Lower semicontinuity of the entropic disturbance and its applications in   quantum information theory
  We prove that for any infinite-dimensional quantum channel the entropic disturbance (defined as difference between the $\chi$-quantity of a generalized ensemble and that of the image of the ensemble under the channel) is lower semicontinuous on the natural set of its definition. We establish a number of useful corollaries of this property, in particular, we prove the continuity of the output $\chi\textrm{-}$quantity and the existence of $\chi$-optimal ensemble for any quantum channel under the energy-type input constraint. 
M.E. Shirokov, A.S. Holevo
quant-ph math-ph math.MP