  The class of possible thermodynamic conversions can be extended by
introducing an auxiliary system called catalyst, which assists state conversion
while remaining its own state unchanged. We reveal a complete characterization
of catalytic state conversion in quantum and single-shot thermodynamics by
allowing an infinitesimal correlation between the system and the catalyst.
Specifically, we prove that a single thermodynamic potential, which provides
the necessary and sufficient condition for the correlated-catalytic state
conversion, is given by the standard nonequilibrium free energy defined with
the Kullback-Leibler divergence. This resolves the conjecture raised by
Wilming, Gallego, and Eisert [Entropy 19, 241 (2017)] and by Lostaglio and
Muller [Phys. Rev. Lett. 123, 020403 (2019)] in positive. Moreover, we show
that, with the aid of the work storage, any quantum state can be converted into
another one by paying the work cost equal to the difference of the
nonequilibrium free energy. Our result would serve as a step towards
establishing resource theories of catalytic state conversion in the fully
quantum regime.
