is
shown by its joint convexity. Moreover, the generalized Peierls-Bogoliubov
inequality is also proven.
