Monotonic multi-state quantum <i>f</i>-divergences

We use the Tomita-Takesaki modular theory and Kubo-Ando operator mean to write down a large class of multi-state quantum $f$-divergences prove that they satisfy data processing inequality. For two states, this includes $(\alpha,z)$-R\'enyi divergences, Petz, measures in \cite{matsumoto2015new} as special cases. The method used is interpolation non-commutative $L^p_\omega$ spaces result applies general von Neumann algebras including local algebra field theory. conjecture these R\'enyi divergences have operational interpretations terms optimal error probabilities asymmetric state discrimination.