  Quantum correlations may be measured by means of the distance of the state to
the subclass of states $\Omega$ having well defined classical properties. In
particular, a geometric measure of asymmetric discord [Dakic et al., Phys. Rev.
Lett. 105, 190502 (2010)] was recently defined as the Hilbert-Schmidt distance
of a given two-qubit state to the closest classical-quantum (CQ) correlated
state. We analyze a geometric measure of symmetric discord defined as the
Hilbert-Schmidt distance of a given state to the closest classical-classical
(CC) correlated state. The optimal member of $\Omega$ is just specially
measured original state both for the CQ and CC discords. This implies that this
measure is equal to quantum deficit of post-measurement purity. We discuss some
general relations between the CC discords and explain why an analytical formula
for the CC discord, contrary to the CQ discord, can hardly be found even for a
general two-qubit state. Instead of such exact formula, we find simple
analytical measurement-based upper bounds for the CC discord which, as we show,
are very efficient in the case of two qubits and may serve as independent
indicators of two-party quantum correlations. In particular, we propose an
adaptive upper bound, which corresponds to the optimal states induced by
single-party measurements: optimal measurement on one of the parties determines
an optimal measurement on the other party. We discuss how to refine the
adaptive upper bound by nonoptimal single-party measurements and by an
iterative procedure which usually rapidly converges to the CC discord. We also
raise the question of optimality of the symmetric measurements realising the CC
discord on symmetric states, and give partial answer for the qubit case.

Quantum Physics