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 Leah 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 earlier distinction between entangled and separable (non-entangled) quantum states.
Definition and mathematical relations
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 ) {\displaystyle I(A;B)=H(A)+H(B)-H(A,B)}
J ( A ; B ) = H ( A ) − H ( A | B ) {\displaystyle 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(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 density matrix (or density operator) ρ;
I ( ρ ) = S ( ρ A ) + S ( ρ B ) − S ( ρ ) {\displaystyle I(\rho )=S(\rho _{A})+S(\rho _{B})-S(\rho )}
J A ( ρ ) = S ( ρ B ) − S ( ρ B | ρ A ) {\displaystyle J_{A}(\rho )=S(\rho _{B})-S(\rho _{B}|\rho _{A})}
The difference between the two expressions defines the basis-dependent quantum discord
D A ( ρ ) = I ( ρ ) − J A ( ρ ) , {\displaystyle {\mathcal {D}}_{A}(\rho )=I(\rho )-J_{A}(\rho ),}
which is asymmetrical in the sense that D A ( ρ ) {\displaystyle {\mathcal {D}}_{A}(\rho )} can differ from D B ( ρ ) {\displaystyle {\mathcal {D}}_{B}(\rho )} . 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:
D A ( ρ ) = I ( ρ ) − max { Π j A } J { Π j A } ( ρ ) = S ( ρ A ) − S ( ρ ) + min { Π j A } S ( ρ B | { Π j A } ) {\displaystyle {\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}\}})}
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