  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.
