  We study quantum correlations beyond entanglement in two-mode Gaussian states
of continuous variable systems, by means of the measurement-induced disturbance
(MID) and its ameliorated version (AMID). In analogy with the recent studies of
the Gaussian quantum discord, we define a Gaussian AMID by constraining the
optimization to all bi-local Gaussian positive operator valued measurements. We
solve the optimization explicitly for relevant families of states, including
squeezed thermal states. Remarkably, we find that there is a finite subset of
two-mode Gaussian states, comprising pure states, where non-Gaussian
measurements such as photon counting are globally optimal for the AMID and
realize a strictly smaller state disturbance compared to the best Gaussian
measurements. However, for the majority of two--mode Gaussian states the
unoptimized MID provides a loose overestimation of the actual content of
quantum correlations, as evidenced by its comparison with Gaussian discord.
This feature displays strong similarity with the case of two qubits. Upper and
lower bounds for the Gaussian AMID at fixed Gaussian discord are identified. We
further present a comparison between Gaussian AMID and Gaussian entanglement of
formation, and classify families of two-mode states in terms of their Gaussian
AMID, Gaussian discord, and Gaussian entanglement of formation. Our findings
provide a further confirmation of the genuinely quantum nature of general
Gaussian states, yet they reveal that non-Gaussian measurements can play a
crucial role for the optimized extraction and potential exploitation of
classical and nonclassical correlations in Gaussian states.
