  Gaussian distribution of a quantum state with continuous spectrum is known to
maximize the Shannon entropy at a fixed variance. Applying it to a pair of
canonically conjugate quantum observables $\hat x$ and $\hat p$, quantum
entropic uncertainty relation can take a suggestive form, where the standard
deviations $\sigma_x$ and $\sigma_p$ are featured explicitly. From the
construction, it follows in a transparent manner that: (i) the entropic
uncertainty relation implies the Kennard-Robertson uncertainty relation in a
modifed form, $\sigma_x\sigma_p\geq\hbar e^{\cal N}/2$; (ii) the additional
factor ${\cal N}$ quantifies the quantum non-Gaussianity of the probability
distributions of two observables; (iii) the lower bound of the entropic
uncertainty relation for non-gaussian continuous variable (CV) mixed state
becomes stronger with purity. Optimality of specific non-gaussian CV states to
the refined uncertainty relation has been investigated and the existance of new
class of CV quantum state is identified.
