  We propose an approach to the study of quantum resource manipulation based on
the basic observation that quantum channels which preserve certain sets of
states are contractive with respect to the base norms induced by those sets. We
forgo the usual physical assumptions on quantum dynamics: instead of enforcing
complete positivity, trace preservation, or resource-theoretic considerations,
we study transformation protocols as norm-contractive maps. This allows us to
apply to this problem a technical toolset from functional analysis, unifying
previous approaches and introducing new families of bounds for the distillable
resources and the resource cost, both one-shot and asymptotic. Since our
expressions lend themselves naturally to single-letter forms, they can often be
calculated in practice; by doing so, we demonstrate with examples that they can
yield the best known bounds on quantities such as the entanglement cost. As
applications, we not only give an alternative derivation of the recent result
of [arXiv:2111.02438] which showed that entanglement theory is asymptotically
irreversible, but also provide the quantities introduced in that work with
explicit operational meaning in the context of entanglement distillation
through a novel generalised form of hypothesis testing relative entropy.
Besides entanglement, we reveal a new irreversible quantum resource: through
improved bounds for state transformations in the resource theory of magic-state
quantum computation, we show that there exist qutrit magic states that cannot
be reversibly interconverted under stabiliser protocols.
