  We study nonclassical correlations beyond entanglement in a family of
two-mode non-Gaussian states which represent the continuous-variable
counterpart of two-qubit Werner states. We evaluate quantum discord and other
quantumness measures obtaining exact analytical results in special instances,
and upper and lower bounds in the general case. Non-Gaussian measurements such
as photon counting are in general necessary to solve the optimization in the
definition of quantum discord, whereas Gaussian measurements are strictly
suboptimal for the considered states. The gap between Gaussian and optimal
non-Gaussian conditional entropy is found to be proportional to a measure of
non-Gaussianity in the regime of low squeezing, for a subclass of
continuous-variable Werner states. We further study an example of a
non-Gaussian state which is positive under partial transposition, and whose
nonclassical correlations stay finite and small even for infinite squeezing.
Our results pave the way to a systematic exploration of the interplay between
nonclassicality and non-Gaussianity in continuous-variable systems, in order to
gain a deeper understanding of -and to draw a bigger advantage from- these two
important resources for quantum technology.
