Comparing quantumness criteria

Measuring the quantumness of a system can be done with a variety of methods. In this article we compare different criteria, namely quantum discord, Bell inequality violation and non-separability, for systems placed in a Gaussian state. When the state is pure, these criteria are equivalent, while we find that they do not necessarily coincide when decoherence takes place. Finally, we prove that these criteria are essentially controlled by the semi-minor axis of the ellipse representing the state's Wigner function in phase space.