  We investigate the energy-constrained (EC) diamond norm distance between
unitary channels acting on possibly infinite-dimensional quantum systems, and
establish a number of results. Firstly, we prove that optimal EC discrimination
between two unitary channels does not require the use of any entanglement.
Extending a result by Ac\'in, we also show that a finite number of parallel
queries suffices to achieve zero error discrimination even in this EC setting.
Secondly, we employ EC diamond norms to study a novel type of quantum speed
limits, which apply to pairs of quantum dynamical semigroups. We expect these
results to be relevant for benchmarking internal dynamics of quantum devices.
Thirdly, we establish a version of the Solovay--Kitaev theorem that applies to
the group of Gaussian unitaries over a finite number of modes, with the
approximation error being measured with respect to the EC diamond norm relative
to the photon number Hamiltonian.
