On equalities in two entropic inequalities

Understand

A simple criterion for local equality between the constrained Holevo capacity and the quantum mutual information of a quantum channel is obtained.

• It implies 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 of existence of such states is given.