Energy-constrained diamond norms and their applications in quantum information theory

Energy-constrained diamond norms on the set of completely bounded linear maps on spaces of trace class operators are introduced. It is proved that these norms generate the strong convergence topology on the set of quantum channels and operations. They allow to obtain new continuity bounds for basic information characteristics of quantum channels depending only on the input dimension or on the mean energy of input states (if the input system is infinite-dimensional). Criterion for differentiability of quantum dynamical semigroups w.r.t. the energy-constrained diamond norms and sufficient conditions for their exponential representation uniformly converging on the set of states with bounded energy are obtained. Some open questions stated by different authors have been solved.