Uniform continuity bounds for information characteristics of quantum channels depending on input dimension and on input energy

We obtain continuity bounds for basic information characteristics of quantum channels depending on their input dimension (if it is finite) and on the input energy bound (if the input dimension is infinite). We pay special attention to the case in which a multimode quantum oscillator is an input system. First, we prove continuity bounds for the output conditional mutual information for a single channel and for n copies of a channel. Then we obtain estimates for the variation of the output Holevo quantity with respect to simultaneous variations of a channel and of an input ensemble. As a result, tight and close-to-tight continuity bounds for basic capacities of quantum channels, which depend on the input dimension, arc obtained. They complement the Leung-Smith continuity bounds, which depend on the output dimension. Finally, we obtain tight and close-to-tight continuity bounds for basic capacities of infinite-dimensional energy-constrained channels with respect to the energy-constrained Bures distance generating the strong convergence of quantum channels.