  We study asymptotic state transformations in continuous variable quantum
resource theories. In particular, we prove that monotones displaying lower
semicontinuity and strong superadditivity can be used to bound asymptotic
transformation rates in these settings. This removes the need for asymptotic
continuity, which cannot be defined in the traditional sense for
infinite-dimensional systems. We consider three applications, to the resource
theories of (I) optical nonclassicality, (II) entanglement, and (III) quantum
thermodynamics. In cases (II) and (III), the employed monotones are the
(infinite-dimensional) squashed entanglement and the free energy, respectively.
For case (I), we consider the measured relative entropy of nonclassicality and
prove it to be lower semicontinuous and strongly superadditive. One of our main
technical contributions, and a key tool to establish these results, is a handy
variational expression for the measured relative entropy of nonclassicality.
Our technique then yields computable upper bounds on asymptotic transformation
rates, including those achievable under linear optical elements. We also prove
a number of results which guarantee that the measured relative entropy of
nonclassicality is bounded on any physically meaningful state and easily
computable for some classes of states of interest, e.g., Fock diagonal states.
We conclude by applying our findings to the problem of cat state manipulation
and noisy Fock state purification.
