  Especially investigated in recent years, the Gaussian discord can be
quantified by a distance between a given two-mode Gaussian state and the set of
all the zero-discord two-mode Gaussian states. However, as this set consists
only of product states, such a distance captures all the correlations (quantum
and classical) between modes. Therefore it is merely un upper bound for the
geometric discord, no matter which is the employed distance. In this work we
choose for this purpose the Hellinger metric that is known to have many
beneficial properties recommending it as a good measure of quantum behaviour.
In general, this metric is determined by affinity, a relative of the Uhlmann
fidelity with which it shares many important features. As a first step of our
work, the affinity of a pair of $n$-mode Gaussian states is written. Then, in
the two-mode case, we succeeded in determining exactly the closest Gaussian
product state and computed the Gaussian discord accordingly. The obtained
general formula is remarkably simple and becomes still friendlier in the
significant case of symmetric two-mode Gaussian states. We then analyze in
detail two special classes of two-mode Gaussian states of theoretical and
experimental interest as well: the squeezed thermal states and the mode-mixed
thermal ones. The former are separable under a well-known threshold of
squeezing, while the latter are always separable. It is worth stressing that
for symmetric states belonging to either of these classes, we find consistency
between their geometric Hellinger discord and the originally defined discord in
the Gaussian approach. At the same time, the Gaussian Hellinger discord of such
a state turns out to be a reliable measure of the total amount of its cross
correlations.
