  Information is instrumental in our understanding of thermodynamics. Their
interplay has been studied through completely degenerate Hamiltonians whereby
the informational contributions to thermodynamic transformations can be
isolated. In this setting, all states other then the maximally mixed state are
considered to be in informational non-equilibrium. An important yet still open
question is: how to characterise the ability of quantum dynamics to maintain
informational non-equilibrium? Here, the dynamical resource theory of
informational non-equilibrium preservability is introduced to begin providing
an answer to this question. A characterisation of the allowed operations is
given for qubit channels and the n dimensional Weyl-covariant channels - a
physically relevant subset of the general channels. An operational
interpretation of a state discrimination game with Bell state measurements is
given. Finally, an explicit link between a channels classical capacity and its
ability to maintain informational non-equilibrium is made.
