On lower semicontinuity of the entropic disturbance and its applications in quantum information theory

We consider an important characteristic of a quantum channel called the entropic disturbance. It is defined as the difference between the $\chi$-quantity of a generalized ensemble and that of the image of the ensemble under the channel. We prove the lower semicontinuity of the entropic disturbance for any infinite-dimensional quantum channel on its natural domain. A number of useful corollaries of this property are established, in particular, the existence of a $\chi$-optimal ensemble for any quantum channel and the continuity of the output $\chi$-quantity under the energy-type input constraint.