# What is quantum discord?

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?

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There are some progress in calculating discord for X states in 2 qubit bipartite systems. Ali, M., Rau, a. R. P., & Alber, G. (2010). Quantum discord for two-qubit X states. Physical Review A, 81(4), 042105. doi:10.1103/PhysRevA.81.042105 Luo, S. (2008). Quantum discord for two-qubit systems. Physical Review A, 77(4), 042303. doi:10.1103/PhysRevA.77.042303 –  Ars3nous Mar 15 '13 at 20:19

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).

Definition

(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.

1. Classically, one can define $H(A|B)$ as the average $H(A|B)=\sum_b {\mathcal P}(B=b)H(A|B=b)$, each $H(A|B=b)$ being the entropy of $A$ given that the random variable has the value $b$. If one generalizes this to the quantum world, the $B=b$ part implies a quantum measurement (a POVM) which should be specified. A natural choice is the "best" measurement, the one which minimizes the entropy. The Shannon $H$ entropy is replaced by the Von Neumann entropy, and we define $S(A|B_c)=\min_{\text{POVM}} \sum_{b}\mathcal{P}(\text{POVM applied to B gives } b) S(A|\text{POVM applied to B gives }b)$.

2. The previous definition leads classically to a redefinition of the conditional entropy as an entropy difference : $H(A|B)=H(A,B)-H(B)$, which is always positive. Its quantum version, $S(A|B)=S(AB)-S(B)$ can be negative (in contrast with $S(A|B_c)$). Its negativity is a sufficient condition for entanglement.

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 $B$.

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.