  Currently available separability criteria for continuous-variable states are
generally based on the covariance matrix of quadrature operators. The
well-known separability criterion of Duan et al. [Phys. Rev. Lett. 84, 2722
(2000)] and Simon [Phys. Rev. Lett. 84, 2726 (2000)] , for example, gives a
necessary and sufficient condition for a two-mode Gaussian state to be
separable, but leaves many entangled non-Gaussian states undetected. Here, we
introduce an improvement of this criterion that enables a stronger entanglement
detection. The improved condition is based on the knowledge of an additional
parameter, namely the degree of Gaussianity, and exploits a connection with
Gaussianity-bounded uncertainty relations [Phys. Rev. A 86, 030102 (2012)]. We
exhibit families of non-Gaussian entangled states whose entanglement remains
undetected by the Duan-Simon criterion.
