Criteria for equality in two entropic inequalities

We obtain a simple criterion for local equality between the constrained Holevo capacity and the quantum mutual information of a quantum channel. This shows that the set of all states for which this equality holds is determined by the kernel of the channel (as a linear map).
Applications to Bosonic Gaussian channels are considered. It is shown that for a Gaussian channel having no completely depolarizing components the above characteristics may coincide only at non-Gaussian mixed states and a criterion for the existence of such states is given.
All the obtained results may be reformulated as conditions for equality between the constrained Holevo capacity of a quantum channel and the input von Neumann entropy.
Bibliography: 20 titles.