What is quantum discord? I stumbled upon this term on Quantum Computing: The power of discord, but have never heard of it before. Can you give a bit more mathematical explanation of the term here?
It is basically a measure of the quantumness of some correlations, which is not vanishing for some separable state. It was introduced by Ollivier and Zurek (PRL/arXiv). It is the difference between two different generalizations of the classical (Shannon) conditional entropy to the quantum world, and is 0 for a pure bipartite separable state. It has been proven to be the amount of entanglement needed in the task of state-merging (PRA/arXiv and PRA/arXiv).
(PRL/arXiv) Classically the conditional entropy $H(A|B)$ is a measure of the uncertainty one has on the variable $A$ once we know the variable $B$. Of course, the definition of "knowing" $B$ becomes problematic when $B$ is quantum.
The discord is defined as $S(A|B_v)-S(A|B)$ and is always positive. You can maybe see it as the amount of correlation between $A$ and $B$ which is destroyed by a classical measurement of $B$.
Link with state merging
The state merging primitive is the following. Suppose Alice, Bob and Charly share a 3-party pure entangled state. Alice want to send her part to Bob without destroying the quantum correlations between $AB$ and $C$. Basically, she has to teleport $A$ to Bob, and the minimal amount of entanglement Alice and Bob need to perform this task is given by the quantum discord.