Quantum discord for general two-qubit states: Analytical progress

We present a reliable algorithm to evaluate quantum discord for general two-qubit states, amending and extending an approach recently put forward for the subclass of $X$ states. A closed expression for the discord of arbitrary states of two qubits cannot be obtained, as the optimization problem for the conditional entropy requires the solution to a pair of transcendental equations in the state parameters. We apply our algorithm to run a numerical comparison between quantum discord and an alternative, computable measure of nonclassical correlations, namely, the geometric discord. We identify the extremally nonclassically correlated two-qubit states according to the (normalized) geometric discord, at a fixed value of the conventional quantum discord. The latter cannot exceed the square root of the former for systems of two qubits.

Figure 1

Example of conditional entropy $\tilde{\mathcal{S}}$ for a random two-qubit state [Eq. (25)]. The angles $\theta$ and $\varphi$ parametrize the measurement: we can appreciate the symmetry properties of $\tilde{\mathcal{S}}$ with respect to such variables, expressed by the invariance $\tilde{\mathcal{S}}(\theta ,\varphi )=\tilde{\mathcal{S}}(\theta \pm \pi /2,\varphi )$. All quantities plotted are dimensionless.

Figure 2

Comparison between normalized geometric discord $2D_G$ and quantum discord $\mathcal{D}$ for ${10}^6$ randomly generated general two-qubit states. The dashed line is obtained by taking the equality sign in Ineq. (35). Refer to the text for details on the other boundary curves. All quantities plotted are dimensionless.