page_title: Quantum discord
section_title: Definition and mathematical relations
hierarchical_section_title: Quantum discord / Definition and mathematical relations
context_page_description: 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 correlations. The notion of quantum discord thus goes beyond the distinction which had been made earlier between entangled versus separable quantum states.
context_section_description: 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:
  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(ρA) the von Neumann entropy, S(ρ) the joint quantum entropy and S(ρA|ρB) a quantum generalization of conditional entropy (not to be confused with conditional quantum entropy), respectively, for probability density function ρ;
  The difference between the two expressions [I(ρ) − JA(ρ)] defines the basis-dependent quantum discord, which is asymmetrical in the sense that  can differ from . The notation 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:
  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. A very recent work has demonstrated that the upper-bound of Gaussian discord indeed coincides with the actual quantum discord of a Gaussian state, when the latter belongs to a suitable large family of Gaussian states.
  Computing quantum discord is NP-complete and hence difficult to compute in the general case. For certain classes of two-qubit states, quantum discord can be calculated analytically.
caption: 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).