A resource theory of quantum discord? Local Operations and Classical Communication (LOCC) is the classic paradigm for studying entanglement. These are things that are `cheap' and unable to produce entanglement as a resource for a quantum information processing task. We can also describe equivalence classes of entangled states if elements of each class can be transformed to another in that class under LOCC. We can discuss entanglement distillation of going from M copies of a noisy state to N copies of a more entangled state by LOCC. Finally, if some states are undistillable (i.e. N=0 for all M), given a different state $\sigma$, the original noisy state $\rho$ can be activated (or catalysed if you want $\sigma$ to be unchanged) this to a more entangled state. 
Recently, a lot of discussion has centred around quantum discord. Quantum discord aims to capture nonclassicality in states, if not necessarily entanglement. Loosely speaking, a quantum state $\rho$ without discord (concordant) is one where there is a basis of product states (e.g. $|\psi_{1}\rangle|\psi_{j}\rangle...|\psi_{n}\rangle$ for $n$ parties) with respect to which $\rho$ is diagonal. Discord (but not entanglement) has been related to mixed state quantum computing as well as quantum state merging.
Interestingly, it has been shown that given two (non-equal) concordant states, there exists a protocol that produces a distillable entangled state as shown by M. Piani et al and some similar results in A. Streltsov et al. I am curious at how far this analogy between entanglement and `nonclassicalness' distillation can be drawn, in particular, can we construct a reasonable resource theory of discord? I doubt I am the first to think of this so if anyone has any background on this, I'd really appreciate it.
We can restrict to being able to produce concordant states and then operations that preserve classicalness. From a paper by B Eastin we know that unitaries that preserve classicalness amount to a permutation of eigenvalues with a change in product basis; we could go beyond the model of local operations. Has anyone produced any results on distillation of discord? 
If this is all trivial to some of you, my apologies. I am trying to understand what discord actually means from the useful resource theoretic point-of-view.
 A: I think the crucial point is the one Norbert has raised, and the question about the set of operations that do not increase discord has not been characterised. Of course, local unitaries don't change discord, but that is a trivial set. There are a couple of papers interpreting quantum discord as a resource in terms of quantum state merging, and more generally, in terms of the mother protocol of quantum information theory. One can also draw some thermodynamic connections, but they are not fully formalised yet.
A: By coincidence I was thinking about exactly this problem myself... actually thinking about why I think it is not a good example of a resource theory.  The basic reason is that the state of states with zero discord is not convex, and so not closed under mixing!  Take 2 zero-discord states which are diagonal in different bases, mix them, and you have a state with positive discord. Since classical mixing is an that operation always available in the lab, is seems difficult to see how to make a resource theory.  In fairness one must all note that the resource theory of non-gaussian states is also non-convex, though often this is fixed up by thinking about a resource theory of continuous variable states where the resource is negativity of the Wigner function (this is a convex resource theory).  All other developed resource theories I can think of have a convex structure!
Bit of a short answer but that is my opinion on the matter!
