How not to Renyi generalize the Quantum Conditional Mutual Information

We study the relation between the quantum conditional mutual information and the quantum $alpha$-Renyi 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 $alpha$-Renyi 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.