  We consider a general resource theory that allows the use of free resource as
a catalyst. We show that the amount of `resource' contained in a given state,
in the asymptotic scenario, is equal to the regularized relative entropy of
resource of that state, which then yields a straightforward operational meaning
to this quantity. Such an answer has been long sought for in any resource
theory since the usefulness of a state in information-processing tasks is
directly related to the amount of resource the state possesses in the
beginning. While we need to place a few assumptions in our resource theoretical
framework, it is still general enough and includes quantum resource theory of
entanglement, coherence, asymmetry, non-uniformity, purity, contextuality,
stabilizer computation and the classical resource theory of randomness
extraction as special cases. Since our resource theoretic framework includes
entanglement theory, our result also implies that the amount of noise one has
to inject locally in order to erase all entanglement contained in an entangled
state is equal to the regularized relative entropy of entanglement, resolving
an open question posted in [Groisman et al., Phys. Rev. A. 72: 032317, 2005].
On the way to prove the main result, we also quantify the amount of resource
contained in a state in the one-shot setting (where one only has a single copy
of the state), in terms of the smooth max-relative entropy. Our one-shot result
employs a recently developed technique of convex-split lemma.
