  I show that non-decreasing entropy provides a necessary and sufficient
condition to convert the state of a physical system into a different state by a
reversible transformation that acts on the system of interest and a further
"catalyst" whose state has to remain invariant exactly in the transition. This
statement is proven both in the case of finite-dimensional quantum mechanics,
where von~Neumann entropy is the relevant entropy, and in the case of systems
whose states are described by probability distributions on finite sample
spaces, where Shannon entropy is the relevant entropy. The results give an
affirmative resolution to the (approximate) "catalytic entropy conjecture"
introduced by Boes et al. [PRL 122, 210402 (2019)]. They provide a complete
single-shot characterization without external randomness of von Neumann entropy
and Shannon entropy. I also compare the results to the setting of
phenomenological thermodynamics and show how they can be used to obtain a
quantitative single-shot characterization of Gibbs states in quantum
statistical mechanics.
