  Non-equilibrium states of quantum systems in contact with thermal baths help
telling environments with different temperatures or different statistics apart.
We extend these studies to a more generic problem that consists in
discriminating between two baths with disparate constituents at unequal
temperatures. Notably there exist temperature regimes in which the presence of
coherence in the initial state preparation is beneficial for the discrimination
capability. We also find that non-equilibrium states are not universally
optimal, and detail the conditions in which it becomes convenient to wait for
complete thermalisation of the probe. These concepts are illustrated in a
linear optical simulation.
