  We present a generalization of the Holevo theorem by means of distances used
in the definition of distinguishability of states, showing that each one leads
to an alternative Holevo theorem. This result involves two quantities: the
generalized Holevo information and the generalized accessible information.
Additionally, we apply the new inequalities to qubits ensembles showing that
for the Kolmogorov notion of distinguishability (for the case of an ensemble of
two qubits) the generalized quantities are equal. On the other hand, by using a
known example, we show that the Bhattacharyya notion captures not only the
non-commutativity of the ensemble but also its purity.
