  We study the relation between the quantum conditional mutual information and
the quantum $\alpha$-R\'enyi divergences. Considering the totally antisymmetric
state we show that it is not possible to attain a proper generalization of the
quantum conditional mutual information by optimizing the distance in terms of
quantum $\alpha$-R\'enyi divergences over the set of all Markov states. The
failure of the approach considered arises from the observation that a small
quantum conditional mutual information does not imply that the state is close
to a quantum Markov state.
