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We prove that for any infinite-dimensional quantum channel the entropicdisturbance (defined as difference between the $\chi$-quantity of a generalizedensemble and that of the image of the ensemble under the channel) is lowersemicontinuous on the natural set of its definition. We establish a number ofuseful corollaries of this property, in particular, we prove the continuity ofthe output $\chi\textrm{-}$quantity and the existence of $\chi$-optimalensemble for any quantum channel under the energy-type input constraint.

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