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Abstract
By extending the concept of energy-constrained diamond norms, we obtain
continuity bounds on the dynamics of both closed and open quantum systems in
infinite-dimensions, which are stronger than previously known bounds. We
extensively discuss applications of our theory to quantum speed limits,
attenuator and amplifier channels, the quantum Boltzmann equation, and quantum
Brownian motion. Next, we obtain explicit log-Lipschitz continuity bounds for
entropies of infinite-dimensional quantum systems, and classical capacities of
infinite-dimensional quantum channels under energy-constraints. These bounds
are determined by the high energy spectrum of the underlying Hamiltonian and
can be evaluated using Weyl's law.