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In quantum information theory, quantum discord is a measure of nonclassical correlations between two subsystems of a quantum system. It includes correlations that are due to quantum physical effects but do not necessarily involve quantum entanglement.

The notion of quantum discord was introduced by Harold Ollivier and Wojciech H. Zurek and, independently by L. Henderson and Vlatko Vedral. Olliver and Zurek referred to it also as a measure of quantumness of correlations. From the work of these two research groups it follows that quantum correlations can be present in certain mixed separable states; In other words, separability alone does not imply the absence of quantum effects. The notion of quantum discord thus goes beyond the distinction which had been made earlier between entangled versus separable (non-entangled) quantum states.

Definition and mathematical relations

Individual (H(X),H(Y)), joint (H(X,Y)), and conditional entropies for a pair of correlated subsystems X,Y with mutual information I(X; Y).

In mathematical terms, quantum discord is defined in terms of the quantum mutual information. More specifically, quantum discord is the difference between two expressions which each, in the classical limit, represent the mutual information. These two expressions are:

$I(A;B)=H(A)+H(B)-H(A,B)$

$J(A;B)=H(A)-H(A|B)$

where, in the classical case, $H(A)$ is the information entropy, $H(A,B)$ the joint entropy and $H(A|B)$ the conditional entropy, and the two expressions yield identical results. In the nonclassical case, the quantum physics analogy for the three terms are used – $S(\rho _{A})$ the von Neumann entropy, $S(\rho )$ the joint quantum entropy and $S(\rho _{A}|\rho _{B})$ the conditional quantum entropy, respectively, for probability density function $\rho$

$I(\rho )=S(\rho _{A})+S(\rho _{B})-S(\rho )$

$J_{A}(\rho )=S(\rho _{B})-S(\rho _{B}|\rho _{A})$

The difference between the two expressions $I(\rho )-J_{A}(\rho )$ defines the basis-dependent quantum discord, which is asymmetrical in the sense that ${\mathcal {D}}_{A}(\rho )$ can differ from ${\mathcal {D}}_{B}(\rho )$. $J$ represents the part of the correlations that can be attributed to classical correlations and varies in dependence on the chosen eigenbasis; therefore, in order for the quantum discord to reflect the purely nonclassical correlations independently of basis, it is necessary that $J$ first be maximized over the set of all possible projective measurements onto the eigenbasis:

${\mathcal {D}}_{A}(\rho )=I(\rho )-\max _{\{\Pi _{j}^{A}\}}J_{\{\Pi _{j}^{A}\}}(\rho )=S(\rho _{A})-S(\rho )+\min _{\{\Pi _{j}^{A}\}}S(\rho _{B|\{\Pi _{j}^{A}\}})$

Nonzero quantum discord indicates the presence of correlations that are due to noncommutativity of quantum operators. For pure states, the quantum discord becomes a measure of quantum entanglement, more specifically, in that case it equals the entropy of entanglement.

Vanishing quantum discord is a criterion for the pointer states, which constitute preferred effectively classical states of a system. It could be shown that quantum discord must be non-negative and that states with vanishing quantum discord can in fact be identified with pointer states. Other conditions have been identified which can be seen in analogy to the Peres–Horodecki criterion and in relation to the strong subadditivity of the von Neumann entropy.

Efforts have been made to extend the definition of quantum discord to continuous variable systems, in particular to bipartite systems described by Gaussian states.

Properties

Zurek provided a physical interpretation for discord by showing that it "determines the difference between the efficiency of quantum and classical Maxwell’s demons...in extracting work from collections of correlated quantum systems".

Discord can also be viewed In operational terms as an "entanglement consumption in an extended quantum state merging protocol". Providing evidence for non-entanglement quantum correlations normally involves elaborate quantum tomography methods; however, in 2011, such correlations could be demonstrated experimentally in a room temperature nuclear magnetic resonance system, using chloroform molecules that represent a two-qubit quantum system.

Quantum discord has been seen as a possible basis for the performance in terms of quantum computation ascribed to certain mixed-state quantum systems, with a mixed quantum state representing a statistical ensemble of pure states (see quantum statistical mechanics).

Evidence has been provided for poignant differences between the properties of quantum entanglement and quantum discord. It has been shown that quantum discord is more resilient to dissipative environments than is quantum entanglement. This has been shown for Markovian environments as well as for non-Markovian environments based on a comparison of the dynamics of discord with that of concurrence, where discord has proven to be more robust. It has been shown that, at least for certain models of a qubit pair which is in thermal equilibrium and form an open quantum system in contact with a heat bath, the quantum discord increases with temperature in certain temperature ranges, thus displaying a behaviour that is quite in contrast with that of entanglement, and that furthermore, surprisingly, the classical correlation actually decreases as the quantum discord increases. Nonzero quantum discord can persist even in the limit of one of the subsystems undergoing an infinite acceleration, whereas under this condition the quantum entanglement drops to zero due to the Unruh effect.

Alternative measures

An operational measure, in terms of distillation of local pure states, the ‘quantum deficit’. The one-way and zero-way versions were shown to be equal to the relative entropy of quantumness.

Other measures of nonclassical correlations include the measurement induced disturbance (MID) measure and the localized noneffective unitary (LNU) distance and various entropy-based measures.

There exists a geometric measure of discord, which obeys a factorization law, can be put in relation to von Neumann measurements, and a measure of ‘measurement-induced nonlocality’ (MIN).

References

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