We discuss the symmetric quantum discord (SQD) for an arbitrary two-qubit state consisting of subsystems A and B and give the analysis formula of the symmetric quantum discord for the arbitrary two-qubit state. We also give the optimization process of the symmetric quantum discord for some states and obtain the symmetric quantum discord. We compare the quantum discord (QD) with the symmetric quantum discord, and find that the symmetric quantum discord is greater than the quantum discord. We also find that the symmetric quantum discord can be unequal to the quantum discord when the right quantum discord (measure on subsystem B) is equal to the left quantum discord (measure on subsystem A).
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.