# 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.

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## 2 Answers

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!

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Nice answer, Earl. My initial thinking was about restricting operations in entanglement resource theory. Since the CC lets us prepare arbitrary convex combinations of states, if we deny ourselves this, we can deny ourselves the ability to go from concordant to discordant. Personally, I am not really convinced of discord in the way I am convinced of entanglement. However, I want to add some more substance to this ill-feeling. It seems contrived to deny ourselves classical communication though. – Matty Hoban Oct 27 '11 at 20:53
Another point worth mentioning is that quantum discord came to prominence, at least to me, in an analysis of the "Power of 1 qubit model".  In this model you have 1 pure qubit, unlimited mixed qubits and any unitary operations and you compute something that looks interesting (the trace of a unitary).  At some point in the middle of the computation the state has discord.  I would argue that in this model the relevant resource is purity, not discord. – Earl Oct 28 '11 at 13:25
Discord is a manifestation of coherence so I can see your reasoning. I think I will accept your answer as it is a fair comment. I think my question was suitably open ended that I'd accept any worthwhile and informative answer. – Matty Hoban Oct 28 '11 at 20:37
I'm not sure non-convexity is a problem.  Generally, a resource theory arises from restrictions on the permitted operations.  Is there a corresponding characterization for discord, such as LOCC for entanglement? (I.e.: Discord cannot be created with restriction X, and it can be used to overcome the restriction?) Then I would think it should be possible to obtain a resource theory for discord. (Although the restriction, and thus the resource theory, might be less natural than one arising from LOCC restrictions.) – Norbert Schuch Oct 30 '11 at 19:56

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.

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Thanks for providing a nice answer, Animesh. I was hoping you'd turn up and provide some input. I agree that essentially any resource theory by definition emerges from a restricted set of operations. I was thinking that the set of operations could be the one that Eastin describes, but can we still not produce discord with a larger set of operations. If we have stochastic local operations but no classical communication, can we produce discord? This may seem a silly question with an obvious answer. – Matty Hoban Nov 2 '11 at 14:02
+1 from me. Welcome to the site Animesh. – Joe Fitzsimons Nov 2 '11 at 14:23
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