  Resource theory is a general, model-independent approach aiming to understand
the qualitative notion of resource quantitatively. In a given resource theory,
free operations are physical processes that do not create the resource and are
considered zero-cost. This brings the following natural question: For a given
free operation, what is its ability to preserve a resource? We axiomatically
formulate this ability as the resource preservability, which is constructed as
a channel resource theory induced by a state resource theory. We provide two
general classes of resource preservability monotones: One is based on state
resource monotones, and another is based on channel distance measures.
Specifically, the latter gives the robustness monotone, which has been recently
found to have an operational interpretation. As examples, we show that
athermality preservability of a Gibbs-preserving channel can be related to the
smallest bath size needed to thermalize all its outputs, and it also bounds the
capacity of a classical communication scenario under certain thermodynamic
constraints. We further apply our theory to the study of entanglement
preserving local thermalization (EPLT) and provide a new family of EPLT which
admits arbitrarily small nonzero entanglement preservability and free
entanglement preservation at the same time. Our results give the first
systematic and general formulation of the resource preservation character of
free operations.
