  Quantum catalysis, the ability to enable previously impossible
transformations by using auxiliary systems without degrading them, has emerged
as a powerful tool in various resource theories. Although catalytically enabled
state transformations have been formally characterized by the monotonic
behaviour of entropic quantifiers (e.g., the von Neumann entropy or
non-equilibrium free energy), such characterizations often rely on unphysical
assumptions, namely the ability of using catalysts of infinitely large
dimension. This approach offers very limited insights into the practical
significance of using catalysis for quantum information processing. Here, we
address this problem across a broad class of quantum resource theories.
Leveraging quantum information tools beyond the asymptotic regime, we establish
sufficient conditions for the existence of catalytic transformations with
finite-size catalysts. We further unveil connections between finite-size
catalysis and multi-copy transformations. Notably, we discover a fascinating
phenomenon of catalytic resonance: by carefully tailoring the catalysts's
state, one can drastically reduce the required dimension of the catalyst, thus
enabling efficient catalytic transformations with minimal resources. Finally,
we illustrate our findings with examples from the resource theories of
entanglement and thermodynamics, as well in the context of catalytic unitary
transformations.
