What is quantum discord?

Question

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?

Answer 1

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 $B$.

Link with state merging

(PRA/arXiv and PRA/arXiv)

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.

Answer 2

One way to put it is that quantum discord quantifies "correlations" that cannot be directly realised into correlations between measurement results. The presence of discord in a given bipartite quantum state signals that the two parties are more "linked together" than what might be observable via the correlations in the outcomes of any choice of local measurements. Another point of view is that quantum discord is related to situations in which measuring one part of the system necessarily disturbs the other side.

Given a bipartite state, one can ask about the maximum amount of correlations observable in local measurement oucomes. Call this the accessible information, which can thus be defined as $$J(\rho) \equiv \max_{\Pi^A} J(\rho|\{\Pi_i^A\}_i), \qquad J(\rho|\{\Pi_i^A\}_i)\equiv S(\operatorname{Tr}_A(\rho)) - S(\rho; B|\{\Pi^A_i\}_i),\\ S(\rho; B|\{\Pi^A_i\}_i) \equiv \sum_i p_i S(\rho(B|A=i)). $$ This looks nasty, so let's unpack:

1. $J(\rho)$ is the accessible information, calculated wrt measurements of $A$. One could also consider the analogous definition when $B$ is measured instead, and potentially obtain different numbers. This means one must be careful about the direction in which discord/accessible information are defined.
2. With $J(\rho|\{\Pi_i^A\}_i)$ I mean the accessible information with respect to a choice of measurement on $A$. Here, $\{\Pi_i^A\}_i$ is a projective measurement on $A$. The accessible information is obtained finding the measurement choice $\{\Pi^A_i\}_i$ that maximises $J(\rho|\{\Pi_i^A\}_i)$.
3. Once we fix a measurement choice $\{\Pi^A_i\}$, we use a definition of mutual information coming from the classical formula $I(X:Y)=H(A)-H(A|B)$. The conditional classical entropy $H(A|B)$ becomes the conditional entropy $S(\rho;B|\{\Pi_i^A\})$, which equals the average (von Neumann) entropy on $B$, conditioned to each measurement outcome of $\rho$ on $A$.
4. More explicitly, the conditional entropy reads $$S(\rho; B|\{\Pi^A_i\}_i) \equiv \sum_i p_i S(\rho(B|A=i)),$$ where $$p_i \equiv \operatorname{Tr}[(\Pi^A_i\otimes I_B)\rho], \qquad \rho(B|A=i) \equiv\frac{1}{p_i} \operatorname{Tr}_A[(\Pi^A_i\otimes I_B)\rho].$$ In words, $p_i$ is the probability of $A$ obtaining the outcome $i$ (again, when using the measurement $\{\Pi_i^A\}$, while $\rho(B|A=i)$ is the residual state on $B$ obtained when $A$ obtains the outcome $i$ (and tells $B$ about it).

The quantum discord $\delta$ is the part of the quantum mutual information $I(\rho)\equiv S(\rho_A)+S(\rho_B)-S(\rho)$, that is not realised as accessible mutual information, i.e. $$I(\rho) = J(\rho) + \delta(\rho).$$ Again, one has to be careful about which system is being measured, so more precisely the definition should be given specifying this information.

A nice characterisation

A bipartite state $\rho$ has zero discord wrt to measurements on $A$ iff it is a classical-quantum state, i.e. it admits a decomposition of the form $$\rho = \sum_i p_i \, |i\rangle\!\langle i|\otimes \rho_i,$$ for some orthonormal basis $\{|i\rangle\}$ and ensemble of states $\rho_i$.

A few examples

For an arbitrary pure state $\rho$, one has $I(\rho)=2J(\rho)$. In particular, the discord is symmetric, and equals the accessible information. For example, a maximally entangled two-qubit state has $I(\rho)=2$ and $J(\rho)=\delta(\rho)=1$.

Consider the two-qubit state $$\rho = \frac12 ( \mathbb{P}_0\otimes \mathbb{P}_0 + \mathbb{P}_1\otimes \mathbb{P}_+), \qquad \mathbb{P}_v\equiv |v\rangle\!\langle v|.$$ This is classical-quantum wrt measurements on $A$, thus has zero discord left-to-right. But it also has nonzero discord wrt measurements on $B$. Note how this is also a separable state, showcasing how discord is a form of nonclassicality that is different than entanglement.

For an example of a two-qubit separable discordant state that is not classical-quantum, one can consider the standard example given in Ollivier and Zurek's original paper: a Werner state of the form $$\rho_z = \frac{1-z}{4}I +z |\Phi^+\rangle\!\langle\Phi^+| =\frac14\begin{pmatrix}1+z & 0&0& 2z \\ 0& 1-z & 0 & 0 \\ 0&0&1-z&0 \\ 2z & 0 & 0 & 1+z\end{pmatrix}, \\ |\Phi^+\rangle\equiv\frac{1}{\sqrt2}(|00\rangle+|11\rangle),$$ which is separable for $z\le 1/3$ but has nonzero discord in both directions.