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The channels, and more generally superoperators acting on the trace classoperators of a quantum system naturally form a Banach space under thecompletely bounded trace norm (aka diamond norm). However, it is well-knownthat in infinite dimension, the norm topology is often "too strong" forreasonable applications. Here, we explore a recently introducedenergy-constrained diamond norm on superoperators (subject to an energy boundon the input states). Our main motivation is the continuity of capacities andother entropic quantities of quantum channels, but we also present anapplication to the continuity of one-parameter unitary groups and certainone-parameter semigroups of quantum channels.

Energy-constrained diamond norm with applications to the uniform continuity of continuous variable channel capacities