  Thermal operations are an operational model of non-equilibrium quantum
thermodynamics. In the absence of coherence between energy levels, exact state
transition conditions under thermal operations are known in terms of a
mathematical relation called thermo-majorization. But incorporating coherence
has turned out to be challenging, even under the relatively tractable model
wherein all Gibbs state-preserving quantum channels are included. Here we find
a mathematical generalization of thermal operations at low temperatures,
"cooling maps", for which we derive the necessary and sufficient state
transition condition. Cooling maps that saturate recently-discovered bounds on
coherence transfer are realizable as thermal operations, motivating us to
conjecture that all cooling maps are thermal operations. Cooling maps, though a
less conservative generalization to thermal operations, are more tractable than
Gibbs-preserving operations, suggesting that cooling map-like models at general
temperatures could be of use in gaining insight about thermal operations.

Quantum Physics