  We consider how to quantify non-Gaussianity for the correlation of a
bipartite quantum state by using various measures such as relative entropy and
geometric distances. We first show that an intuitive approach, i.e.,
subtracting the correlation of a reference Gaussian state from that of a target
non-Gaussian state, fails to yield a non-negative measure with monotonicity
under local Gaussian channels. Our finding clearly manifests that quantum-state
correlations generally have no Gaussian extremality. We therefore propose a
different approach by introducing relevantly averaged states to address
correlation. This enables us to define a non-Gaussianity measure based on,
e.g., the trace-distance and the fidelity, fulfilling all requirements as a
measure of non-Gaussian correlation. For the case of the fidelity-based
measure, we also present readily computable lower bounds of non-Gaussian
correlation.
