Quantum discord, as introduced by Olliver and Zurek ͓Phys. Rev. Lett. 88, 017901 ͑2001͔͒, is a measure of the discrepancy between two natural yet different quantum analogs of the classical mutual information. This notion characterizes and quantifies quantumness of correlations in bipartite states from a measurement perspective, and is fundamentally different from the various entanglement measures in the entanglement vs separability paradigm. The phenomenon of nonzero quantum discord is a manifestation of quantum correlations due to noncommutativity rather than due to entanglement, and has interesting and significant applications in revealing the advantage of certain quantum tasks. We will evaluate analytically the quantum discord for a large family of two-qubit states, and make a comparative study of the relationships between classical and quantum correlations in terms of the quantum discord. We furthermore compare the quantum discord with the entanglement of formation, and illustrate that the latter may be larger than the former, although for separable states, the entanglement of formation always vanishes and thus is less than the quantum discord.

Dakic, Vedral, and Brukner [arXiv:1004.0190 (2010)] introduced a geometric measure of quantum discord and derived an explicit formula for any two-qubit state. This measure is significant in capturing quantum correlations from a geometric perspective. In this brief report, we evaluate the geometric measure of quantum discord for an arbitrary state and obtain an explicit and tight lower bound. Furthermore, we reveal an intrinsic feature of geometric measure of quantum discord by showing that it actually coincides with a simpler quantity based on von Neumann measurements.Correlations in quantum states, with fundamental applications and implications for quantum information processing, are usually studied in the entanglement-versus-separability framework [1,2]. However, entanglement, while widely regarded as nonlocal quantum correlations, is not the only kind of correlation. An alternative classification for correlations based on quantum measurements has arisen in recent years and also plays an important role in quantum information theory [3][4][5][6]. This is the quantum-versus-classical paradigm for correlations. In particular, the quantum discord as a measure of quantum correlations, initially introduced by Ollivier and Zurek [7] and by Henderson and Vedral [8], is attracting increasing interest .Recall that the quantum discord of a bipartite state ρ on a system H a ⊗ H b with marginals ρ a and ρ b can be expressed asHere the minimum is over von Neumann measurements (onedimensional orthogonal projectors summing to the identity) a = { a k } on party a, and a (ρ) := k a k ⊗ 1 b ρ a k ⊗ 1 b is the resulting state after the measurement. I (ρ) := S(ρ a ) + S(ρ b ) − S(ρ) is the quantum mutual information, S(ρ) := −trρlnρ is the von Neumann entropy, and 1 b is the identity operator on H b . The intuitive meaning of quantum discord thus may be interpreted as the minimal loss of correlations (as measured by the quantum mutual information) due to measurement. This formulation of quantum discord is equivalent to the original definition of quantum discord by Ollivier and Zurek [7]. Quite recently, Dakic et al. introduced the following geometric measure of quantum discord [40]: D(ρ) := min χ ||ρ − χ || 2 ,where the minimum is over the set of zero-discord states [i.e., Q(χ ) = 0] and the geometric quantity ||ρ − χ || 2 := tr(ρ − χ ) 2 * luosl@amt.ac.cn is the square of Hilbert-Schmidt norm of Hermitian operators. For any two-qubit statewith {σ i } being the Pauli spin matrices, its geometric measure of quantum discord is evaluated as [40]where x := (x 1 ,x 2 ,x 3 ) t is a column vector, ||x|| 2 := i x 2 i , T := (t ij ) is a matrix, and λ max is the largest eigenvalue of the matrix xx t + T T t . Here the superscript t denotes transpose of vectors or matrices.In this brief report, we first generalize the preceding result by presenting a variational expression for the geometric measure of quantum discord for general bipartite states. We further present an explicit and tight lower bound. Then we show that the geometric measure o...

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