  Catalysts are substances that assist transformation of other resourceful
objects without being consumed in the process. However, the fact that their
`catalytic power' is limited and can be depleted is often overlooked,
especially in the recently developing theories on catalysis of quantum
randomness utilizing building correlation with catalyst. In this work, we
establish a resource theory of one-shot catalytic randomness in which
uncorrelatedness is consumed in catalysis of randomness. We do so by completely
characterizing bipartite unitary operators that can be used to implement
catalysis of randomness using partial transpose. By doing so, we find that
every catalytic channel is factorizable, and therefore there exists a unital
channel that is not catalytic. We define a family of catalytic entropies that
quantifies catalytically extractable entropy within a quantum state and show
how much degeneracy of quantum state can boost the catalytic entropy beyond its
ordinary entropy. Based on this, we demonstrate that a randomness source can be
actually exhausted after a certain amount of randomness is extracted. We apply
this theory to systems under conservation law that forbids superposition of
certain quantum states and find that non-maximally mixed states can yield the
maximal catalytic entropy. We discuss implications of this theory to various
topics including catalytic randomness absorption, the no-secret theorem and the
possibility of multi-party infinite catalysis.
