On quantum conditional entropies defined in terms of the $f$-divergences

We consider a family of quantum conditional entropies based on the concept of quantum $f$-divergences. First, we explicitly formulate conditions under which the notion of quantum conditional entropy is well defined in this way. In particular, we demand that the value of conditional entropy be independent of any extension of the principal Hilbert space. Using fundamental properties of the quantum $f$-divergences, several interesting relations for such conditional entropies are formulated. We separately examine an especially important case of quantum conditional entropies related to the Tsallis divergence.